# Conjugacy-closedness is transitive

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., conjugacy-closed subgroup) satisfying a subgroup metaproperty (i.e., transitive subgroup property)

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Get more facts about conjugacy-closed subgroup |Get facts that use property satisfaction of conjugacy-closed subgroup | Get facts that use property satisfaction of conjugacy-closed subgroup|Get more facts about transitive subgroup property

## Statement

### Statement with symbols

Suppose are groups such that is conjugacy-closed in and is conjugacy-closed in .

## Related facts

## Proof

### Hands-on proof

**Given**: Groups such that is conjugacy-closed in and is conjugacy-closed in .

**To prove**: is conjugacy-closed in .

**Proof**: We need to show that if are conjugate in , then they are conjugate in . First, observe that since , are elements of conjugate in . Since is conjugacy-closed in , are conjugate in .

Thus, are elements of that are conjugate in . Since is conjugacy-closed in , the elements are conjugate in .