# Subnormal stability automorphism-invariant subgroup

From Groupprops

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

### Definition with symbols

A subgroup of a group is termed a **subnormal stability automorphism-invariant subgroup** if, for any subnormal series of , and any stability automorphism of that subnormal series, .

## Relation with other properties

### Stronger properties

### Weaker properties

## Facts

- For a finite group, the stability automorphisms of any subnormal series have no other prime factors to their order than the prime factors of the order of the group.
`For full proof, refer: Stability group of subnormal series of finite group has no other prime factors` - For a group of prime power order, the stability automorphisms of subnormal series are precisely the same as the -automorphisms.
`For full proof, refer: Stability group of subnormal series of p-group is p-group, p-group of automorphisms of p-group is contained in stability group of some normal series`