Stability automorphism of subnormal series

This article defines an automorphism property, viz a property of group automorphisms. Hence, it also defines a function property (property of functions from a group to itself)
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Definition

Symbol-free definition

An automorphism of a group is said to be a stability automorphism if there exists a subnormal series for the group such that the given automorphism is a stability automorphism for that subnormal series.

Definition with symbols

An automorphism $\sigma$ of a group $G$ is termed a stability automorphism if there exists a subnormal series:

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

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

Metaproperties

Group-closedness

This automorphism property is group-closed: it is closed under the group operations on automorphisms (composition, inversion and the identity map). It follows that the subgroup comprising automorphisms with this property, is a normal subgroup of the automorphism group
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The product of two stability automorphisms is a stability automorphism. The idea is to take the subnormal series of the product as a subnormal series that in some sense subsumes the subnormal series of the two stability automorphisms.

For full proof, refer: Stability automorphism is group-closed

Direct product-closedness

This automorphism property is direct product-closed
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