# Conjugacy-closed and Hall not implies retract

From Groupprops

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., conjugacy-closed Hall subgroup) neednotsatisfy the second subgroup property (i.e., Hall retract)

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## Statement

A conjugacy-closed Hall subgroup (i.e., a Hall subgroup that is conjugacy-closed: any two elements in the subgroup that are conjugate in the whole group are conjugate in the subgroup) need not be a Retract (?). Specifically, there need not be a Normal Hall subgroup (?) that is a complement to it.

## Related facts

- Conjugacy-closed and Sylow implies retract
- Conjugacy-closed Abelian Sylow implies retract
- Conjugacy-closed Abelian Hall implies retract

## Proof

### Example of Sylow complements in symmetric groups

`Further information: Symmetric group on subset is conjugacy-closed`

If are sets, then the symmetric group on embeds as a conjugacy-closed subgroup of the symmetric group on . In other words, if two elements of are conjugate in , they are also conjugate in .

Let be a prime equal to or more. Then, let and . Define and . Then:

- conjugacy-closed: The group is a conjugacy-closed subgroup of the group by the above fact.
- Hall: is also a Hall subgroup (in fact, it is a Hall -subgroup) of .
- Not a retract: For this, observe that for , the only normal subgroups of are the whole group, the trivial subgroup, and the alternating group. None of these have order , and thus, cannot be realized as a retract.