# Conjugacy-closedness is not join-closed

From Groupprops

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

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## Statement

We can have a situation where are conjuacy-closed subgroups of but the join is not conjugacy-closed in .

## Proof

### Example of the dihedral group

`Further information: dihedral group:D8`

Let be the dihedral group of order eight:

.

Suppose are subgroups given as follows:

.

Observe that:

- Since and are both subgroups of order two, they are both conjugacy-closed in .
- The join is an Abelian subgroup of order four. It is clearly
*not*conjugacy-closed, because the elements and in this subgroup are conjugate in .