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.

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.