Subgroup Chains

Series involving centres or commutator subgroups are likely to be trivial for abelian groups, but are noted here for completeness.

ChiefSeries(G): GrpAb -> [GrpAb]

CompositionSeries(G): GrpAb -> [GrpAb]

A composition series for the finite abelian group \(G\) returned as a sequence of subgroups.

Agemo(G, i): GrpAb, RngIntElt -> GrpAb

Given a finite \(p\)-group \(G\), return the characteristic subgroup of \(G\) generated by the elements \(x^{p^i}\), \(x \in G\), where \(i\) is a positive integer.

Omega(G, i): GrpAb, RngIntElt -> GrpAb

Given a finite \(p\)-group \(G\), return the characteristic subgroup of \(G\) generated by the elements of order dividing \(p^i\), where \(i\) is a positive integer.

ElementaryAbelianSeries(G): GrpAb -> [ GrpAb ]

A descending series of subgroups of \(G\) where each quotient is elementary abelian.

DerivedSeries(G): GrpAb -> [ GrpAb ]

LowerCentralSeries(G): GrpAb -> [ GrpAb ]

The derived series of \(G\).

UpperCentralSeries(G): GrpAb -> [ GrpAb ]

The upper central series of \(G\).

SubnormalSeries(G, H): GrpAb, GrpAb -> [ GrpAb ]

A subnormal series from \(G\) to the subnormal subgroup \(H\).