# Subgroup for which any join of conjugates is a join of finitely many conjugates

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]

## Definition

Suppose is a subgroup of a group . We say that is a **subgroup for which any join of conjugates is a join of finitely many conjugates** if, for any subset of , there is a finite subset of such that:

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Normal subgroup | equals all of its conjugates | (obvious) | Almost normal subgroup, Nearly normal subgroup|FULL LIST, MORE INFO | |

Subgroup of finite index | has finite index in the whole group | Almost normal subgroup, Nearly normal subgroup|FULL LIST, MORE INFO | ||

Subgroup of finite group | the whole group is a finite group | |FULL LIST, MORE INFO | ||

Nearly normal subgroup | has finite index in its normal closure | |FULL LIST, MORE INFO | ||

Almost normal subgroup | has finitely many conjugate subgroups | |FULL LIST, MORE INFO |