Center-fixing automorphism-balanced subgroup

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This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: center-fixing automorphism-invariant subgroup and subgroup whose center is contained in the center of the whole group
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Definition

The following are equivalent definitions of center-fixing automorphism-balanced subgroup.

No.	Shorthand	A subgroup of a group is termed a center-fixing automorphism-balanced subgroup if ...	A subgroup $H$ of a group $G$ is termed a center-fixing automorphism-balanced subgroup if ...
1	center-fixing automorphism-invariant, and center contained in group's center	it is a center-fixing automorphism-invariant subgroup of the whole group, and its center is contained in the center of the whole group.	$H$ is invariant under any automorphism $\sigma$ of $G$ with the property that $\sigma (x)=x$ for all $x\in Z(G)$, and additionally, $Z(H)\leq Z(G)$. Here $Z(H)$ and $Z(G)$ denote respectively the centers of $H$ and $G$.
2	center-fixing automorphism restricts to center-fixing automorphism	any center-fixing automorphism of the whole group restricts to a center-fixing automorphism of the subgroup.	For any automorphism $\sigma$ of $G$ with the property that $\sigma (x)=x$ for all $x\in Z(G)$, we have that $\sigma (H)\subseteq H$, and also that $\sigma (x)=x$ for all $x\in Z(H)$.

Relation with other properties

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
center-fixing automorphism-invariant subgroup	invariant under all center-fixing automorphisms
normal subgroup whose center is contained in the center of the whole group	normal and its center is contained in the whole group's center.			|FULL LIST, MORE INFO
normal subgroup	every inner automorphism sends the subgroup to itself			|FULL LIST, MORE INFO
subgroup whose center is contained in the center of the whole group	its center is contained in the whole group's center.			|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-balanced subgroup of $G$ if ...	This means that being center-fixing automorphism-balanced is ...	Additional comments
center-fixing automorphism $\to$ center-fixing automorphism	every center-fixing automorphism of $G$ restricts to a center-fixing automorphism of $H$	the balanced subgroup property for center-fixing automorphisms	Hence, it is a t.i. subgroup property, both transitive and identity-true