P-automorphism-invariant subgroup

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Definition

Let $p$ be a prime number. A subgroup $H$ of a p-group $G$ (i.e., a group where the order of every element is a power of $p$) is termed $p$-automorphism-invariant if, for any automorphism $\sigma$ of $G$ such that the order of $\sigma$ is a power of $p$, $\sigma (H)=H$.

For the property the context of finite $p$-groups, refer p-automorphism-invariant subgroup of finite p-group.

Relation with other properties

Equivalent properties

For a finite $p$-group, the property of being $p$-automorphism invariant equals the property of being a subnormal stability automorphism-invariant subgroup. That's because any $p$-automorphism can be realized as the stability automorphism of a subnormal series, and conversely, any stability automorphism of a subnormal series must be a $p$-automorphism. Further information: 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

Stronger properties

• Characteristic subgroup: For proof of the implication, refer Characteristic implies p-automorphism-invariant and for proof of its strictness (i.e. the reverse implication being false) refer p-automorphism-invariant not implies characteristic.

Weaker properties

• Normal subgroup