Subgroup invariant under conjugation by a generating set

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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

• For a finite subgroup, this condition is equivalent to being normal.
• For a subgroup of finite index, this condition is equivalent to being normal.
• The condition is equivalent to normality in a slender group as well as in an Artinian group -- and in particular in a periodic group.

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