# Subgroup invariant under conjugation by a generating set

From Groupprops

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

## Contents

## Definition

A subgroup of a group is termed a **subgroup invariant under conjugation by a generating set** if there is a generating set for such that conjugation by any element of sends to a subset of itself, i.e., for all .

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 |