Subnormal stability automorphism-invariant subgroup

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Definition

Definition with symbols

A subgroup $H$ of a group $G$ is termed a subnormal stability automorphism-invariant subgroup if, for any subnormal series of $G$, and any stability automorphism $\sigma$ of that subnormal series, $\sigma (H)=H$.

Relation with other properties

Stronger properties

• Characteristic subgroup
• Cofactorial automorphism-invariant subgroup for a finite group.

Weaker properties

• Normal stability automorphism-invariant subgroup

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 $p$-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