Autoclinism-invariant subgroup

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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]

Examples

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