# 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 $H$ is a subgroup of a group $G$. We say that $H$ is a subgroup for which any join of conjugates is a join of finitely many conjugates if, for any subset $S$ of $G$, there is a finite subset $T$ of $G$ such that:

$\langle \bigcup_{s \in S} sHs^{-1} \rangle = \langle \bigcup_{t \in T} tHt^{-1}\rangle$

## 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