# P-automorphism-invariant subgroup

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

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