# Central factor is upper join-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., central factor) satisfying a subgroup metaproperty (i.e., upper join-closed subgroup property)

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## Contents

## Statement

### Statement with symbols

Suppose is a group, is a subgroup of , and , are subgroups of all containing . Suppose, further, that is a central factor of each . Then, is also a central factor of the join of the s.

## Related facts

- Central factor satisfies intermediate subgroup condition
- Central factor satisfies image condition
- Central factor is transitive
- Central factor does not satisfy transfer condition

## Proof

### Using function restriction expressions

This subgroup property implication can be proved by using function restriction expressions for the subgroup properties

View other implications proved this way |read a survey article on the topic

**Given**: is a group, is a subgroup of , and , are subgroups of all containing . is a central factor of each .

**To prove**: is a central factor of the join of s. In other words, if is an element in the join of the s, then there exists such that conjugation by agrees with conjugation by on .

**Proof**: Suppose is an element in the join of the s. Then, there exist and such that .

For each , since is a central factor of , there exists such that conjugation by agrees with conjugation by on . Set . Then, since conjugation is a group action, we see that conjugation by agrees with conjugation by on , completing the proof.

### Proof using centralizer definition

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