Center-fixing automorphism-invariant subgroup

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Definition

A subgroup of a group is termed a center-fixing automorphism-invariant subgroup if every center-fixing automorphism of the whole group sends the subgroup to within itself.

Metaproperties

Metaproperty	Satisfied?	Proof	Statement with symbols
strongly intersection-closed subgroup property	Yes	Follows from invariance implies strongly intersection-closed	If $H_{i},i\in I$ are all center-fixing automorphism-invariant subgroups of $G$, then so is the intersection of subgroups $\bigcap _{i\in I}H_{i}$.
strongly join-closed subgroup property	Yes	Follows from endo-invariance implies strongly join-closed	If $H_{i},i\in I$ are all center-fixing automorphism-invariant subgroups of $G$, then so is the join of subgroups $\langle H_{i}\rangle _{i\in I}$.
trim subgroup property	Yes	direct from definition	In any group $G$, both the whole group $G$ and the trivial subgroup are center-fixing automorphism-invariant.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
characteristic subgroup	invariant under all automorphisms	characteristic implies center-fixing automorphism-invariant	center-fixing automorphism-invariant not implies characteristic	|FULL LIST, MORE INFO
central subgroup	contained in the center	central implies center-fixing automorphism-invariant	center-fixing automorphism-invariant not implies central	|FULL LIST, MORE INFO
center-fixing automorphism-balanced subgroup	every center-fixing automorphism of the whole group restricts to a center-fixing automorphism of the subgroup.			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
normal subgroup	invariant under all inner automorphisms			|FULL LIST, MORE INFO

Formalisms

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Function restriction expression

This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
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Function restriction expression	$H$ is a center-fixing automorphism-invariant subgroup of $G$ if ...	This means that being center-fixing automorphism-invariant is ...	Additional comments
center-fixing automorphism $\to$ function	every center-fixing automorphism of $G$ sends every element of $H$ to within $H$	the invariance property for center-fixing automorphisms	On account of this, it is a strongly intersection-closed subgroup property.
center-fixing automorphism $\to$ endomorphism	every center-fixing automorphism of $G$ restricts to an endomorphism of $H$	the endo-invariance property for center-fixing automorphisms; i.e., it is the invariance property for center-fixing automorphism, which is a property stronger than the property of being an endomorphism	On account of this, it is a strongly join-closed subgroup property.
center-fixing automorphism $\to$ automorphism	every center-fixing automorphism of $G$ restricts to an automorphism of $H$	the auto-invariance property for center-fixing automorphisms; i.e., it is the invariance property for center-fixing automorphism, which is a group-closed property of automorphisms