Normal subgroup whose center is contained in the center of the whole group

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and subgroup whose center is contained in the center of the whole group
View other subgroup property conjunctions | view all subgroup properties

Definition

A subgroup of a group is termed a normal subgroup whose center is contained in the center of the whole group if it satisfies the following equivalent conditions.

No.	Shorthand	A subgroup of a group is termed a normal subgroup whose center is contained in the center of the whole group if ...	A subgroup $H$ of a group $G$ is termed a normal subgroup whose center is contained in the center of the whole group if ...
1	normal, and center contained in whole group	it is a normal subgroup and is also a subgroup whose center is contained in the center of the whole group.	$H$ is normal in $G$ and $Z(H) \le Z(G)$, where $Z(H)$ and $Z(G)$ are the respective centers of $H$ and $G$.
2	inner automorphism restricts to center-fixing automorphism	any inner automorphism of the whole group restricts to a center-fixing automorphism of the subgroup.	for any $g \in G$, the automorphism $x \mapsto gxg^{-1}$ restricts to a center-fixing automorphism of the subgroup $H$, i.e., it sends $H$ to itself and fixes every element of $Z(H)$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
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
central factor	every inner automorphism restricts to an inner automorphism			|FULL LIST, MORE INFO
conjugacy-closed normal subgroup	every inner automorphism restricts to a class-preserving automorphism			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
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

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

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.
Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property

Function restriction expression	$H$ is a normal subgroup whose center is contained in the center of the whole group in $G$ if ...	This means that being a normal subgroup whose center is contained in the center of the whole group is ...	Additional comments
inner automorphism $\to$ center-fixing automorphism	every inner automorphism of $G$ restricts to a center-fixing automorphism of $H$	a left-inner subgroup property.