# Subgroup invariant under conjugation by a generating set

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]
This is a variation of normal subgroup|Find other variations of normal subgroup | Read a survey article on varying normal subgroup

## 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

### 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