A number of functions are provided which construct various standard groups. The effect of these functions is to construct the group on some standard set of generators. The group category of the result may be specified as an argument to the function.
Construct the abelian group defined by the sequence Q = [n1, ..., nr] of positive integers. The function constructs the direct product of cyclic groups Zn1 x Zn2 x ... x Znr. In some categories, ni may also be 0, denoting the infinite cyclic group Z. If the single-argument version of the function is used, the group will be constructed in the category GrpAb; otherwise, its category will be C, where C may be GrpAb, FINITELY PRESENTED GROUPS, GrpGPC, GrpPC or GrpPerm.
Construct the alternating group on n letters. If the single-argument version of the function is used, the group will be constructed in the category GrpPerm; otherwise, its category will be C, where C may be FINITELY PRESENTED GROUPS or GrpPerm.
Construct the cyclic group of order n. If the single-argument version of the function is used, the group will be constructed in the category GrpPerm; otherwise, its category will be C, where C may be GrpAb, FINITELY PRESENTED GROUPS, GrpGPC, GrpPC or GrpPerm.
Construct the dihedral group of order 2 * n. If the single-argument version of the function is used, the group will be constructed in the category GrpPerm; otherwise, its category will be C, where C may be FINITELY PRESENTED GROUPS, GrpGPC, GrpPC or GrpPerm.
The first intrinsic constructs the dicyclic group of order 4n. The second, when given an abelian group A and an element a of order 2, constructs the associated dicyclic group generated by A and an x with x2=a and ax=a - 1 for all x∈A.
Construct the symmetric group on n letters. If the single-argument version of the function is used, the group will be constructed in the category GrpPerm; otherwise, its category will be C, where C may be FINITELY PRESENTED GROUPS or GrpPerm.
Given a prime p and a small positive integer n, construct an extra-special group G of order p2n + 1. The isomorphism type of G can be selected using the parameter Type described below.If the two-argument version of the function is used, the group will be constructed in the category GrpPerm; otherwise, its category will be C, where C may be FINITELY PRESENTED GROUPS, GrpGPC, GrpPC or GrpPerm. If C is FINITELY PRESENTED GROUPS, GrpPC or GrpPerm, the prime p must be small.
Type: MonStgElt Default: "+"Possible values for this parameter are "+" (default) and "-".If Type is set to "+", the function returns for p = 2 the central product of n copies of the dihedral group of order 8, and for p > 2 it returns the unique extra-special group of order p2n + 1 and exponent p.
If Type is set to "-", the function returns for p = 2 the central product of a quaternion group of order 8 and n - 1 copies of the dihedral group of order 8, and for p > 2 it returns the unique extra-special group of order p2n + 1 and exponent p2.
> A := AbelianGroup([6, 2, 7]);
> A;
Abelian Group isomorphic to Z/2 + Z/42
Defined on 3 generators
Relations:
6*A.1 = 0
2*A.2 = 0
7*A.3 = 0
> A6 := Alt(6);
> A6;
Permutation group A6 acting on a set of cardinality 6
Order = 360 = 2^3 * 3^2 * 5
(1, 2)(3, 4, 5, 6)
(1, 2, 3)
> D8 := DihedralGroup(GrpPC, 4); > D8; GrpPC : D8 of order 8 = 2^3 PC-Relations: D8.2^2 = D8.3, D8.2^D8.1 = D8.2 * D8.3
> S7<a, b> := SymmetricGroup(GrpFP, 7);
> S7;
Finitely presented group S7 on 2 generators
Relations
a^7 = Id(S7)
b^2 = Id(S7)
(a * b)^6 = Id(S7)
(a^-1 * b * a * b)^3 = Id(S7)
(b * a^-2 * b * a^2)^2 = Id(S7)
(b * a^-3 * b * a^3)^2 = Id(S7)
Given two groups G and H belonging to the category C, construct the direct product of G and H as a group in C.
Given a sequence Q of n groups belonging to the category C, construct the direct product Q[1] x Q[2] x ... x Q[n] as a group in the category C.
Given two groups K and H and a homomorphism f:H to Aut(K), construct the semidirect product of K and H where the elements of H act on K via the map f. The semidirect product S is returned together with maps embedding K and H into S, and the projection S to H.MaxDeg: RngIntElt Default: 1000000The maximum degree permutation representation the algorithm will attempt.Print: BoolElt Default: falseUseRegular: BoolElt Default: falseSetting UseRegular to true forces the algorithm to go via the regular representations of K and H.
> K := CyclicGroup(3);
> H := CyclicGroup(4);
> A := AutomorphismGroup(K);
> f := hom< H->A | [ hom<K->K | [K.1^2]> ] >;
> S := SemidirectProduct(K,H,f);
> S;
Permutation group S acting on a set of cardinality 7
(1, 2, 3)
(2, 3)(4, 5, 6, 7)
Given a module M over a finite group ring KG, construct the semidirect product of M and G, where the elements of G act on M as in the module. The semidirect product S is returned together with maps embedding M and G into S, and the projection S to G.The module M will act as a regular normal subgroup on the first orbit of S. If the action of G on M is faithful, then S will be transitive, and otherwise there will be a second orbit on which H acts faithfully.
MaxDeg: RngIntElt Default: 1000000The maximum degree permutation representation the algorithm will attempt.Print: BoolElt Default: false
> G := SL(2,3);
> I := IrreducibleModules(G,GF(2));
> I[2];
GModule of dimension 2 over GF(2)
> E := AffineSplitExtension(I[2]);
> E;
Permutation group E acting on a set of cardinality 12
Order = 96 = 2^5 * 3
(1, 2)(3, 4)
(1, 3)(2, 4)
(2, 3, 4)(6, 9, 11)(8, 10, 12)
(5, 6, 7, 8)(9, 12, 10, 11)
> G := SymmetricGroup(4);
> H := DihedralGroup(3);
> D := DirectProduct(G, H);
> D;
Permutation group D acting on a set of cardinality 7
(1, 2, 3, 4)
(1, 2)
(5, 6, 7)
(5, 6)
> Order(D);
144