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Overview of the GroupTheory Package

Calling Sequence

Description

List of GroupTheory Package Commands

Group Constructors Palette

Context-Sensitive Operations

Examples

Compatibility

	Calling Sequence
	GroupTheory[command]( arguments ) command( arguments )

Description

•	The GroupTheory package provides a collection of commands for computing with, and visualizing, finitely generated (especially finite) groups. There are several classes of groups that are implemented.

permutation groups	groups given by a set of generating permutations
finitely presented groups	groups given by generators and defining relations
Cayley table groups	groups whose binary operation is specified by a Cayley table
custom groups	"black-box" user-defined groups whose elements are of an unspecified nature
symbolic groups	abstract groups depending on symbolic parameters

•	The package contains a variety of constructors that allow you to easily create groups in common families. Furthermore, several databases of groups exist in the package, and interfaces to these databases are provided.

Tutorials

WorkingWithFPGroups	Working with Finitely Presented Groups
WorkingWithPermutations	Working with Permutations

List of GroupTheory Package Commands

•	The following is a list of the commands in the GroupTheory package.

Constructors

AbelianGroup	construct a finitely generated Abelian group
AllAbelianGroups	return a list of all the Abelian groups of a given order
AlternatingGroup	construct the alternating group of a given order
BabyMonster	construct the Baby Monster sporadic finite simple group
CayleyTableGroup	construct a Cayley table group
ChevalleyF4	construct a Chevalley group of type F4
ChevalleyG2	construct a Chevalley group of type G2
ConwayGroup	construct a Conway sporadic simple group
CustomGroup	construct a custom group, given its operations
CyclicGroup	construct a cyclic group of a given order
DicyclicGroup	construct a dicyclic group
DihedralGroup	construct the dihedral group of a given degree
DirectProduct	construct a direct product of groups
ElementaryGroup	construct an elementary Abelian group
ExceptionalGroup	construct one of the exceptional finite simple groups
FischerGroup	construct one of the Fischer groups
FPGroup	construct a finitely presented group from generators and defining relators
FreeGroup	construct a free group
FrobeniusGroup	construct a Frobenius group from the database
GaloisGroup	construct the Galois group of a polynomial
GeneralLinearGroup	construct the general linear group over a finite field
GeneralOrthogonalGroup	construct the general orthogonal group over a finite field
GeneralUnitaryGroup	construct a general unitary group over a finite field
GL	construct the general linear group over a finite field
Group	construct a group from various data
HamiltonianGroup	construct a finite Hamiltonian group
HaradaNortonGroup	construct the Harada-Norton simple group
HeldGroup	construct the Held simple group
HigmanSimsGroup	construct the Higman-Sims simple group
JankoGroup	construct one of the Janko sporadic finite simple groups
LyonsGroup	construct the Lyons simple group
MathieuGroup	construct one of the Mathieu finite simple groups
McLaughlinGroup	construct the McLaughlin simple group
MetacyclicGroup	construct a metacyclic group
Monster	construct the Monster simple group
ONanGroup	construct the O'Nan simple group
Orbit	construct the orbit of an element under a permutation group
OrthogonalGroup	construct an orthogonal group
PerfectGroup	construct a perfect group from the database
Perm	create a permutation
PermutationGroup	construct a permutation group given generating permutations
PGL	construct a projective linear group over a finite field
PGU	construct a projective general unitary group over a finite field
ProjectiveGeneralLinearGroup	construct a projective general linear group over a finite field
ProjectiveGeneralUnitaryGroup	construct a projective general unitary group over a finite field
ProjectiveSpecialLinearGroup	construct a projective special linear group over a finite field
ProjectiveSpecialUnitaryGroup	construct a projective special unitary group over a finite field
ProjectiveSymplecticGroup	construct a projective symplectic group over a finite field
PSL	construct a projective special linear group over a finite field
PSp	construct a projective symplectic group over a finite field
PSU	construct a projective special unitary group over a finite field
QuaternionGroup	construct a generalised quaternion group
Ree2F4	construct a large Ree group of Lie type
Ree2G2	construct a small Ree group of Lie type
RubiksCubeGroup	construct the group of the Rubik's Cube
RudvalisGroup	construct the Rudvalis simple group
SL	construct a special linear group over a finite field
SmallGroup	construct a specific group of small order
SpecialLinearGroup	construct a special linear group over a finite field
SpecialOrthogonalGroup	construct a special orthogonal group over a finite field
SpecialUnitaryGroup	construct a special unitary group over a finite field
Stabilizer	compute the stabilizer of a point under a permutation group
Steinberg2E6	construct a Steinberg group of type 2E6
Steinberg3D4	construct a Steinberg group of type G2
Subgroup	construct a subgroup of a given group
Supergroup	construct a supergroup of a given group
Suzuki2B2	construct a Suzuki group of Lie type
SuzukiGroup	construct the Suzuki simple group
Symm	construct the symmetric group of a given degree
SymmetricGroup	construct the symmetric group of a given degree
SymplecticGroup	construct a symplectic group over a finite field
ThompsonGroup	construct the Thompson simple group
TitsGroup	construct the Tits simple group
TransitiveGroup	construct a specific transitive permutation group
TrivialGroup	construct the trivial group
TrivialSubgroup	construct the trivial subgroup of a given group

Subgroups

Center	compute the center of a group
Centraliser	compute the centraliser of an element of a group
Centralizer	compute the centralizer of an element of a group
Centre	compute the centre of a group
Commutator	compute the commutator of two subgroups
Core	compute the core of a subgroup of a group
DerivedSubgroup	compute the derived (commutator) subgroup of a group
DirectFactors	compute the directly indecomposable direct factors of a finite group
FittingSubgroup	compute the Fitting subgroup of a group
FrattiniSubgroup	compute the Frattini subgroup of a group
FrobeniusComplement	compute a representative Frobenius complement of a Frobenius group
FrobeniusKernel	compute the Frobenius kernel of a Frobenius group
FrobeniusProduct	compute the product of two complexes in a finite group
HallSubgroup	compute a Hall pi-subgroup of a finite soluble group
Hypercenter	compute the hypercenter residual of a group
Index	compute the index of a subgroup of a group
Intersection	compute the intersection of two subgroups of a group
IsDirectlyIndecomposable	test whether a group is directly indecomposable
IsMalnormal	test whether a subgroup of a group is malnormal
IsNormal	test whether a subgroup of a group is normal
IsQuasinormal	test whether a subgroup of a group is quasi-normal
IsSubnormal	test whether a subgroup of a group is subnormal
NilpotentResidual	compute the nilpotent residual of a group
NormalClosure	compute the normal closure of a subgroup or set of group elements
Normaliser	compute the normaliser of a subgroup of a group
NormaliserSubgroup	compute the normaliser of a subgroup of a group
NormalizerSubgroup	compute the normalizer of a subgroup of a group
NormalSubgroups	compute the normal subgroups of a finite group
PCore	compute the p-core of a subgroup of a group
Socle	compute the socle of a group
SolubleResidual	compute the soluble residual of a group
SolvableResidual	compute the solvable residual of a group
SubgroupLattice	compute the lattice of subgroups of a group
SylowBasis	compute a Sylow basis for a finite soluble group
SylowSubgroup	compute a Sylow p-subgroup of a finite group

Databases

AllFrobeniusGroups	return a list of the known Frobenius groups of a given order
AllHamiltonianGroups	return a list of all the Hamiltonian groups of a given order
AllPerfectGroups	return a list of the perfect groups of a given order
AllSmallGroups	return a list of all the groups of a given order
AllTransitiveGroups	return a list of all the transitive groups of a given degree
FrobeniusGroup	construct a Frobenius group from the database
IdentifyFrobeniusGroup	locate a given Frobenius group in the database of Frobenius groups
NumFrobeniusGroups	return the number of known Frobenius groups of a given order
NumHamiltonianGroups	return the number of Hamiltonian groups of a given order
NumPerfectGroups	return the number of perfect groups of a given order
NumTransitiveGroups	return the number of transitive groups of a given degree
PerfectGroup	construct a perfect group from the database
RandomSmallGroup	return a random group from the database of small groups
SearchFrobeniusGroups	search the Frobenius Groups database
SearchPerfectGroups	search the Perfect Groups database
SearchSmallGroups	search the Small Groups database
SearchTransitiveGroups	search the Transitive Groups database
SmallGroup	construct a specific group of small order
TransitiveGroup	construct a specific transitive permutation group

Invariants

AbelianInvariants	compute the Abelian invariants of a finitely presented group
CompositionLength	compute the composition length of a group
DerivedLength	compute the derived length of a group
Exponent	compute the exponent of a group
FittingLength	compute the nilpotent (Fitting) length of a group
FrattiniLength	compute the Frattini length of a group
GroupOrder	compute the order of a group
NilpotencyClass	compute the class of nilpotence of a group
NilpotentLength	compute the nilpotent (Fitting) length of a group
PermGroupRank	compute the rank of a permutation group
Transitivity	compute the transitivity of a permutation group

Predicates

AreConjugate	check whether two group elements are conjugate
AreIsomorphic	test whether two groups are isomorphic
IsAbelian	test whether a group is Abelian
IsAbelianSylowGroup	test whether a group has Abelian Sylow subgroups
IsAlmostSimple	test whether a group is almost simple
IsAlternating	test (probabilistically) whether a permutation group is an alternating group in its natural action
IsCAGroup	test whether a group is a (CA)-group
IsCCGroup	test whether a group is a (CC)-group
IsCNGroup	test whether a group is a (CN)-group
IsCommutative	test whether a group is commutative
IsCyclic	test whether a group is cyclic
IsCyclicSylowGroup	test whether a group has cyclic Sylow subgroups
IsDedekind	test whether a group is Dedekind
IsDihedral	test whether a permutation group is a dihedral group
IsElementary	test whether a group is elementary Abelian
IsFinite	test whether a group is finite
IsFrobeniusGroup	test whether a group is a Frobenius group
IsFrobeniusPermGroup	test whether a group is a Frobenius permutation group
IsGCLTGroup	test whether a group is a GCLT-group
IsHallPaigeGroup	test whether a group has a complete mapping
IsHamiltonian	test whether a group is Hamiltonian
IsHomocyclic	test whether a group is homocyclic
IsLagrangian	test whether a group is Lagrangian
IsMalnormal	test whether a subgroup of a group is malnormal
IsMetabelian	test whether a group is Abelian
IsMetacyclic	test whether a group is Metacyclic
IsNilpotent	test whether a group is nilpotent
IsNormal	test whether a subgroup of a group is normal
IsOrderedSylowTowerGroup	test whether a group has a Sylow tower
IsPerfect	test whether a group is perfect
IsPerfectOrderClassesGroup	test whether a group has perfect order classes
IsPermutable	test whether a subgroup of a group is permutable
IsPGroup	test whether a group is a p-group
IsPrimitive	test whether a permutation group is primitive
IsRegular	test whether a permutation group is regular
IsRegularPGroup	test whether a group is a regular p-group
IsSimple	test whether a group is simple
IsSoluble	test whether a group is soluble
IsSolvable	test whether a group is solvable
IsStemGroup	test whether a group is a stem group
IsSubgroup	test whether one group is a subgroup of another
IsSubnormal	test whether a subgroup of a group is subnormal
IsSupersoluble	test whether a group is supersoluble
IsSylowTowerGroup	test whether a group has a Sylow tower
IsSymmetric	test (probabilistically) whether a permutation group is a symmetric group in its natural action
IsTransitive	test whether a permutation group is transitive
IsTrivial	test whether a group is trivial
SubgroupMembership	test whether an element belongs to a given subgroup of a group

Permutation Groups

CycleIndexPolynomial	compute the cycle index polynomial of a permutation group
Degree	return the degree of a permutation group
FrobeniusPermRep	construct a Frobenius permutation group isomorphic to a given Frobenius group
IsAlternating	test (probabilistically) whether a permutation group is an alternating group in its natural action
IsPrimitive	test whether a permutation group is primitive
IsRegular	test whether a permutation group is regular
IsSymmetric	test (probabilistically) whether a permutation group is a symmetric group in its natural action
IsTransitive	test whether a permutation group is transitive
MinimumPermutationRepresentationDegree	compute the minimum degree of a faithful permutation representation for a group
Orbit	construct the orbit of an element under a permutation group
Orbits	compute the orbits of a permutation group
PermGroupRank	compute the rank of a permutation group
Stabilizer	compute the stabilizer of a point under a permutation group
Transitivity	compute the transitivity of a permutation group

Finitely Presented Groups

AbelianInvariants	compute the Abelian invariants of a finitely presented group
PresentationComplexity	compute a measure of the complexity of a finitely presented group
Relators	return the relators of a finitely presented group
Simplify	simplify the presentation of a finitely presented group

Visualization

DrawCayleyTable	draw the Cayley table of a finite group
DrawSubgroupLattice	draw the lattice of subgroups of a finite group

Series

CompositionSeries	compute a composition series of a group
DerivedSeries	compute the derived series of a group
FrattiniSeries	compute the Frattini series of a group
LowerCentralSeries	compute the lower central series of a group
LowerFittingSeries	compute the lower Fitting series of a group
LowerPCentralSeries	compute the lower p-central series of a group
OrderedSylowTower	compute a Sylow tower for a Sylow tower group
SylowTower	compute a Sylow tower for a Sylow tower group
UpperCentralSeries	compute the upper central series of a group

Dessins

DecomposeDessin	find all decompositions of a Belyi map represented by a dessin
FindDessins	find all dessins d'enfants with a specified branch pattern

Elements

RandomElement	compute a random element of a group
RandomImvolution	compute a random involution of a group
RandomPElement	compute a random p-element of a group
RandomPPrimeElement	compute a random element of a group with order relatively prime to p

Numbers

IsAbelianNumber	test whether every group of a given order is Abelian
IsCyclicNumber	test whether every group of a given order is cyclic
IsGCLTNumber	test whether every group of a given order is a GCLT group
IsIntegrableNumber	test whether every group of a given order is integrable
IsLagrangianNumber	test whether every group of a given order is Lagrangian
IsMetabelianNumber	test whether every group of a given order is metabelian
IsMetacyclicNumber	test whether every group of a given order is metacyclic
IsNilpotentNumber	test whether every group of a given order is nilpotent
IsOrderedSylowTowerNumber	test whether every group of a given order has an ordered Sylow tower
IsSimpleNumber	test whether every group of a given order is supersoluble
IsSolubleNumber	test whether every group of a given order is soluble
IsSupersolubleNumber	test whether every group of a given order is supersoluble

Other

CayleyGraph	return the Cayley graph of a finite group
CayleyTable	return the Cayley table of a finite group
CFSG	finite simple group classifier object
Character	construct a character from a character table
CharacterTable	compute the character table of a finite group
ClassifyFiniteSimpleGroup	classify a finite simple group
ConjugacyClass	compute the conjugacy class of a group element
ConjugacyClasses	compute all the conjugacy classes of a finite group
Conjugator	compute an element conjugating one group element to another
ElementOrder	compute the order of a group element
ElementOrderSum	compute the sum of the element orders of a finite group
ElementPower	compute an integer power of a group element
Elements	compute the elements of a finite group, orbit, coset or conjugacy class
Factor	expression a group element as a product of a coset representative and a subgroup element
Generators	return the set of generators of a group
GruenbergKegelGraph	return the Gruenberg-Kegel graph of a finite group
IdentifySmallGroup	compute the Small Group ID of a small group
Labels	return the set of generator labels of a group
LeftCoset	compute a left coset of a group element
LeftCosets	compute the left cosets of a subgroup of a group
MaximumElementOrder	compute the largest order of an element of a finite group
NonRedundantGenerators	return a set of non-redundant generators of a group
NumSimpleGroups	count the number of simple groups of a given finite order
Operations	return the operations record of a group
OrderClassNumber	compute the number of order classes of a finite group
OrderClassPolynomial	compute the order class polynomial of a finite group
OrderClassProfile	compute the element order profile of a finite group
OrderRank	compute the number of order class lengths greater than unity of a finite group
PresentationComplexity	compute a measure of the complexity of a finitely presented group
PrimePowerFactors	factor a group element into a product of elements of prime power order
RightCoset	compute a right coset of a group element
RightCosets	compute the right cosets of a subgroup of a group
TabulateSimpleGroups	list the simple groups with orders in a given range

Group Constructors Palette

•	The Group Constructors palette contains buttons for constructing groups.

•	Palettes are displayed in the left pane of the Maple window. (If it is not visible, from the main menu, select View > Palettes > Show Palette > Group Constructors)

•	Some palette items have placeholders. Fill in the placeholders, using Tab to navigate to the next placeholder.

Context-Sensitive Operations

•	In the Standard Worksheet interface, you can apply operations to a group through the Context Panel under the Group operations submenu.

Examples

The following command enables you to use the commands in the GroupTheory package without having to prefix each command with "GroupTheory:-".

>	$with (GroupTheory) &colon;$

Create a symmetric group of degree 4.

>	$G ≔ SymmetricGroup (4)$

$GroupTheory :- SymmetricGroup (4)$

(1)

Visualize the lattice of subgroups of G.

>	$DrawSubgroupLattice (G)$

Create a dihedral group of degree 4 (and order 8).

>	$H ≔ DihedralGroup (4)$

$GroupTheory :- DihedralGroup (4, form = permgroup)$

(2)

Visualize the Cayley (operation) table for H.

>	$DrawCayleyTable (H)$

Compute the orders (cardinalities) of G and H.

>	$GroupOrder (G), GroupOrder (H)$

$24, 8$

(3)

Compute the character table of H.

>	$Display (CharacterTable (H))$

C	1a	2a	2b	2c	4a
\|C\|	1	1	2	2	2

$χ__1$	$1$	$1$	$1$	$1$	$1$
$χ__2$	$1$	$1$	$−1$	$−1$	$1$
$χ__3$	$1$	$1$	$−1$	$1$	$−1$
$χ__4$	$1$	$1$	$1$	$−1$	$−1$
$χ__5$	$2$	$−2$	$0$	$0$	$0$

Form the direct product in two ways.

>	$U ≔ DirectProduct (G, H)$

$GroupTheory :- DirectProduct (module () ... end module, module () ... end module)$

(4)

>	$V ≔ DirectProduct (H, G)$

$GroupTheory :- DirectProduct (module () ... end module, module () ... end module)$

(5)

Check that these are isomorphic.

>	$AreIsomorphic (U, V)$

$true$

(6)

Notice that the order of the direct product is equal to the product of the orders of the factors.

>	$GroupOrder (U) = GroupOrder (G) GroupOrder (H)$

$192 = 192$

(7)

Since the order of U does not exceed 511, we can identify the group explicitly.

>	$id ≔ IdentifySmallGroup (U)$

$id ≔ 192, 1472$

(8)

Retrieve the identified group from the database, and check the isomorphism.

>	$W ≔ SmallGroup (id) &colon;$

>	$AreIsomorphic (U, W)$

$true$

(9)

Compatibility

•	The GroupTheory package was introduced in Maple 17.

•	For more information on Maple 17 changes, see Updates in Maple 17.

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