# P-core-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. Let $G$ be a p-group, i.e., a (possibly infinite) group in which every element has order equal to a power of $p$. A subgroup $H$ of $G$ is termed $p$-core-automorphism-invariant if for any normal $p$-subgroup of $\operatorname{Aut}(G)$, $H$ is invariant under all automorphisms in that subgroup.

In the finite case, this is equivalent to the following: $H$ is invariant under the subgroup $O_p(\operatorname{Aut}(G))$, where $O_p$ denotes the $p$-core or Sylow-core, i.e., the unique largest normal $p$-subgroup.

For this property in the context of a finite p-group, refer p-core-automorphism-invariant subgroup of finite p-group.