# Associated direct sum of a subnormal series

From Groupprops

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## Definition

Let be a group and consider:

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

(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`