 GroupTheory - Maple Help

Overview of the GroupTheory Package 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

 Working with Finitely Presented Groups Working with Permutations Working with Symbolic Groups List of GroupTheory Package Commands

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

 construct a finitely generated Abelian group construct the affine general linear group over a finite field construct the affine special linear group over a finite field construct the affine general linear group over a finite field return all the Abelian groups of a given order construct the alternating group of a given order construct the affine special linear group over a finite field construct the Baby Monster sporadic finite simple group construct a Cayley table group construct a Chevalley group of type E6 construct a Chevalley group of type E7 construct a Chevalley group of type E8 construct a Chevalley group of type F4 construct a Chevalley group of type G2 construct a Conway sporadic simple group construct a custom group, given its operations construct a cyclic group of a given order construct a dicyclic group construct the dihedral group of a given degree construct a direct product of groups construct an elementary Abelian group construct one of the exceptional finite simple groups construct one of the Fischer groups construct a finitely presented group from generators and defining relators construct a free group construct a Frobenius group from the database construct the Galois group of a polynomial construct the general semi-linear group over a finite field construct the general linear group over a finite field construct the general orthogonal group over a finite field construct the general semi-linear group over a finite field construct a general unitary group over a finite field construct the general linear group over a finite field construct a group from various data construct a finite Hamiltonian group construct the Harada-Norton simple group construct the Held simple group construct the Higman-Sims simple group construct one of the Janko sporadic finite simple groups construct the Lyons simple group construct one of the Mathieu finite simple groups construct the McLaughlin simple group construct a metacyclic group construct the Monster simple group construct the O'Nan simple group construct the orbit of an element under a permutation group construct an orthogonal group construct a perfect group from the database create a permutation construct a permutation group given generating permutations construct the general semi-linear group over a finite field construct a projective linear group over a finite field construct the projective general orthogonal group over a finite field construct a projective general unitary group over a finite field construct a projective general linear group over a finite field construct the projective general orthogonal group over a finite field construct the general semi-linear group over a finite field construct a projective general unitary group over a finite field construct a projective special linear group over a finite field construct the projective special orthogonal group over a finite field construct the special semi-linear group over a finite field construct a projective special unitary group over a finite field construct a projective symplectic group over a finite field construct the projective symplectic semi-linear group over a finite field construct the special semi-linear group over a finite field construct the projective symplectic semi-linear group over a finite field construct a projective special linear group over a finite field construct the projective special orthogonal group over a finite field construct a projective symplectic group over a finite field construct a projective special unitary group over a finite field construct a quasicyclic p-group construct a quasi-dihedral group construct a generalized quaternion group construct a large Ree group of Lie type construct a small Ree group of Lie type construct the group of the Rubik's Cube construct the Rudvalis simple group construct a semi-dihedral group construct the special semi-linear group over a finite field construct the symplectic semi-linear group over a finite field construct a special linear group over a finite field construct a specific group of small order construct a special linear group over a finite field construct a special orthogonal group over a finite field construct the special semi-linear group over a finite field construct a special unitary group over a finite field compute the stabilizer of a point, list or set under a permutation group construct a Steinberg group of type 2E6 construct a Steinberg group of type G2 construct a subgroup of a given group construct a supergroup of a given group construct a Suzuki group of Lie type construct the Suzuki simple group construct the symmetric group of a given degree construct the symmetric group of a given degree construct a symplectic group over a finite field construct the symplectic semi-linear group over a finite field construct the Thompson simple group construct the Tits simple group construct a specific transitive permutation group construct the trivial group construct the trivial subgroup of a given group construct a wreath product of permutation groups Subgroups

 compute the center of a group compute the centraliser of an element of a group compute the centralizer of an element of a group compute the centre of a group compute the commutator of two subgroups compute the core of a subgroup of a group compute the cosocle of a group compute the derived (commutator) subgroup of a group compute the directly indecomposable direct factors of a finite group compute the Fitting subgroup of a group compute the Frattini subgroup of a group compute a representative Frobenius complement of a Frobenius group compute the Frobenius kernel of a Frobenius group compute the product of two complexes in a finite group compute a Hall pi-subgroup of a finite soluble group compute a Hall system for a finite soluble group compute the hypercenter residual of a group compute the index of a subgroup of a group compute the intersection of two subgroups of a group test whether a group is directly indecomposable test whether a subgroup of a group is malnormal test whether a subgroup of a group is normal test whether a subgroup of a group is quasi-normal test whether a subgroup of a group is subnormal compute the maximal normal subgroups of a permutation group compute the minimal normal subgroups of a permutation group compute the nilpotent residual of a group compute the normal closure of a subgroup or set of group elements compute the normaliser of a subgroup of a group compute the normaliser of a subgroup of a group compute the normalizer of a subgroup of a group compute the normal subgroups of a finite group compute the p-core of a subgroup of a group compute the socle of a group compute the soluble residual of a group compute the solvable residual of a group compute the lattice of subgroups of a group compute a Sylow basis for a finite soluble group compute a Sylow p-subgroup of a finite group Databases

 return a list of the known Frobenius groups of a given order return a list of all the Hamiltonian groups of a given order return a list of the perfect groups of a given order return a list of all the groups of a given order return a list of all the transitive groups of a given degree construct a Frobenius group from the database locate a given Frobenius group in the database of Frobenius groups return the number of known Frobenius groups of a given order return the number of Hamiltonian groups of a given order return the number of perfect groups of a given order return the number of transitive groups of a given degree construct a perfect group from the database return a random group from the database of small groups search the Frobenius Groups database search the Perfect Groups database search the Small Groups database search the Transitive Groups database construct a specific group of small order construct a specific transitive permutation group Invariants

 compute the abelian invariants of a group compute the number of conjugacy classes of a finite group compute the composition length of a group compute the conjugate rank of a finite group compute the derived length of a group compute the exponent of a group compute the nilpotent (Fitting) length of a group compute the Frattini length of a group compute the order of a group compute the class of nilpotence of a group compute the nilpotent (Fitting) length of a group compute the number of order classes of a finite group compute the number of order class lengths greater than unity of a finite group compute the rank of a permutation group compute the rank of a finite p-group compute the primary invariants of a group compute the transitivity of a permutation group Predicates

 check whether two group elements are conjugate test whether two groups are isomorphic test whether a group is Abelian test whether a group has Abelian Sylow subgroups test whether a group is almost simple test (probabilistically) whether a permutation group is an alternating group in its natural action test whether a group is a (CA)-group test whether a group is a Camina group test whether a group is a (CC)-group test whether a group is characteristically simple test whether a group is a (CN)-group test whether a group is commutative test whether a group is a (CP1)-group test whether a group is a (CP)-group test whether a group is cyclic test whether a group has cyclic Sylow subgroups test whether a group is Dedekind test whether a permutation group is a dicyclic group test whether a permutation group is a dihedral group test whether a group is elementary Abelian test whether a group is an extraspecial p-group test whether a group is finite test whether a group is finitely generated test whether a group is a Frobenius group test whether a group is a Frobenius permutation group test whether a group is a GCLT-group test whether a group has a complete mapping test whether a group is Hamiltonian test whether a group is homocyclic test whether a group is Lagrangian test whether a subgroup of a group is malnormal test whether a group is Abelian test whether a group is Metacyclic test whether a group is nilpotent test whether a subgroup of a group is normal test whether a group has a Sylow tower test whether a group is perfect test whether a group has perfect order classes test whether a subgroup of a group is permutable test whether a group is a p-group test whether a permutation group is primitive test whether a permutation group is quasi-primitive test whether a group is quasi-simple test whether a permutation group is a quaternion group test whether a permutation group is regular test whether a group is a regular p-group test whether a permutation group is semi-primitive test whether a permutation group is semi-regular test whether a group is simple test whether a group is soluble test whether a group is solvable test whether a group is a special p-group test whether a group is a stem group test whether one group is a subgroup of another test whether a subgroup of a group is subnormal test whether a group is supersoluble test whether a group has a Sylow tower test (probabilistically) whether a permutation group is a symmetric group in its natural action test whether a permutation group is transitive test whether a group is trivial test whether an element belongs to a given subgroup of a group Permutation Groups

 compute the abelian invariants of a permutation group return a permutation group equivalent to the action of a permutation on a system of blocks return a block system for a permutation group, non-trivial if possible compute the cycle index polynomial of a permutation group return the degree of a permutation group return an EARNS of a primitive group if it has one construct a Frobenius permutation group isomorphic to a given Frobenius group test (probabilistically) whether a permutation group is an alternating group in its natural action test whether a permutation group is primitive test whether a permutation group is quasi-primitive test whether a permutation group is regular test whether a permutation group is semi-primitive test whether a permutation group is semi-regular test (probabilistically) whether a permutation group is a symmetric group in its natural action test whether a permutation group is transitive return the largest element displaced by a permutation group return a minimal block system for a permutation group, non-trivial if possible compute the minimum degree of a faithful permutation representation for a group return the smallest element displaced by a permutation group construct the orbit of an element under a permutation group compute the orbits of a permutation group compute the rank of a permutation group compute the primary invariants of a permutation group return an isomorphic permutation group of possibly smaller degree return the restriction of a permutation group to a stable subset compute the stabilizer of a point, list or set under a permutation group return the support of a permutation group return the number of elements displaced by a permutation group compute the transitivity of a permutation group Finitely Presented Groups

 compute the abelian invariants of a finitely presented group compute a measure of the complexity of a finitely presented group compute the primary invariants of a finitely presented group return the relators of a finitely presented group simplify the presentation of a finitely presented group Visualization

 draw the Cayley table of a finite group draw the lattice of normal subgroups of a finite group draw the lattice of subgroups of a finite group Series

 compute the series of agemo subgroups of a $p$-group compute a composition series of a group compute the derived series of a group compute the Frattini series of a group compute the lower central series of a group compute the lower Fitting series of a group compute the lower p-central series of a group compute the series of omega subgroups of a $p$-group compute a Sylow tower for a Sylow tower group compute a Sylow tower for a Sylow tower group compute the upper central series of a group Dessins

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

 compute the order of a group element compute the sum of the element orders of a finite group compute an integer power of a group element compute the elements of a finite group, orbit, coset or conjugacy class compute the largest order of an element of a finite group compute the order class polynomial of a finite group compute the element order profile of a finite group factor a group element into a product of elements of prime power order compute a random element of a group compute a random involution of a group compute a random p-element of a group compute a random element of a gr