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, $Z(H)\le Z(G)$. 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
normal 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
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