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.

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Examples

• Every group is autoclinism-invariant in itself.
• The trivial subgroup is autoclinism-invariant in any group.
• All members of the upper central series and all members of the lower central series are autoclinism-invariant.
• For finite p-groups, the ZJ-subgroup and D*-subgroup are autoclinism-invariant.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
endoclinism-invariant subgroup				|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
characteristic subgroup	invariant under all automorphisms			|FULL LIST, MORE INFO
normal subgroup	invariant under all inner automorphisms			|FULL LIST, MORE INFO