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Subgroup invariant under conjugation by a generating set

Definition

A subgroup H of a group G is termed a subgroup invariant under conjugation by a generating set if there is a generating set S for G such that conjugation by any element of S sends H to a subset of itself, i.e., sHs^{-1} \subseteq H for all s \in S.

Note that the definition is in terms of the existence of a generating set, and does not say that the condition must hold for every generating set.

Relation with other properties

Collapse to normality

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Normal subgroup invariant under conjugation by all elements Conjugate-comparable subgroup|FULL LIST, MORE INFO
Conjugate-comparable subgroup comparable with all its conjugate subgroups conjugate-comparable implies invariant under conjugation by a generating set invariant under conjugation by a generating set not implies conjugate-comparable |FULL LIST, MORE INFO
Subgroup invariant under a generating set of the automorphism group invariant under a generating set of the automorphism group invariant under a generating set of the automorphism group implies invariant under conjugation by a generating set any finite example of normal not implies characteristic