Standard Constructions and Conversions

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

AbelianGroup(Q): [ RngIntElt ] -> GrpAb

Let \(Q = [ a_1, \ldots, a_r]\) be a sequence of non-negative integers. This function creates the abelian group \({\mathbb{Z}}_1 + \cdots + {\mathbb{Z}}_r\), where \(Z_i\) is the cyclic group of order \(|a_i|\) if \(a_i\neq0\) or the infinite cyclic group \({\mathbb{Z}}\) otherwise, \(i = 1, \ldots, r\).

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 \(\phi: G \ar A\) 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 \(\phi: G \ar A\) 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 \(\phi: A\rightarrow G\) 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 \(\phi: G\rightarrow A\) is also returned.

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

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

CommutatorSubgroup(G): GrpAb -> GrpAb

DerivedSubgroup(G): GrpAb -> GrpAb

DerivedGroup(G): GrpAb -> GrpAb

The derived subgroup of \(G\), that is the trivial group, since \(G\) is abelian.

CommutatorSubgroup(H, K): GrpAb, GrpAb -> GrpAb

CommutatorSubgroup(G, H, K): GrpAb, GrpAb, GrpAb -> GrpAb

The commutator subgroup of groups \(H\) and \(K\) in their common overgroup \(G\).

Centralizer(G, a): GrpAb, GrpAbElt -> GrpAb

Centraliser(G, a): GrpAb, GrpAbElt -> GrpAb

The centraliser of \(a\) in \(G\).

Core(G, H): GrpAb, GrpAb -> GrpAb

The maximal normal subgroup of \(G\) that is contained in the subgroup \(H\) of \(G\). Since \(G\) is abelian, this is \(H\) itself.

Centre(G): GrpAb -> GrpAb

Center(G): GrpAb -> GrpAb

The center of \(G\), ie. \(G\) itself.

FittingGroup(G): GrpAb -> GrpAb

FittingSubgroup(G): GrpAb -> GrpAb

The Fitting subgroup of \(G\).

Hypercentre(G): GrpAb -> GrpAb

Hypercenter(G): GrpAb -> GrpAb

The hypercentre of \(G\).