Finite group structures¶

Elements of finite groups all have a type described by a FiniteGroupType. For example, permutations acting on a certain domain, matrices of a given dimension; direct products of those.

FiniteGroupType : A group of elements from which finite subgroups can be constructed.
TotalOrder: RepLAB defines a total ordering on any finite group type.
FiniteGroup: A group with a finite number of elements, is generated by a finite number of generators.
MatrixGroup: A finite group of matrices whose coefficients are in the cyclotomic field (characteristic zero, includes the rational numbers)

FiniteGroupType¶

class replab.FiniteGroupType¶

Bases: replab.Group, replab.TotalOrder

Describes a type of finite groups

It is a possibly infinite group, and this object is able to construct finite subgroups of itself.

It also defines a (somewhat arbitrarily chosen) total order used, for example, to define minimal canonical representatives of conjugacy classes and cosets.

Examples of group types include:

permutations of a given domain 1..n (the set of such permutations is finite),
invertible matrices of a given size with coefficients in the cyclotomic field (the set of such matrices is not finite, but we assume that any explicitly constructed group is finite),
direct products of finite group types.

Note

For implementations purposes, there are several families of group types in RepLAB.

The first is replab.perm.PermutationGroupType where computational group theory algorithms are implemented and used directly.
The second is replab.gen.FiniteGroupType, in which the constructed objects delegate all computations to objects where the algorithms are actually implemented, through a “nice isomorphism”.

For now, those two families are the only ones implemented. Natural extensions would be:

Implementing algorithms for solvable groups through the polycyclic presentation as in e.g. GAP System.
Implementing black box group algorithms, either through a naive enumeration approach (to validate existing algorithms), or through recent algorithms proposed for black box groups.
Implementing matrix groups through a BSGS approach using the action of matrices on vectors; possibly also through recent algorithms based on composition trees.

Class members list

Properties

identity – Monoid identity element

Prettyprinting

additionalFields – Returns the name/value pairs corresponding to additional fields to be printed
disp – Standard MATLAB/Octave display method
headerStr – Tiny single line description of the current object type
hiddenFields – Returns the names of the fields that are not printed as a row vector
longStr – Multi-line description of the current object
shortStr – Single line text description of the current object

Property cache

cache – Sets the value of the designated property in the cache
cached – Returns the cached property if it exists, computing it if necessary
cachedOrDefault – Returns the cached property if it exists, or the provided default value if it is unknown yet
cachedOrEmpty – Returns the cached property if it exists, or [] if it is unknown yet
inCache – Returns whether the value of the given property has already been computed

Laws

check – Checks the consistency of this object
checkAndContinue – Checks the consistency of this object
laws – Returns the laws that this object obeys

Unique ID

eq – Equality test
id – Returns the unique ID of this object (deprecated)
isequal – Equality test
ne – Non-equality test

Test helpers

assertEqv – Compares two elements for equality
assertNotEqv – Compares two elements for inequality

Sampling and equality test

eqv – Tests domain elements for equality/equivalence
sample – Samples an element from this set

Monoid operations

areCommuting – Returns whether the two given elements commute
compose – Composes two monoid/group elements
composeAll – Composes a sequence of monoid elements
composeN – Computes the power of a given element by repeated squaring
isIdentity – Tests if the given element is the identity

Comparison

compare – Compares two elements of a domain

General

groupWithGenerators – Constructs a group with the given generators
isSameTypeAs – Tests whether this type is the same as another type
subgroupWithGenerators – Constructs a subgroup of a given group with the given parameters

Sort algorithm

sort – Returns a permutation that sorts the given array

Morphisms

automorphismGroup – Returns the automorphism group of this group
commutingMorphismsMorphism – Constructs a morphism from morphisms with commuting images
innerAutomorphism – Returns the inner automorphism given using conjugation by the given element
isomorphismByFunctions – Constructs a group isomorphism using preimage/image functions
morphismByFunction – Constructs a group morphism using an image function

Group methods

commutator – Returns the commutator of two group elements
composeFlat – Returns the composition of elements from an array picked according to the given indices/letters
composeWithInverse – Returns the composition of an element with the inverse of another element
inverse – Computes the inverse of an element
leftConjugate – Returns the left conjugate of a group element

Inherited elements

additionalFields(self)¶: Documentation in replab.Str.additionalFields()

areCommuting(self, x, y)¶: Documentation in replab.Monoid.areCommuting()

assertEqv(self, x, y, context)¶: Documentation in replab.Domain.assertEqv()

assertNotEqv(self, x, y, context)¶: Documentation in replab.Domain.assertNotEqv()

automorphismGroup(self)¶: Documentation in replab.Group.automorphismGroup()

cache(self, name, value, handleExisting)¶: Documentation in replab.Obj.cache()

cached(self, name, fun)¶: Documentation in replab.Obj.cached()

cachedOrDefault(self, name, defaultValue)¶: Documentation in replab.Obj.cachedOrDefault()

cachedOrEmpty(self, name)¶: Documentation in replab.Obj.cachedOrEmpty()

check(self)¶: Documentation in replab.Obj.check()

checkAndContinue(self)¶: Documentation in replab.Obj.checkAndContinue()

commutator(self, x, y, startWithInverse)¶: Documentation in replab.Group.commutator()

commutingMorphismsMorphism(self, target, morphisms)¶: Documentation in replab.Group.commutingMorphismsMorphism()

compare(self, x, y)¶: Documentation in replab.TotalOrder.compare()

compose(self, x, y)¶: Documentation in replab.Monoid.compose()

composeAll(self, elements)¶: Documentation in replab.Monoid.composeAll()

composeFlat(self, array, letters)¶: Documentation in replab.Group.composeFlat()

composeN(self, x, n)¶: Documentation in replab.Monoid.composeN()

composeWithInverse(self, x, y)¶: Documentation in replab.Group.composeWithInverse()

disp(self)¶: Documentation in replab.Str.disp()

eq(self, rhs)¶: Documentation in replab.Obj.eq()

eqv(self, t, u)¶: Documentation in replab.Domain.eqv()

headerStr(self)¶: Documentation in replab.Str.headerStr()

hiddenFields(self)¶: Documentation in replab.Str.hiddenFields()

id(self)¶: Documentation in replab.Obj.id()

identity¶: Documentation in replab.Monoid.identity

inCache(self, name)¶: Documentation in replab.Obj.inCache()

innerAutomorphism(self, by)¶: Documentation in replab.Group.innerAutomorphism()

inverse(self, x)¶: Documentation in replab.Group.inverse()

isIdentity(self, x)¶: Documentation in replab.Monoid.isIdentity()

isequal(lhs, rhs)¶: Documentation in replab.Obj.isequal()

isomorphismByFunctions(self, target, preimageElementFun, imageElementFun, torusMap)¶: Documentation in replab.Group.isomorphismByFunctions()

laws(self)¶: Documentation in replab.Obj.laws()

leftConjugate(self, by, on)¶: Documentation in replab.Group.leftConjugate()

longStr(self, maxRows, maxColumns)¶: Documentation in replab.Str.longStr()

morphismByFunction(self, target, imageElementFun, torusMap)¶: Documentation in replab.Group.morphismByFunction()

ne(self, rhs)¶: Documentation in replab.Obj.ne()

sample(self)¶: Documentation in replab.Domain.sample()

shortStr(self, maxColumns)¶: Documentation in replab.Str.shortStr()

sort(self, array)¶: Documentation in replab.TotalOrder.sort()

Own members

groupWithGenerators(self, generators, varargin)¶

Constructs a group with the given generators

Parameters

generators (cell(1,*) of group elements) – Group generators
generatorNames – Names of the generators
order – Group order
relators – Relators

Kwtype generatorNames

cell(1,*) of charstring, optional

Kwtype order

vpi or integer, optional

Kwtype relators

cell(1,*) of charstring or integer(1,*), optional

Returns

Constructed group

Return type

FiniteGroup

isSameTypeAs(self, otherType)¶

Tests whether this type is the same as another type

Parameters: otherType (FiniteGroupType) – Other type to test with
Returns: True if the types are the same
Return type: logical

subgroupWithGenerators(self, group, generators, varargin)¶

Constructs a subgroup of a given group with the given parameters

Parameters

group (FiniteGroup) – Finite group whose subgroup we are constructing
generators (cell(1,*) of group elements) – Subgroup generators
generatorNames – Names of the generators
order – Group order
relators – Relators

Kwtype generatorNames

cell(1,*) of charstring

Kwtype order

vpi or integer

Kwtype relators

cell(1,*) of charstring or integer(1,*)

Returns

Constructed group

Return type

FiniteGroup

TotalOrder¶

class replab.TotalOrder¶

Bases: replab.Domain

Describes a domain equipped with a total order

Class members list

Properties

Prettyprinting

additionalFields – Returns the name/value pairs corresponding to additional fields to be printed
disp – Standard MATLAB/Octave display method
headerStr – Tiny single line description of the current object type
hiddenFields – Returns the names of the fields that are not printed as a row vector
longStr – Multi-line description of the current object
shortStr – Single line text description of the current object

Property cache

cache – Sets the value of the designated property in the cache
cached – Returns the cached property if it exists, computing it if necessary
cachedOrDefault – Returns the cached property if it exists, or the provided default value if it is unknown yet
cachedOrEmpty – Returns the cached property if it exists, or [] if it is unknown yet
inCache – Returns whether the value of the given property has already been computed

Laws

check – Checks the consistency of this object
checkAndContinue – Checks the consistency of this object
laws – Returns the laws that this object obeys

Unique ID

eq – Equality test
id – Returns the unique ID of this object (deprecated)
isequal – Equality test
ne – Non-equality test

Test helpers

assertEqv – Compares two elements for equality
assertNotEqv – Compares two elements for inequality

Sampling and equality test

eqv – Tests domain elements for equality/equivalence
sample – Samples an element from this set

Comparison

compare – Compares two elements of a domain

Sort algorithm

sort – Returns a permutation that sorts the given array

Inherited elements

additionalFields(self)¶: Documentation in replab.Str.additionalFields()

assertEqv(self, x, y, context)¶: Documentation in replab.Domain.assertEqv()

assertNotEqv(self, x, y, context)¶: Documentation in replab.Domain.assertNotEqv()

cache(self, name, value, handleExisting)¶: Documentation in replab.Obj.cache()

cached(self, name, fun)¶: Documentation in replab.Obj.cached()

cachedOrDefault(self, name, defaultValue)¶: Documentation in replab.Obj.cachedOrDefault()

cachedOrEmpty(self, name)¶: Documentation in replab.Obj.cachedOrEmpty()

check(self)¶: Documentation in replab.Obj.check()

checkAndContinue(self)¶: Documentation in replab.Obj.checkAndContinue()

disp(self)¶: Documentation in replab.Str.disp()

eq(self, rhs)¶: Documentation in replab.Obj.eq()

eqv(self, t, u)¶: Documentation in replab.Domain.eqv()

headerStr(self)¶: Documentation in replab.Str.headerStr()

hiddenFields(self)¶: Documentation in replab.Str.hiddenFields()

id(self)¶: Documentation in replab.Obj.id()

inCache(self, name)¶: Documentation in replab.Obj.inCache()

isequal(lhs, rhs)¶: Documentation in replab.Obj.isequal()

laws(self)¶: Documentation in replab.Obj.laws()

longStr(self, maxRows, maxColumns)¶: Documentation in replab.Str.longStr()

ne(self, rhs)¶: Documentation in replab.Obj.ne()

sample(self)¶: Documentation in replab.Domain.sample()

shortStr(self, maxColumns)¶: Documentation in replab.Str.shortStr()

Own members

compare(self, x, y)¶

Compares two elements of a domain

The method returns:

-1 if x < y
0 if x == y
1 if x > y

Parameters

x (element) – First element
y (element) – Second element

Returns

Result of comparison

Return type

-1,0,1

sort(self, array)¶

Returns a permutation that sorts the given array

Returns a permutation I such that array(I) is sorted.

Parameters: array (cell(1,*) of elements of this domain) – Array to sort
Returns: Permutation that sorts the array
Return type: integer(1,*)

FiniteGroup¶

class replab.FiniteGroup¶

Bases: replab.CompactGroup, replab.FiniteSet

Describes a group with a finite number of elements

A finite group is generated by a finite set of generators.

Groups can be compared for equality using the == operator.

RepLAB implements group algorithms optimized for permutation groups using the BSGS (base and strong generating set) construction.

Generators and equality

The generating set of a group is a property of the group, and is used, for example:

to factor a group element as a word in the generators,
as the default preimages set for representation/morphism construction,
to construct an abstract form of the group (see relatorsWord).

However, groups that differ only by their generators are equal as sets. Thus, two groups that are equal under == can nevertheless return different values for the methods that work with the group generators.

For display purposes, and to describe group elements as words in the generators, the group generators have names generatorNames. Again, groups that differ by the names of their generators are considered mathematically equal.

The exception is AbstractGroup because the generator names are used in the group elements themselves.

Finite group types

Each finite group has a type (FiniteGroupType), that describes the most general group embedding its elements. The following are finite group types:

Permutations acting on 1 .. n (see PermutationGroup)
Signed permutations acting on +/- 1 .. n (see SignedPermutationGroup)
Cyclotomic matrices of a given size (note: we restrict the corresponding matrix groups to the finite ones)
Product groups of the above (direct, semidirect, wreath)

Finite group methods such as contains can be called with arguments from the group type.

Types can be compared with the == operator.

Implementations of finite groups

FiniteGroup has two main subclasses:

PermutationGroup, where all the methods are implemented directly using algorithms based on the BSGS construction,
replab.gen.FiniteGroup, which delegate the computations to another group type using an isomorphism.

As of July 2021, generic group implementations delegate computations to permutation groups, but this could evolve in the future.

Currently, some delegations of computations assume that the destination group is a permutation group, which is why we have permutationGroup and permutationIsomorphism methods on finite groups.

Class members list

Properties

generators – Group generators
identity – Monoid identity element
type – Type of the contained elements

Prettyprinting

additionalFields – Returns the name/value pairs corresponding to additional fields to be printed
disp – Standard MATLAB/Octave display method
headerStr – Tiny single line description of the current object type
hiddenFields – Returns the names of the fields that are not printed as a row vector
longStr – Multi-line description of the current object
shortStr – Single line text description of the current object

Test helpers

assertEqv – Compares two elements for equality
assertNotEqv – Compares two elements for inequality

Sampling and equality test

eqv – Tests domain elements for equality/equivalence
sample – Samples an element from this set

Laws

check – Checks the consistency of this object
checkAndContinue – Checks the consistency of this object
laws – Returns the laws that this object obeys

Property cache

cache – Sets the value of the designated property in the cache
cached – Returns the cached property if it exists, computing it if necessary
cachedOrDefault – Returns the cached property if it exists, or the provided default value if it is unknown yet
cachedOrEmpty – Returns the cached property if it exists, or [] if it is unknown yet
inCache – Returns whether the value of the given property has already been computed

Unique ID

eq – Equality test
id – Returns the unique ID of this object (deprecated)
isequal – Tests finite sets for equality
ne – Non-equality test

Monoid operations

areCommuting – Returns whether the two given elements commute
compose – Composes two monoid/group elements
composeAll – Composes a sequence of monoid elements
composeN – Computes the power of a given element by repeated squaring
isIdentity – Tests if the given element is the identity

Morphisms

abstractIsomorphism – Returns an isomorphism to an abstract group
automorphismGroup – Returns the automorphism group of this group
commutingMorphismsMorphism – Constructs a morphism from morphisms with commuting images
conjugatingAutomorphism – Returns the morphism that corresponds to left conjugation by an element
findIsomorphism – Finds an isomorphism from this finite group to another finite group, if it exists
findIsomorphisms – Finds all the isomorphisms from this finite group to another finite group
findMorphisms – Finds all the morphisms from this finite group to another finite group
innerAutomorphism – Returns the inner automorphism given using conjugation by the given element
isIsomorphicTo – Returns whether this group is isomorphic to another group
isMorphismByImages – Checks whether the given images describe a group morphism
isomorphismByFunction – Constructs an isomorphism to a finite group using an image function
isomorphismByFunctions – Constructs a group isomorphism using preimage/image functions
isomorphismByImages – Constructs an isomorphism to a group using images of generators
morphismByFunction – Constructs a group morphism using an image function
morphismByImages – Constructs a morphism to a group using images of generators
orderPreservingPermutationIsomorphism – Returns an isomorphism from this group to a permutation group, which preserves element order
permutationIsomorphism – Returns an isomorphism from this group to a permutation group
regularIsomorphism – Returns an isomorphism from this group to a permutation group whose degree matches the order of this group

Group methods

commutator – Returns the commutator of two group elements
composeFlat – Returns the composition of elements from an array picked according to the given indices/letters
composeWithInverse – Returns the composition of an element with the inverse of another element
inverse – Computes the inverse of an element
leftConjugate – Returns the left conjugate of a group element

Elements

contains – Tests whether this set contains the given element
elementOrder – Returns the order of a group element
elements – Returns a cell array containing all the elements of this set
elementsSequence – Returns a sequence corresponding to this set
nElements – Returns the size of this set
representative – Returns the minimal element of this set under the type ordering
setProduct – Returns a description of this set as a product of sets
smallGeneratingSet – Returns a small set of elements generating the group

Relations to other sets

hasSameTypeAs – Returns if this finite set has the same type as the given finite set

Image under isomorphism

imap – Returns the image of this finite set under an isomorphism

General

mtimes –

Representations

commutingRepsRep – Constructs a representation from commuting representations
directSumRep – Computes the direct sum of representations on this group
permutationRep – Constructs a permutation representation of this group
regularRep – Returns the left regular representation of this group
repByImages – Constructs a finite dimensional representation of this group from preimages/images pairs
signedPermutationRep – Returns a real signed permutation representation of this group
tensorRep – Computes the tensor product of representations
trivialRep – Returns the trivial representation of this group on a finite dimensional vector space

Group construction

directPower – Returns the direct product of this group with itself a number of times
directProduct – Returns the direct product of groups
semidirectProduct – Describes an external semidirect product of groups

Group properties

abelianInvariants – Returns the group abelian invariants
exponent – Returns the group exponent
fastRecognize – Attempts to recognize this group in the standard atlas
hasReconstruction – Returns whether the group has a “reconstruction”
isCommutative – Returns whether this group is commutative
isCyclic – Returns whether this group is a cyclic group
isSimple – Returns whether this group is simple
isTrivial – Tests whether this group is trivial
knownOrder – Returns whether the order of this group has already been computed
maximalTorusDimension – Returns, if available, the dimension of the maximal torus contained in the connected component of this group
order – Returns the group order
presentation – Returns a presentation of this group as a string
recognize – Attempts to recognize this group in the standard atlas
reconstruction – Returns a reconstruction of the group used to speed up group averaging

Isomorphic groups

abstractGroup – Returns an abstract group isomorphic to this finite gorup
permutationGroup – Returns the permutation group isomorphic to this finite group

Subgroups

center – Returns the center of this group
centralizer – Returns the centralizer of the given object in this group
derivedSeries – Returns the derived series of this group
derivedSubgroup – Computes the derived subgroup of this group
intersection – Computes the intersection of two groups
randomProperSubgroup – Constructs a random proper subgroup of this group
randomSubgroup – Constructs a random subgroup of this group
subgroup – Constructs a subgroup of the current group from elements
subgroupWithGenerators – Constructs a subgroup of the current group from generators
trivialSubgroup – Returns the trivial subgroup of this group

Conjugacy classes and character table

characterTable – Alias for complexCharacterTable
complexCharacterTable – Returns the complex character table of this group
conjugacyClass – Returns the conjugacy class corresponding to the given element
conjugacyClasses – Returns the conjugacy classes of this group
findLeftConjugations – Returns the set of all elements that left conjugates an element to another element
knownComplexCharacterTable – Returns whether the complex character table of this group is already known or computed
knownRealCharacterTable – Returns whether the real character table of this group is already known or computed
realCharacterTable – Returns the real character table of this group
setComplexCharacterTable – Sets the complex character table of this group
setConjugacyClasses – Sets the conjugacy classes of this group
setRealCharacterTable – Sets the real character table of this group

Construction of groups

closure – Computes the group generated by the elements of this group and other elements/groups
leftConjugateGroup – Returns the left conjugate of the current group by the given element
normalClosure – Computes the normal closure of an object in the closure of this group and this object
withGeneratorNames – Returns a modified copy of this finite group with renamed generators

Cosets

doubleCoset – Returns a double coset
doubleCosets – Returns the set of double cosets in this group by the given groups
isNormalizedBy – Returns whether a given element/group normalizes this group
leftCoset – Returns a left coset
leftCosets – Returns the set of left cosets of the given subgroup in this group
mldivide – Shorthand for rightCosets
mrdivide – Shorthand for leftCosets
normalCoset – Returns a normal coset
normalCosets – Returns the set of normal cosets of the given subgroup in this group
rightCoset – Returns a right coset
rightCosets – Returns the set of right cosets of the given subgroup in this group

Generators-related methods

factorizeFlat – Factorizes an element or a coset as a flat sequence in the generators
factorizeWord – Factorizes an element or a coset as a word in the generators
flatToWord – Prints a word described by letters into a string
generator – Returns the i-th group generator
generatorInverse – Returns the inverse of the i-th group generator
generatorNames – Returns the names of the group generators
imageFlat – Returns the image of a flat word in the group generators
imageWord – Computes the image of a word in the group generators
knownRelators – Returns whether the relators of this group are known
nGenerators – Returns the number of group generators
relatorsFlat – Returns the relators of a presentation of this finite group in the flat format
relatorsWord – Returns the relators of a presentation of this finite group
wordToFlat – Parses a word into generator letters

Relations to other groups

isNormalSubgroupOf – Returns whether this group is a normal subgroup of another group
isSubgroupOf – Returns whether this group is a subgroup of another group

Inherited elements

additionalFields(self)¶: Documentation in replab.Str.additionalFields()

areCommuting(self, x, y)¶: Documentation in replab.Monoid.areCommuting()

assertEqv(self, x, y, context)¶: Documentation in replab.Domain.assertEqv()

assertNotEqv(self, x, y, context)¶: Documentation in replab.Domain.assertNotEqv()

automorphismGroup(self)¶: Documentation in replab.Group.automorphismGroup()

cache(self, name, value, handleExisting)¶: Documentation in replab.Obj.cache()

cached(self, name, fun)¶: Documentation in replab.Obj.cached()

cachedOrDefault(self, name, defaultValue)¶: Documentation in replab.Obj.cachedOrDefault()

cachedOrEmpty(self, name)¶: Documentation in replab.Obj.cachedOrEmpty()

check(self)¶: Documentation in replab.Obj.check()

checkAndContinue(self)¶: Documentation in replab.Obj.checkAndContinue()

commutator(self, x, y, startWithInverse)¶: Documentation in replab.Group.commutator()

commutingMorphismsMorphism(self, target, morphisms)¶: Documentation in replab.Group.commutingMorphismsMorphism()

commutingRepsRep(self, field, dimension, reps)¶: Documentation in replab.CompactGroup.commutingRepsRep()

compose(self, x, y)¶: Documentation in replab.Monoid.compose()

composeAll(self, elements)¶: Documentation in replab.Monoid.composeAll()

composeFlat(self, array, letters)¶: Documentation in replab.Group.composeFlat()

composeN(self, x, n)¶: Documentation in replab.Monoid.composeN()

composeWithInverse(self, x, y)¶: Documentation in replab.Group.composeWithInverse()

contains(self, el)¶: Documentation in replab.FiniteSet.contains()

directPower(self, n)¶: Documentation in replab.CompactGroup.directPower()

directProduct(varargin)¶: Documentation in replab.CompactGroup.directProduct()

directSumRep(self, field, reps)¶: Documentation in replab.CompactGroup.directSumRep()

disp(self)¶: Documentation in replab.Str.disp()

elements(self)¶: Documentation in replab.FiniteSet.elements()

elementsSequence(self)¶: Documentation in replab.FiniteSet.elementsSequence()

eq(self, rhs)¶: Documentation in replab.Obj.eq()

eqv(self, t, u)¶: Documentation in replab.Domain.eqv()

hasReconstruction(self)¶: Documentation in replab.CompactGroup.hasReconstruction()

hasSameTypeAs(self, rhs)¶: Documentation in replab.FiniteSet.hasSameTypeAs()

headerStr(self)¶: Documentation in replab.Str.headerStr()

hiddenFields(self)¶: Documentation in replab.Str.hiddenFields()

id(self)¶: Documentation in replab.Obj.id()

identity¶: Documentation in replab.Monoid.identity

imap(self, f)¶: Documentation in replab.FiniteSet.imap()

inCache(self, name)¶: Documentation in replab.Obj.inCache()

innerAutomorphism(self, by)¶: Documentation in replab.Group.innerAutomorphism()

inverse(self, x)¶: Documentation in replab.Group.inverse()

isIdentity(self, x)¶: Documentation in replab.Monoid.isIdentity()

isequal(self, rhs)¶: Documentation in replab.FiniteSet.isequal()

isomorphismByFunctions(self, target, preimageElementFun, imageElementFun, torusMap)¶: Documentation in replab.Group.isomorphismByFunctions()

laws(self)¶: Documentation in replab.Obj.laws()

leftConjugate(self, by, on)¶: Documentation in replab.Group.leftConjugate()

longStr(self, maxRows, maxColumns)¶: Documentation in replab.Str.longStr()

maximalTorusDimension(self)¶: Documentation in replab.CompactGroup.maximalTorusDimension()

morphismByFunction(self, target, imageElementFun, torusMap)¶: Documentation in replab.Group.morphismByFunction()

mtimes()¶: No documentation

nElements(self)¶: Documentation in replab.FiniteSet.nElements()

ne(self, rhs)¶: Documentation in replab.Obj.ne()

reconstruction(self)¶: Documentation in replab.CompactGroup.reconstruction()

representative(self)¶: Documentation in replab.FiniteSet.representative()

sample(self)¶: Documentation in replab.Domain.sample()

semidirectProduct(self, N, phi)¶: Documentation in replab.CompactGroup.semidirectProduct()

setProduct(self)¶: Documentation in replab.FiniteSet.setProduct()

shortStr(self, maxColumns)¶: Documentation in replab.Str.shortStr()

tensorRep(self, field, reps)¶: Documentation in replab.CompactGroup.tensorRep()

trivialRep(self, field, dimension)¶: Documentation in replab.CompactGroup.trivialRep()

type¶: Documentation in replab.FiniteSet.type

Own members

generators¶

Group generators

Type: cell(1,*) of type elements

abelianInvariants(self)¶

Returns the group abelian invariants

It computes the decomposition of the factor group of this group by its derived subgroup (which is abelian), and returns the primary decomposition of that abelian group.

The abelian invariants are sorted in weakly increasing order.

Example

>>> G = replab.PermutationGroup.cyclic(100);
>>> isequal(G.abelianInvariants, [4 25])
    1

Returns: Group abelian invariants
Return type: integer(1,*)

abstractGroup(self)¶

Returns an abstract group isomorphic to this finite gorup

The generators of the abstract group must be in one-to-one correspondance with the generators of this group, and their names are taken from generatorNames

Returns: An abstract group isomorphic to this group
Return type: replab.AbstractGroup

abstractIsomorphism(self)¶

Returns an isomorphism to an abstract group

Example

>>> G = replab.S(3);
>>> f = G.abstractIsomorphism;
>>> f.imageElement([2 3 1])
    'x1'

Returns: Isomorphism to an abstract group
Return type: FiniteIsomorphism

center(self)¶

Returns the center of this group

Returns: The center
Return type: FiniteGroup

centralizer(self, arg)¶

Returns the centralizer of the given object in this group

Example

>>> G = replab.S(4);
>>> C = G.centralizer([2 3 1 4]);
>>> C == replab.PermutationGroup.of([2 3 1 4])
  1

Example

>>> G = replab.S(4);
>>> C = G.centralizer(G.subgroup({[2 3 1 4]}));
>>> C == replab.PermutationGroup.of([2 3 1 4])
  1

Parameters: rhs (FiniteGroup or group element) – Element to compute the centralizer of
Returns: The centralizer
Return type: FiniteGroup

characterTable(self)¶

Alias for complexCharacterTable

Returns: Complex character table
Return type: ComplexCharacterTable

closure(self, varargin)¶

Computes the group generated by the elements of this group and other elements/groups

Arguments can be either elements of type or FiniteGroup of the same type.

Example

>>> G = replab.PermutationGroup.cyclic(3);
>>> S3 = replab.S(3);
>>> H = S3.subgroup({[2 1 3]});
>>> C = G.closure(H);
>>> C == S3
    1

Returns: The closure
Return type: FiniteGroup

complexCharacterTable(self)¶

Returns the complex character table of this group

The complex character table is the standard, textbook character table of the group.

Note that randomized techniques are used to find group isomorphisms, and thus the output of this method may not be deterministic.

Raises: An error if the character table cannot be computed. –
Returns: Complex character table
Return type: ComplexCharacterTable

conjugacyClass(self, g, varargin)¶

Returns the conjugacy class corresponding to the given element

Parameters: g (element) – Arbitrary group element

Keywords Args:: isCanonical (logical, optional): Whether the given g is the class canonical representative, default: false centralizer (FiniteGroup, optional): Centralizer of the given element

Returns: The conjugacy class containing the given element
Return type: ConjugacyClass

conjugacyClasses(self)¶

Returns the conjugacy classes of this group

Returns: Conjugacy classes
Return type: ConjugacyClasses

conjugatingAutomorphism(self, by)¶

Returns the morphism that corresponds to left conjugation by an element

Parameters: by (element of type) – Element to conjugate the group with
Returns: Conjugating automorphism
Return type: replab.FiniteMorphism

derivedSeries(self)¶

Returns the derived series of this group

Returns
Return type: cell(1,*) of FiniteGroup

derivedSubgroup(self)¶

Computes the derived subgroup of this group

Example

>>> S4 = replab.S(4);
>>> S4.order
    24
>>> D = S4.derivedSubgroup;
>>> D == replab.PermutationGroup.alternating(4)
    1

Returns: The derived subgroup
Return type: replab.FiniteGroup

doubleCoset(self, element, rightSubgroup, varargin)¶

Returns a double coset

Returns the set self * element * rightSubgroup.

Parameters

element (group element) – Group element
rightSubgroup (FiniteGroup) – Finite group
group – Group containing both self and element, computed if omitted
isCanonical – Whether the given element is canonical w.r.t. to the group ordering

Kwtype group

FiniteGroup, optional

Kwtype isCanonical

logical, optional

Returns

The constructed double coset

Return type

replab.DoubleCoset

doubleCosets(self, leftSubgroup, rightSubgroup)¶

Returns the set of double cosets in this group by the given groups

Example

>>> n = 5;
>>> G = replab.S(n);
>>> H = G.pointwiseStabilizer(n);
>>> cosets = G.doubleCosets(H, H);
>>> cosets.nElements
    2

Example

>>> G = replab.S(3);
>>> H = replab.PermutationGroup.of([2 1 3]);
>>> K = replab.PermutationGroup.of([3 2 1]);
>>> cosets = G.doubleCosets;
>>> cosets.nElements
    2

Parameters

leftSubgroup (replab.FiniteGroup) – First subgroup
rightSubgroup (replab.FiniteGroup) – Second subgroup

Returns

The set of double cosets

Return type

replab.DoubleCosets

elementOrder(self, g)¶

Returns the order of a group element

Parameters: g (element of this group) – Group element
Returns: The order of g, i.e. the smallest o such that g^o == identity
Return type: vpi

exponent(self)¶

Returns the group exponent

The group exponent is the smallest integer e such that g^e == identity for all g in G.

Returns: The group exponent
Return type: vpi

factorizeFlat(self, elementOrCoset)¶

Factorizes an element or a coset as a flat sequence in the generators

See imageFlat for the convention.

If a coset is given, this method tries to pick a coset element with a short factorization.

Parameters: elementOrCoset (element or coset of this group) – Element or coset to factorize
Returns: Flat sequence of the word in the generators
Return type: integer(1,*)

factorizeWord(self, elementOrCoset)¶

Factorizes an element or a coset as a word in the generators

If a coset is given, this method tries to pick a coset element with a short factorization.

Example

>>> G = replab.S(3);
>>> G.factorizeWord([2 3 1])
    'x1'
>>> H = G.withGeneratorNames({'s', 't'});
>>> H.factorizeWord([2 3 1])
    's'

Parameters: elementOrCoset (element or coset of this group) – Element to factorize
Returns: Word representing the element
Return type: charstring

fastRecognize(self)¶

Attempts to recognize this group in the standard atlas

Returns: A result in case the group is identified; or [] if unrecognized.
Return type: FiniteIsomorphism or []

findIsomorphism(self, to)¶

Finds an isomorphism from this finite group to another finite group, if it exists

Parameters: to (replab.FiniteGroup) – Target of the isomorphism
Returns: The isomorphism if it exists, or an empty array
Return type: FiniteIsomorphism or []

findIsomorphisms(self, to, varargin)¶

Finds all the isomorphisms from this finite group to another finite group

Example

>>> G = replab.S(6);
>>> m = G.findIsomorphisms(G, 'upToConjugation', true);
>>> length(m)
    2

Parameters

to (replab.FiniteGroup) – Target of the isomorphism
upToConjugation – Whether to list morphisms up to conjugation of the image group, default: false
single – Whether to return maximum a single result, default: false

Kwtype upToConjugation

logical, optional

Kwtype single

logical, optional

Returns

The isomorphisms

Return type

cell(1,*) of FiniteIsomorphism

findLeftConjugations(self, s, t, varargin)¶

Returns the set of all elements that left conjugates an element to another element

Let s and t be two elements of this group. This returns the set of all elements b such that t = b s b^-1 or t = leftConjugate(b, s).

When no such b exists, this returns [].

Parameters

s (group element) – Source element
t (group element) – Target element
sCentralizer – Centralizer of s in this group
tCentralizer – Centralizer of t in this group

Kwtype sCentralizer

FiniteGroup, optional

Kwtype tCentralizer

FiniteGroup, optional

Returns

Set of all elements of this group that left conjugates s to t if it exists

Return type

replab.LeftCoset or []

findMorphisms(self, to, varargin)¶

Finds all the morphisms from this finite group to another finite group

Example

>>> S3 = replab.S(3);
>>> S4 = replab.S(4);
>>> m = S4.findMorphisms(S3, 'upToConjugation', true, 'surjective', false);
>>> length(m)
    3

Parameters

to (replab.FiniteGroup) – Target of the morphism
upToConjugation – Whether to list morphisms up to conjugation of the image group, default: false
surjective – Whether to consider only surjective morphisms (or epimorphisms), whose image span to, default: false
single – Whether to return maximum a single result, default: false

Kwtype upToConjugation

logical, optional

Kwtype surjective

logical, optional

Kwtype single

logical, optional

Returns

The morphisms

Return type

cell(1,*) of FiniteMorphism

flatToWord(self, letters)¶

Prints a word described by letters into a string

Parameters: letters (integer(1,*)) – Letters composing the word
Returns: Word in the generators
Return type: charstring

generator(self, i)¶

Returns the i-th group generator

Parameters: i (integer, charstring) – Generator index or name
Returns: i-th group generator
Return type: element

generatorInverse(self, i)¶

Returns the inverse of the i-th group generator

Parameters: i (integer, charstring) – Generator index or name
Returns: Inverse of the i-th group generator
Return type: element

generatorNames(self)¶

Returns the names of the group generators

Returns: Generator names
Return type: cell(1,*) of charstring

imageFlat(self, letters)¶

Returns the image of a flat word in the group generators

The letters take the values 1...nG which represent the group generators, and also the values -nG...-1 which represent the inverses of those generators.

Parameters: letters (integer(1,*)) – Letters composing the word.

Returns;: element: Element of the group corresponding to the given word

imageWord(self, word)¶

Computes the image of a word in the group generators

Example

>>> G = replab.S(3);
>>> G.imageWord('x1')
    2 3 1

Parameters: word (charstring) – Word written using mathematical notation
Returns: Element of the group corresponding to the given word
Return type: element

intersection(self, G)¶

Computes the intersection of two groups

Example

>>> D12 = replab.PermutationGroup.of([1 6 5 4 3 2], [2 1 6 5 4 3]);
>>> G = replab.PermutationGroup.of([2 1 3 4 5 6], [2 3 4 5 1 6]);
>>> I = D12.intersection(G);
>>> I == replab.PermutationGroup.of([5 4 3 2 1 6])
    1

Parameters: G (replab.FiniteGroup) – Group with the same type as this group
Returns: Subgroup representing the intersection
Return type: replab.FiniteGroup

isCommutative(self)¶: Returns whether this group is commutative

isCyclic(self)¶

Returns whether this group is a cyclic group

Example

>>> C3 = replab.PermutationGroup.cyclic(3);
>>> C3.isCyclic
    1
>>> C3_C3 = C3.directProduct(C3);
>>> C3_C3.isCommutative
    1
>>> C3_C3.isCyclic
    0

Returns: True if the group is cyclic
Return type: logical

isIsomorphicTo(self, target)¶

Returns whether this group is isomorphic to another group

Example

>>> h1 = [2 1 3 4];
>>> h2 = [3 4 1 2];
>>> H = replab.PermutationGroup.of(h1, h2);
>>> D = replab.D(4);
>>> H.isIsomorphicTo(D)
    1

Parameters: target (replab.FiniteGroup) – Finite group to test for isomorphism
Returns: Whether this group is isomorphic to the given group
Return type: logical

isMorphismByImages(self, target, varargin)¶

Checks whether the given images describe a group morphism

The calling convention of this method is the same as morphismByImages, except that nChecks is missing.

Parameters

target (replab.Group) – Target group
preimages – Preimages of the morphism which generate self, defaults to self.generators
images – Images of the given preimages, defaults to target.generators if target is a FiniteGroup

Kwtype preimages

cell(1, *) of self elements

Kwtype images

cell(1, *) of target elements

Returns

True if the given images define a morphism

Return type

logical

isNormalSubgroupOf(self, rhs)¶

Returns whether this group is a normal subgroup of another group

Example

>>> S3 = replab.S(3);
>>> A3 = replab.PermutationGroup.alternating(3);
>>> A3.isNormalSubgroupOf(S3)
    1

Example

>>> S3 = replab.S(3);
>>> G = S3.subgroup({[2 1 3]});
>>> G.isNormalSubgroupOf(S3)
    0

Parameters: rhs (replab.FiniteGroup) – Other group with the same type as this one
Returns: True if this group is a normal subgroup of rhs
Return type: logical

isNormalizedBy(self, element)¶

Returns whether a given element/group normalizes this group

This is true when element * group * == group * element.

The same definition when element is a group; then we ask that all this group elements satisfy that property.

Parameters: element (group element or FiniteGroup) – Group element or group
Returns: True if the given element/group normalizes this group
Return type: logical

isSimple(self)¶

Returns whether this group is simple

Example

>>> S5 = replab.S(5);
>>> S5.isSimple
    0
>>> A5 = S5.derivedSubgroup;
>>> A5.isSimple
    1

Returns: True if the group is simple
Return type: logical

isSubgroupOf(self, rhs)¶

Returns whether this group is a subgroup of another group

Example

>>> S3 = replab.S(3);
>>> G = S3.subgroup({[2 1 3]});
>>> G.isSubgroupOf(S3)
    1

Parameters: rhs (replab.FiniteGroup) – Other group with the same type as this one
Returns: True if this group is a subgroup of rhs
Return type: logical

isTrivial(self)¶

Tests whether this group is trivial

Returns: True if this group is trivial (i.e. has only one element)
Return type: logical

isomorphismByFunction(self, target, imageElementFun, varargin)¶

Constructs an isomorphism to a finite group using an image function

Due to the additional structure on target, this method returns a FiniteIsomorphism, and does not need an explicit preimage function.

Parameters

target (FiniteGroup or FiniteGroupType) – Finite group isomorphic to the present group
imageElementFun (function_handle) – Function that computes the image of a source element
preimageElementFun – Function the computes the preimage of a target element

Kwtype preimageElementFun

function_handle, optional

Returns

The constructed isomorphism

Return type

FiniteIsomorphism

isomorphismByImages(self, target, varargin)¶

Constructs an isomorphism to a group using images of generators

Parameters

target (FiniteGroup or FiniteGroupType) – Target of the isomorphism
preimages – Preimages of the isomorphism which generate self, defaults to self.generators
images – Images of the given preimages, defaults to target.generators
imageElementFun – Image function, default: []
nChecks – Number of randomized image checks to perform, if inf computes and verifies a presentation of self

Kwtype preimages

cell(1, *) of self elements

Kwtype images

cell(1, *) of target elements

Kwtype imageElementFun

function_handle or [], optional

Kwtype nChecks

integer or inf

knownComplexCharacterTable(self)¶

Returns whether the complex character table of this group is already known or computed

Returns: Whether the table is already computed
Return type: logical

knownOrder(self)¶

Returns whether the order of this group has already been computed

Returns: True if the order is known
Return type: logical

knownRealCharacterTable(self)¶

Returns whether the real character table of this group is already known or computed

Returns: Whether the table is already computed
Return type: logical

knownRelators(self)¶

Returns whether the relators of this group are known

Returns: True if the relators are known already
Return type: logical

leftConjugateGroup(self, by)¶

Returns the left conjugate of the current group by the given element

res = self.leftConjugateGroup(by)

In particular, it ensures that res.generator(i) = self.type.leftConjugate(by, self.generator(i))

Parameters: by (element of type) – Element to conjugate the group with
Returns: The conjugated group
Return type: replab.FiniteGroup

leftCoset(self, element, varargin)¶

Returns a left coset

Returns the set element * self.

Parameters

element (group element) – Group element
group – Group containing both self and element, computed if omitted
isCanonical – Whether the given element is canonical w.r.t. to the group ordering

Kwtype group

FiniteGroup, optional

Kwtype isCanonical

logical, optional

Returns

The constructed right coset

Return type

replab.LeftCoset

leftCosets(self, subgroup)¶

Returns the set of left cosets of the given subgroup in this group

Parameters: subgroup (replab.FiniteGroup) – Subgroup of this group
Returns: Left cosets
Return type: replab.LeftCosets

mldivide(self, supergroup)¶: Shorthand for rightCosets

morphismByImages(self, target, varargin)¶

Constructs a morphism to a group using images of generators

The type of the morphism depends on the type of target:

if target is a FiniteGroup or a FiniteGroupType, this method returns a FiniteMorphism,
if target is only a Group, this method returns a Morphism.

Example

>>> S4 = replab.S(4);
>>> m = S4.morphismByImages(replab.S(3), 'images', {[1 3 2] [3 2 1]});
>>> m.laws.checkSilent
    1

Parameters

target (Group or FiniteGroup) – Target of the morphism, the morphism image is a subgroup of this
preimages – Preimages of the morphism which generate self, defaults to self.generators
images – Images of the given preimages, defaults to target.generators if target is a FiniteGroup
imageElementFun – Image function, default: []
nChecks – Number of randomized image checks to perform, if inf computes and verifies a presentation of self

Kwtype preimages

cell(1, *) of self elements

Kwtype images

cell(1, *) of target elements

Kwtype imageElementFun

function_handle or [], optional

Kwtype nChecks

integer or inf

Returns

The constructed morphism

Return type

Morphism or FiniteMorphism

mrdivide(self, subgroup)¶: Shorthand for leftCosets

nGenerators(self)¶

Returns the number of group generators

Returns: Number of group generators
Return type: integer

normalClosure(self, obj)¶

Computes the normal closure of an object in the closure of this group and this object

Example

>>> S5 = replab.S(5);
>>> S4 = S5.subgroup({[2 3 4 1 5] [2 1 3 4 5]});
>>> nc = S4.normalClosure([1 2 4 5 3]); % demonstrates both syntaxes
>>> nc = S4.normalClosure(S5.subgroup({[1 2 4 5 3]}));
>>> nc == S5.subgroup({[1 2 4 5 3] [5 2 3 1 4] [2 5 3 4 1]})
    1

Example

>>> S4 = replab.S(4);
>>> nc = S4.normalClosure(S4.subgroup({[2 3 1 4]}));
>>> nc == S4.subgroup({[2 3 1 4] [1 3 4 2]})
    1

Parameters: obj (FiniteGroup or element of type) – Group or element to compute the normal closure of
Returns: The normal closure of obj in this group
Return type: replab.FiniteGroup

normalCoset(self, element, varargin)¶

Returns a normal coset

Returns the set element * self == self * element.

Parameters

element (group element) – Group element
group – Group containing both self and element, computed if omitted
isCanonical – Whether the given element is canonical w.r.t. to the group ordering

Kwtype group

FiniteGroup, optional

Kwtype isCanonical

logical, optional

Returns

The constructed normal coset

Return type

replab.NormalCoset

normalCosets(self, subgroup)¶

Returns the set of normal cosets of the given subgroup in this group

Parameters: subgroup (replab.FiniteGroup) – Normal subgroup of this group
Returns: The set of normal cosets
Return type: NormalCosets

order(self)¶

Returns the group order

Returns: The group order
Return type: vpi

orderPreservingPermutationIsomorphism(self)¶

Returns an isomorphism from this group to a permutation group, which preserves element order

Returns: Isomorphism with replab.FiniteIsomorphism.preservesTypeOrder set
Return type: replab.FiniteIsomorphism

permutationGroup(self)¶

Returns the permutation group isomorphic to this finite group

The generators of the permutation group are in one-to-one correspondance with the generators of this group.

This is the image of permutationIsomorphism.

Returns: A permutation group isomorphic to this group
Return type: replab.PermutationGroup

permutationIsomorphism(self)¶

Returns an isomorphism from this group to a permutation group

Compared to orderPreservingPermutationIsomorphism, this isomorphism may be more efficient to compute or use.

Returns: Isomorphism
Return type: replab.FiniteIsomorphism

permutationRep(self, dimension, varargin)¶

Constructs a permutation representation of this group

The returned representation is real. Use replab.Rep.complexification to obtain a complex representation.

Example

>>> C3 = replab.PermutationGroup.cyclic(3);
>>> rep = C3.permutationRep(3, 'preimages', {[2 3 1]}, 'images', {[3 1 2]});
>>> rep.dimension
    3

Parameters

dimension (integer) – Dimension of the representation
preimages – Preimages of the representation map which generate self, defaults to self.generators
images – Images of the generators as permutations of size dimension

Kwtype preimages

cell(1,n) of self elements, optional

Kwtype images

cell(1,*) of permutations

Returns

The constructed group representation

Return type

replab.Rep

presentation(self)¶

Returns a presentation of this group as a string

Returns: Presentation
Return type: charstring

randomProperSubgroup(self, nSteps)¶

Constructs a random proper subgroup of this group

The call fails if no proper subgroup exists.

Parameters: nSteps (integer, optional) – How many steps of reduction to perform, default 1
Returns: A proper subgroup
Return type: FiniteGroup

randomSubgroup(self)¶

Constructs a random subgroup of this group

Compared to randomProperSubgroup, this method can return the trivial group or self. This method never fails.

Returns: A subgroup
Return type: FiniteGroup

realCharacterTable(self)¶

Returns the real character table of this group

Note that randomized techniques are used to find group isomorphisms, and thus the output of this method may not be deterministic.

Raises: An error if the character table cannot be computed. –
Returns: Real character table
Return type: RealCharacterTable

recognize(self)¶

Attempts to recognize this group in the standard atlas

Returns: A result in case the group is identified; or [] if unrecognized.
Return type: FiniteIsomorphism or []

regularIsomorphism(self)¶

Returns an isomorphism from this group to a permutation group whose degree matches the order of this group

Returns: Isomorphism
Return type: replab.FiniteIsomorphism

regularRep(self)¶

Returns the left regular representation of this group

Returns: The left regular representation as a real permutation representation
Return type: Rep

relatorsFlat(self)¶

Returns the relators of a presentation of this finite group in the flat format

See AbstractGroup for a description of the presentation of a finite group

Returns: Relators in flat form
Return type: cell(1,*) of integer(1,*)

relatorsWord(self)¶

Returns the relators of a presentation of this finite group

Returns: Relators in word form
Return type: cell(1,*) of charstring

repByImages(self, field, dimension, varargin)¶

Constructs a finite dimensional representation of this group from preimages/images pairs

The preimages need to generate this group. If the preimages are omitted, they default to the generators of this group.

The images can either be exact, in which case they should be provided as cyclotomic matrices, or as double matrices – the latter only if they have integer entries. In the case of exact images, the keyword argument imagesErrorBound can be omitted. When images are exact, RepLAB has the freedom of performing matrix products containing an arbitrary number of factors without loss of precision, and thus a wider range of algorithm can be used. Also, the constructed representation can return exact images using rep.image(g, 'exact')

Otherwise, the provided images will be considered inexact. In that case, RepLAB uses a factorization algorithm to decompose group elements as short words in the preimages. The keyword argument imagesErrorBound should be given as well; it corresponds to an upper bound on the error of the matrix images in the Frobenius norm. Either a single number can be provided, as an upper bound on all images in the provided set; or an individual bound for each image. However, this error bound will be ignored for each image provided as an interval matrix. If inexact floating-point images are provided but imagesErrorBound is omitted, an ad-hoc error is estimated and a warning is emitted.

If the images are exact, the isUnitary keyword parameter can be omitted.

Example

>>> S4 = replab.S(4);
>>> m = S4.repByImages('R', 1, 'images', {-1 -1});
>>> m.laws.checkSilent
    1

Parameters

field ({'R', 'C'}) – Whether the representation is real (R) or complex (C)
dimension (integer) – Representation dimension
preimages – Preimages of the representation map which generate self, defaults to self.generators
images – Images of the given preimages
imagesErrorBound – Error
isUnitary – Value of the constructed Rep.isUnitary; a “true” value accelerates computations
isIrreducible – Value of the constructed Rep.isIrreducible
trivialDimension – Value of the constructed Rep.trivialDimension
frobeniusSchurIndicator – Value of the constructed Rep.frobeniusSchurIndicator
'H->R (divisionAlgebraName % ('R->C', 'C->R', 'H->C',) – rep’, ‘’, optional): Name of the division algebra encoding this representation respects, default ''

Kwtype preimages

cell(1,n) of self elements, optional

Kwtype images

cell(1,n) of double(d,d) or cyclotomic(d,d), may be sparse

Kwtype imagesErrorBound

double or double(1,d) or []

Kwtype isUnitary

logical or [], optional

Kwtype isIrreducible

logical or [], optional

Kwtype trivialDimension

integer or [], optional

Kwtype frobeniusSchurIndicator

integer or [], optional

Returns

The constructed group representation

Return type

replab.RepByImages

rightCoset(self, element, varargin)¶

Returns a right coset

Returns the set self * element.

Parameters

element (group element) – Group element
group – Group containing both self and element, computed if omitted
isCanonical – Whether the given element is canonical w.r.t. to the group ordering

Kwtype group

FiniteGroup, optional

Kwtype isCanonical

logical, optional

Returns

The constructed right coset

Return type

replab.RightCoset

rightCosets(self, subgroup)¶

Returns the set of right cosets of the given subgroup in this group

Parameters: subgroup (replab.FiniteGroup) – Subgroup of this group
Returns: Right cosets
Return type: replab.RightCosets

setComplexCharacterTable(self, table)¶

Sets the complex character table of this group

Throws an error if this method or complexCharacterTable has already been called before.

Parameters: table (ComplexCharacterTable) – Complex character table

setConjugacyClasses(self, classes)¶

Sets the conjugacy classes of this group

Throws an error if this method or conjugacyClasses has already been called before.

Parameters: classes (ConjugacyClasses) – Conjugacy classes

setRealCharacterTable(self, table)¶

Sets the real character table of this group

Throws an error if this method or realCharacterTable has already been called before.

Parameters: table (RealCharacterTable) – Complex character table

signedPermutationRep(self, dimension, varargin)¶

Returns a real signed permutation representation of this group

The returned representation is real. Use rep.complexification to obtain a complex representation.

>>> S2 = replab.S(2);
>>> rep = S2.signedPermutationRep(1, 'preimages', {[2 1]}, 'images', {[-1]});
>>> rep.image([2 1])
    -1

Parameters

dimension – Dimension of the representation
preimages – Preimages of the representation map which generate self, defaults to self.generators
images – Images of the generators as signed permutations of size dimension

Kwtype preimages

cell(1,n) of self elements, optional

Kwtype images

cell(1,*) of signed permutations

Returns

The constructed group representation

Return type

replab.Rep

smallGeneratingSet(self)¶

Returns a small set of elements generating the group

Returns: Generating set
Return type: cell(1,*) of group elements

subgroup(self, elements, varargin)¶

Constructs a subgroup of the current group from elements

The generators of the result are the given elements with the identity filtered out.

Example

>>> S5 = replab.S(5);
>>> G = S5.subgroup({[1 2 3 4 5]});
>>> isempty(G.generators)
    1

Parameters

elements (cell(1,*) of elements of this group) – List of elements
generatorNames – Names of the generators
order – Group order
relators – Relators

Kwtype generatorNames

cell(1,*) of charstring, optional

Kwtype order

vpi or integer, optional

Kwtype relators

cell(1,*) of charstring or integer(1,*), optional

Returns

The subgroup generated by generators

Return type

replab.FiniteGroup

subgroupWithGenerators(self, generators, varargin)¶

Constructs a subgroup of the current group from generators

Guarantees that the constructed subgroup preserves the order of the generators. However, the given generators cannot contain the identity element.

Example

>>> S5 = replab.S(5);
>>> G = S5.subgroupWithGenerators({[2 3 4 5 1]});
>>> isequal(G.generators{1}, [2 3 4 5 1])
    1

Parameters

generators (cell(1,*) of elements of this group) – List of generators, does not contain the identity
generatorNames – Names of the generators
order – Group order
relators – Relators

Kwtype generatorNames

cell(1,*) of charstring, optional

Kwtype order

vpi or integer, optional

Kwtype relators

cell(1,*) of charstring or integer(1,*), optional

Returns

The subgroup generated by generators

Return type

replab.FiniteGroup

trivialSubgroup(self)¶

Returns the trivial subgroup of this group

Returns: The trivial subgroup
Return type: replab.FiniteGroup

withGeneratorNames(self, newNames)¶

Returns a modified copy of this finite group with renamed generators

Parameters: newNames (cell(1,*) of charstring) – New generator names
Returns: Updated copy
Return type: FiniteGroup

wordToFlat(self, word)¶

Parses a word into generator letters

Parameters: word (charstring) – Word written using mathematical notation
Returns: Letters of the word
Return type: integer(1,*)

MatrixGroup¶

class replab.MatrixGroup(matrixSize, generators, varargin)¶

Bases: replab.gen.FiniteGroup

Finite group of matrices with coefficients in the cyclotomic field

The cyclotomic field includes the rational numbers, square roots of rational numbers, cosines and sines of fractions of pi and complex roots of unity.

The use of MatrixGroup requires the availability of the Java Virtual Machine, as computations are done in exact precision using a Scala/Java library.

RepLAB implements matrix groups by enumerating all their elements explicitly, which is adequate for groups with a few hundred elements.

Matrix groups are constructed by calling the constructor of MatrixGroup as below, specifying the dimension and the matrix generators. Note that generators are specified in any way that could be passed to the cyclotomic constructor, which includes a string representation. However, one should not pass a floating-point inexact matrix such as [1/sqrt(2), 1/sqrt(2); -1/sqrt(2), 1/sqrt(2)], because RepLAB is not able to guess the exact representation of 0.7071... then.

Example

>>> m1 = [0 1; 1 0]; % doctest: +cyclotomic
>>> m2 = '[1/sqrt(2), 1/sqrt(2); -1/sqrt(2), 1/sqrt(2)]';
>>> M = replab.MatrixGroup(2, {m1, m2});
>>> M.order
    16

The natural representation of a matrix group is given by the matrix itself, see naturalRep .

Class members list

Properties

generators – Group generators
identity – Monoid identity element
nice – Nice object where computations are done
niceIsomorphism – Order-preserving isomorphism from a group containing self to a group containing nice
type – Type of the contained elements

Prettyprinting

additionalFields – Returns the name/value pairs corresponding to additional fields to be printed
disp – Standard MATLAB/Octave display method
headerStr – Tiny single line description of the current object type
hiddenFields – Returns the names of the fields that are not printed as a row vector
longStr – Multi-line description of the current object
shortStr – Single line text description of the current object

Test helpers

assertEqv – Compares two elements for equality
assertNotEqv – Compares two elements for inequality

Sampling and equality test

eqv – Tests domain elements for equality/equivalence
sample – Samples an element from this set

Laws

check – Checks the consistency of this object
checkAndContinue – Checks the consistency of this object
laws – Returns the laws that this object obeys

Property cache

cache – Sets the value of the designated property in the cache
cached – Returns the cached property if it exists, computing it if necessary
cachedOrDefault – Returns the cached property if it exists, or the provided default value if it is unknown yet
cachedOrEmpty – Returns the cached property if it exists, or [] if it is unknown yet
inCache – Returns whether the value of the given property has already been computed

Unique ID

eq – Equality test
id – Returns the unique ID of this object (deprecated)
ne – Non-equality test

Monoid operations

areCommuting – Returns whether the two given elements commute
compose – Composes two monoid/group elements
composeAll – Composes a sequence of monoid elements
composeN – Computes the power of a given element by repeated squaring
isIdentity – Tests if the given element is the identity

Morphisms

abstractIsomorphism – Returns an isomorphism to an abstract group
automorphismGroup – Returns the automorphism group of this group
commutingMorphismsMorphism – Constructs a morphism from morphisms with commuting images
conjugatingAutomorphism – Returns the morphism that corresponds to left conjugation by an element
findIsomorphism – Finds an isomorphism from this finite group to another finite group, if it exists
findIsomorphisms – Finds all the isomorphisms from this finite group to another finite group
findMorphisms – Finds all the morphisms from this finite group to another finite group
innerAutomorphism – Returns the inner automorphism given using conjugation by the given element
isIsomorphicTo – Returns whether this group is isomorphic to another group
isMorphismByImages – Checks whether the given images describe a group morphism
isomorphismByFunction – Constructs an isomorphism to a finite group using an image function
isomorphismByFunctions – Constructs a group isomorphism using preimage/image functions
isomorphismByImages – Constructs an isomorphism to a group using images of generators
morphismByFunction – Constructs a group morphism using an image function
morphismByImages – Constructs a morphism to a group using images of generators
orderPreservingPermutationIsomorphism – Returns an isomorphism from this group to a permutation group, which preserves element order
permutationIsomorphism – Returns an isomorphism from this group to a permutation group
regularIsomorphism – Returns an isomorphism from this group to a permutation group whose degree matches the order of this group

Group methods

commutator – Returns the commutator of two group elements
composeFlat – Returns the composition of elements from an array picked according to the given indices/letters
composeWithInverse – Returns the composition of an element with the inverse of another element
inverse – Computes the inverse of an element
leftConjugate – Returns the left conjugate of a group element

Elements

elementOrder – Returns the order of a group element
smallGeneratingSet – Returns a small set of elements generating the group

Relations to other sets

hasSameTypeAs – Returns if this finite set has the same type as the given finite set

Image under isomorphism

imap – Returns the image of this finite set under an isomorphism

General

MatrixGroup – Constructs a finite matrix group
compatibleWithNiceIsomorphism – Returns whether the given finite isomorphism is compatible with this object isomorphism
mtimes –

Representations

commutingRepsRep – Constructs a representation from commuting representations
directSumRep – Computes the direct sum of representations on this group
naturalRep – Returns the natural representation of this matrix group
permutationRep – Constructs a permutation representation of this group
regularRep – Returns the left regular representation of this group
repByImages – Constructs a finite dimensional representation of this group from preimages/images pairs
signedPermutationRep – Returns a real signed permutation representation of this group
tensorRep – Computes the tensor product of representations
trivialRep – Returns the trivial representation of this group on a finite dimensional vector space

Group construction

directPower – Returns the direct product of this group with itself a number of times
directProduct – Returns the direct product of groups
semidirectProduct – Describes an external semidirect product of groups

Group properties

abelianInvariants – Returns the group abelian invariants
exponent – Returns the group exponent
fastRecognize – Attempts to recognize this group in the standard atlas
hasReconstruction – Returns whether the group has a “reconstruction”
isCommutative – Returns whether this group is commutative
isCyclic – Returns whether this group is a cyclic group
isSimple – Returns whether this group is simple
isTrivial – Tests whether this group is trivial
knownOrder – Returns whether the order of this group has already been computed
maximalTorusDimension – Returns, if available, the dimension of the maximal torus contained in the connected component of this group
order – Returns the group order
presentation – Returns a presentation of this group as a string
recognize – Attempts to recognize this group in the standard atlas
reconstruction – Returns a reconstruction of the group used to speed up group averaging

Isomorphic groups

abstractGroup – Returns an abstract group isomorphic to this finite gorup
permutationGroup – Returns the permutation group isomorphic to this finite group

Subgroups

center – Returns the center of this group
centralizer – Returns the centralizer of the given object in this group
derivedSeries – Returns the derived series of this group
derivedSubgroup – Computes the derived subgroup of this group
intersection – Computes the intersection of two groups
randomProperSubgroup – Constructs a random proper subgroup of this group
randomSubgroup – Constructs a random subgroup of this group
subgroup – Constructs a subgroup of the current group from elements
subgroupWithGenerators – Constructs a subgroup of the current group from generators
trivialSubgroup – Returns the trivial subgroup of this group

Conjugacy classes and character table

characterTable – Alias for complexCharacterTable
complexCharacterTable – Returns the complex character table of this group
conjugacyClass – Returns the conjugacy class corresponding to the given element
conjugacyClasses – Returns the conjugacy classes of this group
findLeftConjugations – Returns the set of all elements that left conjugates an element to another element
knownComplexCharacterTable – Returns whether the complex character table of this group is already known or computed
knownRealCharacterTable – Returns whether the real character table of this group is already known or computed
realCharacterTable – Returns the real character table of this group
setComplexCharacterTable – Sets the complex character table of this group
setConjugacyClasses – Sets the conjugacy classes of this group
setRealCharacterTable – Sets the real character table of this group

Construction of groups

closure – Computes the group generated by the elements of this group and other elements/groups
leftConjugateGroup – Returns the left conjugate of the current group by the given element
normalClosure – Computes the normal closure of an object in the closure of this group and this object
withGeneratorNames – Returns a modified copy of this finite group with renamed generators

Cosets

doubleCoset – Returns a double coset
doubleCosets – Returns the set of double cosets in this group by the given groups
isNormalizedBy – Returns whether a given element/group normalizes this group
leftCoset – Returns a left coset
leftCosets – Returns the set of left cosets of the given subgroup in this group
mldivide – Shorthand for rightCosets
mrdivide – Shorthand for leftCosets
normalCoset – Returns a normal coset
normalCosets – Returns the set of normal cosets of the given subgroup in this group
rightCoset – Returns a right coset
rightCosets – Returns the set of right cosets of the given subgroup in this group

Generators-related methods

factorizeFlat – Factorizes an element or a coset as a flat sequence in the generators
factorizeWord – Factorizes an element or a coset as a word in the generators
flatToWord – Prints a word described by letters into a string
generator – Returns the i-th group generator
generatorInverse – Returns the inverse of the i-th group generator
generatorNames – Returns the names of the group generators
imageFlat – Returns the image of a flat word in the group generators
imageWord – Computes the image of a word in the group generators
knownRelators – Returns whether the relators of this group are known
nGenerators – Returns the number of group generators
relatorsFlat – Returns the relators of a presentation of this finite group in the flat format
relatorsWord – Returns the relators of a presentation of this finite group
wordToFlat – Parses a word into generator letters

Relations to other groups

isNormalSubgroupOf – Returns whether this group is a normal subgroup of another group
isSubgroupOf – Returns whether this group is a subgroup of another group

Bugfix for Octave method selection

contains – Tests whether this set contains the given element
elements – Returns a cell array containing all the elements of this set
elementsSequence – Returns a sequence corresponding to this set
isequal – Tests finite sets for equality
nElements – Returns the size of this set
representative – Returns the minimal element of this set under the type ordering
setProduct – Returns a description of this set as a product of sets

Inherited elements

abelianInvariants(self)¶: Documentation in replab.FiniteGroup.abelianInvariants()

abstractGroup(self)¶: Documentation in replab.FiniteGroup.abstractGroup()

abstractIsomorphism(self)¶: Documentation in replab.FiniteGroup.abstractIsomorphism()

additionalFields(self)¶: Documentation in replab.Str.additionalFields()

areCommuting(self, x, y)¶: Documentation in replab.Monoid.areCommuting()

assertEqv(self, x, y, context)¶: Documentation in replab.Domain.assertEqv()

assertNotEqv(self, x, y, context)¶: Documentation in replab.Domain.assertNotEqv()

automorphismGroup(self)¶: Documentation in replab.Group.automorphismGroup()

cache(self, name, value, handleExisting)¶: Documentation in replab.Obj.cache()

cached(self, name, fun)¶: Documentation in replab.Obj.cached()

cachedOrDefault(self, name, defaultValue)¶: Documentation in replab.Obj.cachedOrDefault()

cachedOrEmpty(self, name)¶: Documentation in replab.Obj.cachedOrEmpty()

center(self)¶: Documentation in replab.FiniteGroup.center()

centralizer(self, arg)¶: Documentation in replab.FiniteGroup.centralizer()

characterTable(self)¶: Documentation in replab.FiniteGroup.characterTable()

check(self)¶: Documentation in replab.Obj.check()

checkAndContinue(self)¶: Documentation in replab.Obj.checkAndContinue()

closure(self, varargin)¶: Documentation in replab.FiniteGroup.closure()

commutator(self, x, y, startWithInverse)¶: Documentation in replab.Group.commutator()

commutingMorphismsMorphism(self, target, morphisms)¶: Documentation in replab.Group.commutingMorphismsMorphism()

commutingRepsRep(self, field, dimension, reps)¶: Documentation in replab.CompactGroup.commutingRepsRep()

compatibleWithNiceIsomorphism(self, iso)¶: Documentation in replab.gen.FiniteSet.compatibleWithNiceIsomorphism()

complexCharacterTable(self)¶: Documentation in replab.FiniteGroup.complexCharacterTable()

compose(self, x, y)¶: Documentation in replab.Monoid.compose()

composeAll(self, elements)¶: Documentation in replab.Monoid.composeAll()

composeFlat(self, array, letters)¶: Documentation in replab.Group.composeFlat()

composeN(self, x, n)¶: Documentation in replab.Monoid.composeN()

composeWithInverse(self, x, y)¶: Documentation in replab.Group.composeWithInverse()

conjugacyClass(self, g, varargin)¶: Documentation in replab.FiniteGroup.conjugacyClass()

conjugacyClasses(self)¶: Documentation in replab.FiniteGroup.conjugacyClasses()

conjugatingAutomorphism(self, by)¶: Documentation in replab.FiniteGroup.conjugatingAutomorphism()

contains(self, el)¶: Documentation in replab.FiniteSet.contains()

derivedSeries(self)¶: Documentation in replab.FiniteGroup.derivedSeries()

derivedSubgroup(self)¶: Documentation in replab.FiniteGroup.derivedSubgroup()

directPower(self, n)¶: Documentation in replab.CompactGroup.directPower()

directProduct(varargin)¶: Documentation in replab.CompactGroup.directProduct()

directSumRep(self, field, reps)¶: Documentation in replab.CompactGroup.directSumRep()

disp(self)¶: Documentation in replab.Str.disp()

doubleCoset(self, element, rightSubgroup, varargin)¶: Documentation in replab.FiniteGroup.doubleCoset()

doubleCosets(self, leftSubgroup, rightSubgroup)¶: Documentation in replab.FiniteGroup.doubleCosets()

elementOrder(self, g)¶: Documentation in replab.FiniteGroup.elementOrder()

elements(self)¶: Documentation in replab.FiniteSet.elements()

elementsSequence(self)¶: Documentation in replab.FiniteSet.elementsSequence()

eq(self, rhs)¶: Documentation in replab.Obj.eq()

eqv(self, t, u)¶: Documentation in replab.Domain.eqv()

exponent(self)¶: Documentation in replab.FiniteGroup.exponent()

factorizeFlat(self, elementOrCoset)¶: Documentation in replab.FiniteGroup.factorizeFlat()

factorizeWord(self, elementOrCoset)¶: Documentation in replab.FiniteGroup.factorizeWord()

fastRecognize(self)¶: Documentation in replab.FiniteGroup.fastRecognize()

findIsomorphism(self, to)¶: Documentation in replab.FiniteGroup.findIsomorphism()

findIsomorphisms(self, to, varargin)¶: Documentation in replab.FiniteGroup.findIsomorphisms()

findLeftConjugations(self, s, t, varargin)¶: Documentation in replab.FiniteGroup.findLeftConjugations()

findMorphisms(self, to, varargin)¶: Documentation in replab.FiniteGroup.findMorphisms()

flatToWord(self, letters)¶: Documentation in replab.FiniteGroup.flatToWord()

generator(self, i)¶: Documentation in replab.FiniteGroup.generator()

generatorInverse(self, i)¶: Documentation in replab.FiniteGroup.generatorInverse()

generatorNames(self)¶: Documentation in replab.FiniteGroup.generatorNames()

generators¶: Documentation in replab.FiniteGroup.generators

hasReconstruction(self)¶: Documentation in replab.CompactGroup.hasReconstruction()

hasSameTypeAs(self, rhs)¶: Documentation in replab.FiniteSet.hasSameTypeAs()

headerStr(self)¶: Documentation in replab.Str.headerStr()

hiddenFields(self)¶: Documentation in replab.Str.hiddenFields()

id(self)¶: Documentation in replab.Obj.id()

identity¶: Documentation in replab.Monoid.identity

imageFlat(self, letters)¶: Documentation in replab.FiniteGroup.imageFlat()

imageWord(self, word)¶: Documentation in replab.FiniteGroup.imageWord()

imap(self, f)¶: Documentation in replab.FiniteSet.imap()

inCache(self, name)¶: Documentation in replab.Obj.inCache()

innerAutomorphism(self, by)¶: Documentation in replab.Group.innerAutomorphism()

intersection(self, G)¶: Documentation in replab.FiniteGroup.intersection()

inverse(self, x)¶: Documentation in replab.Group.inverse()

isCommutative(self)¶: Documentation in replab.FiniteGroup.isCommutative()

isCyclic(self)¶: Documentation in replab.FiniteGroup.isCyclic()

isIdentity(self, x)¶: Documentation in replab.Monoid.isIdentity()

isIsomorphicTo(self, target)¶: Documentation in replab.FiniteGroup.isIsomorphicTo()

isMorphismByImages(self, target, varargin)¶: Documentation in replab.FiniteGroup.isMorphismByImages()

isNormalSubgroupOf(self, rhs)¶: Documentation in replab.FiniteGroup.isNormalSubgroupOf()

isNormalizedBy(self, element)¶: Documentation in replab.FiniteGroup.isNormalizedBy()

isSimple(self)¶: Documentation in replab.FiniteGroup.isSimple()

isSubgroupOf(self, rhs)¶: Documentation in replab.FiniteGroup.isSubgroupOf()

isTrivial(self)¶: Documentation in replab.FiniteGroup.isTrivial()

isequal(self, rhs)¶: Documentation in replab.FiniteSet.isequal()

isomorphismByFunction(self, target, imageElementFun, varargin)¶: Documentation in replab.FiniteGroup.isomorphismByFunction()

isomorphismByFunctions(self, target, preimageElementFun, imageElementFun, torusMap)¶: Documentation in replab.Group.isomorphismByFunctions()

isomorphismByImages(self, target, varargin)¶: Documentation in replab.FiniteGroup.isomorphismByImages()

knownComplexCharacterTable(self)¶: Documentation in replab.FiniteGroup.knownComplexCharacterTable()

knownOrder(self)¶: Documentation in replab.FiniteGroup.knownOrder()

knownRealCharacterTable(self)¶: Documentation in replab.FiniteGroup.knownRealCharacterTable()

knownRelators(self)¶: Documentation in replab.FiniteGroup.knownRelators()

laws(self)¶: Documentation in replab.Obj.laws()

leftConjugate(self, by, on)¶: Documentation in replab.Group.leftConjugate()

leftConjugateGroup(self, by)¶: Documentation in replab.FiniteGroup.leftConjugateGroup()

leftCoset(self, element, varargin)¶: Documentation in replab.FiniteGroup.leftCoset()

leftCosets(self, subgroup)¶: Documentation in replab.FiniteGroup.leftCosets()

longStr(self, maxRows, maxColumns)¶: Documentation in replab.Str.longStr()

maximalTorusDimension(self)¶: Documentation in replab.CompactGroup.maximalTorusDimension()

mldivide(self, supergroup)¶: Documentation in replab.FiniteGroup.mldivide()

morphismByFunction(self, target, imageElementFun, torusMap)¶: Documentation in replab.Group.morphismByFunction()

morphismByImages(self, target, varargin)¶: Documentation in replab.FiniteGroup.morphismByImages()

mrdivide(self, subgroup)¶: Documentation in replab.FiniteGroup.mrdivide()

mtimes()¶: No documentation

nElements(self)¶: Documentation in replab.FiniteSet.nElements()

nGenerators(self)¶: Documentation in replab.FiniteGroup.nGenerators()

ne(self, rhs)¶: Documentation in replab.Obj.ne()

nice¶: Documentation in replab.gen.FiniteSet.nice

niceIsomorphism¶: Documentation in replab.gen.FiniteSet.niceIsomorphism

normalClosure(self, obj)¶: Documentation in replab.FiniteGroup.normalClosure()

normalCoset(self, element, varargin)¶: Documentation in replab.FiniteGroup.normalCoset()

normalCosets(self, subgroup)¶: Documentation in replab.FiniteGroup.normalCosets()

order(self)¶: Documentation in replab.FiniteGroup.order()

orderPreservingPermutationIsomorphism(self)¶: Documentation in replab.FiniteGroup.orderPreservingPermutationIsomorphism()

permutationGroup(self)¶: Documentation in replab.FiniteGroup.permutationGroup()

permutationIsomorphism(self)¶: Documentation in replab.FiniteGroup.permutationIsomorphism()

permutationRep(self, dimension, varargin)¶: Documentation in replab.FiniteGroup.permutationRep()

presentation(self)¶: Documentation in replab.FiniteGroup.presentation()

randomProperSubgroup(self, nSteps)¶: Documentation in replab.FiniteGroup.randomProperSubgroup()

randomSubgroup(self)¶: Documentation in replab.FiniteGroup.randomSubgroup()

realCharacterTable(self)¶: Documentation in replab.FiniteGroup.realCharacterTable()

recognize(self)¶: Documentation in replab.FiniteGroup.recognize()

reconstruction(self)¶: Documentation in replab.CompactGroup.reconstruction()

regularIsomorphism(self)¶: Documentation in replab.FiniteGroup.regularIsomorphism()

regularRep(self)¶: Documentation in replab.FiniteGroup.regularRep()

relatorsFlat(self)¶: Documentation in replab.FiniteGroup.relatorsFlat()

relatorsWord(self)¶: Documentation in replab.FiniteGroup.relatorsWord()

repByImages(self, field, dimension, varargin)¶: Documentation in replab.FiniteGroup.repByImages()

representative(self)¶: Documentation in replab.FiniteSet.representative()

rightCoset(self, element, varargin)¶: Documentation in replab.FiniteGroup.rightCoset()

rightCosets(self, subgroup)¶: Documentation in replab.FiniteGroup.rightCosets()

sample(self)¶: Documentation in replab.Domain.sample()

semidirectProduct(self, N, phi)¶: Documentation in replab.CompactGroup.semidirectProduct()

setComplexCharacterTable(self, table)¶: Documentation in replab.FiniteGroup.setComplexCharacterTable()

setConjugacyClasses(self, classes)¶: Documentation in replab.FiniteGroup.setConjugacyClasses()

setProduct(self)¶: Documentation in replab.FiniteSet.setProduct()

setRealCharacterTable(self, table)¶: Documentation in replab.FiniteGroup.setRealCharacterTable()

shortStr(self, maxColumns)¶: Documentation in replab.Str.shortStr()

signedPermutationRep(self, dimension, varargin)¶: Documentation in replab.FiniteGroup.signedPermutationRep()

smallGeneratingSet(self)¶: Documentation in replab.FiniteGroup.smallGeneratingSet()

subgroup(self, elements, varargin)¶: Documentation in replab.FiniteGroup.subgroup()

subgroupWithGenerators(self, generators, varargin)¶: Documentation in replab.FiniteGroup.subgroupWithGenerators()

tensorRep(self, field, reps)¶: Documentation in replab.CompactGroup.tensorRep()

trivialRep(self, field, dimension)¶: Documentation in replab.CompactGroup.trivialRep()

trivialSubgroup(self)¶: Documentation in replab.FiniteGroup.trivialSubgroup()

type¶: Documentation in replab.FiniteSet.type

withGeneratorNames(self, newNames)¶: Documentation in replab.FiniteGroup.withGeneratorNames()

wordToFlat(self, word)¶: Documentation in replab.FiniteGroup.wordToFlat()

Own members

MatrixGroup(matrixSize, generators, varargin)¶

Constructs a finite matrix group

Additional keyword arguments (type, nice and niceIsomorphism) are used internally but are not part of the public API.

Parameters

matrixSize (integer) – Siye of the matrices
generators (cell(1,*) of replab.cyclotomic(matrixSize,matrixSize)) – Group generators
generatorNames – Names of the generators
order – Order of the group
relators – Relators given either in word or letter format

Kwtype generatorNames

cell(1,*) of charstring

Kwtype order

vpi, optional

Kwtype relators

cell(1,*) of charstring

naturalRep(self, varargin)¶

Returns the natural representation of this matrix group

The natural representation rho has images rho(g) = g.

Example

>>> M = replab.MatrixGroup(2, {[0 1; 1 0], '[1/sqrt(2), 1/sqrt(2); -1/sqrt(2), 1/sqrt(2)]'}); % doctest: +cyclotomic
>>> M.naturalRep.isIrreducible
    1

Parameters: field – Field over which to define the representation, default: ‘C’
Kwtype field: {‘R’ or ‘C’}
Returns: The natural representation of this group
Return type: Rep