P-core-automorphism-invariant subgroup

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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.

Relation with other properties

Stronger properties

• Characteristic subgroup of p-group

Weaker properties

• p-automorphism-invariant subgroup
• Normal subgroup of p-group