Subsections

• Construction of a Standard Group
• Construction of Extensions

A number of functions are provided which construct presentations for various standard finite groups.

AbelianGroup(GrpFP, [n_1,...,n_r]): Cat, [ RngIntElt ] -> GrpFP

Construct the abelian group defined by the sequence [n_1, ..., n_r] of non-negative integers as an fp-group. The function returns the direct product of cyclic groups C_(n_1) x C_(n_2) x ... x C_(n_r), where C_0 is interpreted as an infinite cyclic group.

Alt(GrpFP, n) : Cat, RngIntElt -> GrpFP

Construct the alternating group of degree n as an fp-group, where the generators correspond to the permutations (3, 4, ... , n) and (1, 2, 3), for n odd, or (1, 2)(3, 4, ..., n) and (1, 2, 3), for n even.

Construct the cyclic group of order n as an fp-group.

Construct the dihedral group of degree n and order 2n as an fp-group.

Sym(GrpFP, n) : Cat, RngIntElt -> GrpFP

Construct the symmetric group of degree n as an fp-group, where the generators correspond to the permutations (1, 2, ..., n) and (1, 2).

> S8 := SymmetricGroup(GrpFP, 8);
> S8;
Finitely presented group S8 on 2 generators
Relations
      S8.1^8 = Id(S8)
      S8.2^2 = Id(S8)
      (S8.1 * S8.2)^7 = Id(S8)
      (S8.1^-1 * S8.2 * S8.1 * S8.2)^3 = Id(S8)
      (S8.2 * S8.1^-2 * S8.2 * S8.1^2)^2 = Id(S8)
      (S8.2 * S8.1^-3 * S8.2 * S8.1^3)^2 = Id(S8)
      (S8.2 * S8.1^-4 * S8.2 * S8.1^4)^2 = Id(S8)

Given an fp-group G, construct a maximal central extension tilde G of G. The group tilde G is created as an fp-group.

Given two fp-groups G and H, construct the direct product of G and H.

DirectProduct(Q) : [ GrpFP ] -> GrpFP

Given a sequence Q of r fp-groups, construct the direct product Q[1] x ... x Q[r].

Given two fp-groups G and H, construct the free product of G and H.

FreeProduct(Q) : [ GrpFP ] -> GrpFP

Given a sequence Q of r fp-groups, construct the free product of the groups Q[1], ..., Q[r].

> G<x1, x2> := Group<x1, x2 | x1^4,(x1*x2^-1)^2,x2^4,(x1*x2)^3>;
> G;

Finitely presented group G on 2 generators
Relations
       x1^4 = Id(G)
       (x1 * x2^-1)^2 = Id(G)
       x2^4 = Id(G)
       (x1 * x2)^3 = Id(G)

> D := Darstellungsgruppe(G);
> D;

Finitely presented group D on 4 generators
Relations
       D.1^4 * D.3^-1 * D.4^2 = Id(D)
       D.1 * D.2^-1 * D.1 * D.2^-1 * D.4 = Id(D)
       D.2^4 = Id(D)
       D.1 * D.2 * D.1 * D.2 * D.1 * D.2 * D.4 = Id(D)
       (D.1, D.3) = Id(D)
       (D.2, D.3) = Id(D)
       (D.1, D.4) = Id(D)
       (D.2, D.4) = Id(D)
       (D.3, D.4) = Id(D)

> Index(D, sub< D | >);

108

> // Thus a maximal central extension of G has order 108

> A5 := Group<a, b | a^2, b^3, (a*b)^5 >;
> Z2 := quo< FreeGroup(1) | $.1^2 >;
> G := DirectProduct(A5, Z2);
> G;

Finitely presented group G on 3 generators
Relations
      G.1^2 = Id(G)
      G.2^3 = Id(G)
      (G.1 * G.2)^5 = Id(G)
      G.3^2 = Id(G)
      G.1 * G.3 = G.3 * G.1
      G.2 * G.3 = G.3 * G.2