# 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
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## 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.
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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.