# 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 $H$.