Stability automorphism of subnormal series

This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]

VIEW: Definitions built on this | Facts about this: (facts closely related to Stability automorphism of subnormal series, all facts related to Stability automorphism of subnormal series) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View other semistandard definitions in group theory

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.

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).