Associated direct sum of a subnormal series

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Definition

Let $G$ be a group and consider:

$1=K_{0}\triangleleft K_{1}\triangleleft \ldots \triangleleft K_{n}=G$

a subnormal series for $G$. The associated direct sum for this series is the abstract group:

$\bigoplus _{i=0}^{n-1}K_{i+1}/K_{i}$

(direct sum here means the same thing as direct product).

Facts

Automorphisms

• Every automorphism of the subnormal series defines an automorphism of the associated direct sum. Thus, there is a homomorphism from the automorphism group of the subnormal series, to the automorphism group of the associated direct sum. The kernel of this homomorphism is termed the stability group.
• If the subnormal series is a normal series, then every inner automorphism becomes an automorphism of the subgroup series, and hence gives rise to an automorphism of the direct sum.
• If the subnormal series is a central series, then every inner automorphism is a stability automorphism of the subgroup series, and hence gives rise to the identity map on the direct sum.
• If the subnormal series is a characteristic series, then every automorphism of the group is an automorphism of the subgroup series, hence we have a map from the automorphism group of the whole group to the automorphism group of the associated direct sum.

Additional structure

When the subgroup series is a strongly central series, the associated direct sum acquires the structure of a Lie ring.

Further information: associated Lie ring for a strongly central series