The Subgroup Structure

General Subgroup Constructions

The operators and functions which construct a subgroup of a polycyclic group always return the subgroup as a polycyclic group.

H ^ g: GrpGPC, GrpGPCElt -> GrpGPC

Conjugate(H, g): GrpGPC, GrpGPCElt -> GrpGPC

Construct the conjugate \(g^{-1}*H*g\) of the group \(H\) under the action of the element \(g\). The group \(H\) and the element \(g\) must belong to a common group.

H ^ G: GrpGPC, GrpGPC -> GrpGPC

ncl< G | H >: GrpGPC, GrpGPC -> GrpGPC

NormalClosure(G, H): GrpGPC, GrpGPC -> GrpGPC

Given a subgroup \(H\) of the group \(G\), construct the normal closure of \(H\) in \(G\).

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

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

Construct the commutator subgroup of groups \(H\) and \(K\), where \(H\) and \(K\) are subgroups of a common group \(G\).

Subgroup Constructions Requiring a Nilpotent Covering Group

The operators and functions described in this section require the existence of a nilpotent covering group. They are based on algorithms published in [Lo, 1998]. Again, the constructed subgroup is returned as a polycyclic group.

H meet K: GrpGPC, GrpGPC -> GrpGPC

Given two groups \(H\) and \(K\), contained in some common group \(G\) which is nilpotent, construct the intersection of \(H\) and \(K\).

H meet:= K: GrpGPC, GrpGPC -> GrpGPC

Given two groups \(H\) and \(K\), contained in some common group \(G\) which is nilpotent, replace \(H\) with the intersection of \(H\) and \(K\).

Centraliser(G, g): GrpGPC, GrpGPCElt -> GrpGPC

Centralizer(G, g): GrpGPC, GrpGPCElt -> GrpGPC

The subgroup of \(G\) centralising \(g\). Both \(g\) and \(G\) must be contained in some common nilpotent group.

Centraliser(G, H): GrpGPC, GrpGPC -> GrpGPC

Centralizer(G, H): GrpGPC, GrpGPC -> GrpGPC

The subgroup of \(G\) centralising \(H\). Both \(H\) and \(G\) must be subgroups of some common nilpotent group.

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

The maximal normal subgroup of the nilpotent group \(G\) that is contained in the subgroup \(H\) of \(G\).

Normaliser(G, H): GrpGPC, GrpGPC -> GrpGPC

Normalizer(G, H): GrpGPC, GrpGPC -> GrpGPC

The subgroup of \(G\) normalising \(H\). Both \(H\) and \(G\) must be subgroups of some common nilpotent group.