Autoclinism-invariant subgroup: Difference between revisions

Latest revision as of 00:49, 3 February 2012

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 $σ : G \to G$ satisfying all three of these conditions:

$σ$ induces an automorphism modulo the center, i.e., $σ (x y) = σ (x) σ (y)$ mod $Z (G)$ .
$σ$ restricts to an automorphism on the derived subgroup, i.e., $σ$ sends the derived subgroup to itself and the restriction to the derived subgroup is an automorphism of the derived subgroup.
$σ ([x, y]) = [σ (x), σ (y)]$ 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:

$H$ 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.
$H$ 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

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-{{subgroup property}}
 ==Definition==
 ===As an invariance property===
-A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed an '''autoclinism-invariant subgroup''' if it is invariant under any set map <math>\sigma:G \to G</math> satisfying all three of these conditions:
+A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed an '''autoclinism-invariant subgroup''' if it is invariant under any bijective set map <math>\sigma:G \to G</math> satisfying all three of these conditions:
 * <math>\sigma</math> induces an automorphism modulo the center, i.e., <math>\sigma(xy) = \sigma(x)\sigma(y)</math> mod <math>Z(G)</math>.