Standard Constructions and Conversions

AbelianGroup(GrpAb, Q) : Cat, [ RngIntElt ] -> GrpAb

AbelianGroup(Q) : [ RngIntElt ] -> GrpAb

Let Q = [ a_1, ..., a_r] be a sequence of non-negative integers. This function creates the abelian group Z_1 + ... + Z_r, where Z_i is the cyclic group of order |a_i| if a_i neq0 or the infinite cyclic group Z otherwise, i = 1, ..., r.

AbelianGroup(G) : Grp -> GrpAb, Hom

Given an abelian permutation, matrix or polycyclic group G, represent it as an abelian group A. The function also returns the isomorphism phi: G -> A as its second value.

AbelianQuotient(G) : Grp -> GrpAb, Hom

Given a finitely presented, permutation, matrix or polycyclic group G, return the maximal abelian quotient A of G. The function returns the natural homomorphism phi: G -> A as its second value.

DirectSum(A, B) : GrpAb, GrpAb -> GrpAb

The direct sum of abelian groups A and B.

PCGroup(A) : GrpAb -> GrpPC, Hom(Grp)

A pc-group representation G of A. The isomorphism phi: A -> G is also returned.

PermutationGroup(A) : GrpAb -> GrpPerm, Hom(Grp)

A permutation group representation of A. The particular group G is generated by disjoint cycles whose lengths are the abelian invariants of A. The isomorphism phi: G -> A is also returned.

FPGroup(A) : GrpAb -> GrpFP, Hom(Grp)

A fp-group group representation of A. The particular group G is generated by commuting generators whose orders are the abelian invariants of A. The isomorphism phi: G -> A is also returned.