# Center-fixing automorphism-balanced subgroup

## Contents

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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 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 Center-fixing automorphism-invariant subgroup, Normal subgroup whose center is contained in the center of the whole group|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. Normal subgroup whose center is contained in the center of the whole group|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 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