Associated direct sum of a subnormal series

From Groupprops
Jump to: navigation, search
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

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