P-Sylow-automorphism-invariant subgroup of finite p-group

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Definition

Suppose $p$ is a prime number and $P$ is a finite $p$-group, i.e., a group of prime power order where the underlying prime is $p$. A subgroup $H$ of $P$ is termed a $p$-Sylow-automorphism-invariant subgroup of $P$ if there exists a $p$-Sylow subgroup $S$ of $\operatorname {Aut} (G)$ such that $H$ is invariant under $S$.

Relation with other properties

Stronger properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
p-automorphism-invariant subgroup of finite p-group
Characteristic subgroup of group of prime power order

Weaker properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
p-core-automorphism-invariant subgroup of finite p-group
Normal subgroup of group of prime power order