Retraction-invariant characteristic subgroup

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This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: retraction-invariant subgroup and characteristic subgroup
View other subgroup property conjunctions | view all subgroup properties

Definition

A subgroup of a group is termed a retraction-invariant characteristic subgroup if it satisfies the following two conditions:

  1. It is a retraction-invariant subgroup, i.e., any retraction of the whole group sends the subgroup to itself.
  2. It is a characteristic subgroup, i.e., any automorphism of the whole group sends the subgroup to itself.

Relation with other properties

Stronger properties

property quick description proof of implication proof of strictness (reverse implication failure) intermediate notions
Fully invariant subgroup invariant under all endomorphisms |FULL LIST, MORE INFO

Weaker properties

property quick description 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 Retraction-invariant normal subgroup|FULL LIST, MORE INFO
Retraction-invariant normal subgroup normal and retraction-invariant |FULL LIST, MORE INFO