Stability automorphism of subnormal series

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Definition

Symbol-free definition

An automorphism of a group is said to be a stability automorphism with respect to a subnormal series if it induces the identity map on each successive quotient for the subnormal series.

The stability automorphisms of any fixed subnormal series form a group, called the stability group of that subnormal series. This group lives as a subgroup of the automorphism group.

Definition with symbols

An automorphism $\sigma$ of a group $G$ is termed a stability automorphism with respect to the subnormal series:

$\{e\}=H_{0}\triangleleft H_{1}\triangleleft \ldots \triangleleft H_{n}=G$

if $\sigma (H_{i}x)=H_{i}x$ for any $x\in H_{i+1}$, or equivalently, $\sigma$ acts as identity on $H_{i+1}/H_{i}$.

(An analogous definition can be given for subnormal series indexed by infinite sets).

Facts

• Stability group of subnormal series of p-group is p-group