# Autoclinism-invariant subgroup

## Definition

### As an invariance property

A subgroup $H$ of a group $G$ is termed an autoclinism-invariant subgroup if it is invariant under any bijective set map $\sigma:G \to G$ satisfying all three of these conditions:

• $\sigma$ induces an automorphism modulo the center, i.e., $\sigma(xy) = \sigma(x)\sigma(y)$ mod $Z(G)$.
• $\sigma$ restricts to an automorphism on the derived subgroup, i.e., $\sigma$ sends the derived subgroup to itself and the restriction to the derived subgroup is an automorphism of the derived subgroup.
• $\sigma([x,y]) = [\sigma(x),\sigma(y)]$ for all (possibly equal, possibly distinct) $x,y \in G$.

### As a two-case property

A subgroup $H$ of a group $G$ is termed an autoclinism-invariant subgroup if it satisfies either of these conditions:

1. $H$ contains the center $Z(G)$ and $H/Z(G)$ is invariant under any automorphism of $G/Z(G)$ that is the inner automorphism group part of the data specifying an autoclinism.
2. $H$ is contained in the derived subgroup $[G,G]$ and it is invariant under any automorphism of $[G,G]$ that is the derived subgroup part of the data specifying an autoclinism.
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]
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## Relation with other properties

### Stronger properties

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