# Center-fixing automorphism-invariant subgroup

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

A subgroup of a group is termed a center-fixing automorphism-invariant subgroup if every center-fixing automorphism of the whole group sends the subgroup to within itself.

## Metaproperties

Metaproperty Satisfied? Proof Statement with symbols
strongly intersection-closed subgroup property Yes Follows from invariance implies strongly intersection-closed If $H_i, i \in I$ are all center-fixing automorphism-invariant subgroups of $G$, then so is the intersection of subgroups $\bigcap_{i \in I} H_i$.
strongly join-closed subgroup property Yes Follows from endo-invariance implies strongly join-closed If $H_i, i \in I$ are all center-fixing automorphism-invariant subgroups of $G$, then so is the join of subgroups $\langle H_i \rangle_{i \in I}$.
trim subgroup property Yes direct from definition In any group $G$, both the whole group $G$ and the trivial subgroup are center-fixing automorphism-invariant.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
characteristic subgroup invariant under all automorphisms characteristic implies center-fixing automorphism-invariant center-fixing automorphism-invariant not implies characteristic |FULL LIST, MORE INFO
central subgroup contained in the center central implies center-fixing automorphism-invariant center-fixing automorphism-invariant not implies central |FULL LIST, MORE INFO
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

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Function restriction expression $H$ is a center-fixing automorphism-invariant subgroup of $G$ if ... This means that being center-fixing automorphism-invariant is ... Additional comments
center-fixing automorphism $\to$ function every center-fixing automorphism of $G$ sends every element of $H$ to within $H$ the invariance property for center-fixing automorphisms On account of this, it is a strongly intersection-closed subgroup property.
center-fixing automorphism $\to$ endomorphism every center-fixing automorphism of $G$ restricts to an endomorphism of $H$ the endo-invariance property for center-fixing automorphisms; i.e., it is the invariance property for center-fixing automorphism, which is a property stronger than the property of being an endomorphism On account of this, it is a strongly join-closed subgroup property.
center-fixing automorphism $\to$ automorphism every center-fixing automorphism of $G$ restricts to an automorphism of $H$ the auto-invariance property for center-fixing automorphisms; i.e., it is the invariance property for center-fixing automorphism, which is a group-closed property of automorphisms