Stability automorphism of subnormal series

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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]

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 $\sigma(H_ix) = H_ix$ 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).