P-core-automorphism-invariant subgroup

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.

Stronger properties

 * Weaker than::Characteristic subgroup of p-group

Weaker properties

 * Stronger than::p-automorphism-invariant subgroup
 * Stronger than::Normal subgroup of p-group