# Lattice-complemented not implies permutably complemented

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., lattice-complemented subgroup) neednotsatisfy the second subgroup property (i.e., permutably complemented subgroup)

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## Contents

## Statement

A lattice-complemented subgroup of a group need not be permutably complemented.

## Definitions used

### Lattice-complemented subgroup

`Further information: Lattice-complemented subgroup`

A subgroup of a group is termed **lattice-complemented** in if there exists a subgroup of such that is trivial and .

### Permutably complemented subgroup

`Further information: Permutably complemented subgroup`

A subgroup of a group is termed **permutably complemented** in if there exists a subgroup of such that is trivial and .

## Proof

### Example of the alternating group

`Further information: alternating group:A4`

Let be the alternating group on the set . Consider the two-element subgroup:

.

- is lattice-complemented in : The cyclic subgroup generated by is a lattice complement to in .
- is not permutably complemented in : This can be seen by inspection. In fact, there is no subgroup of order six in , and any permutable complement to must have order six.

### Example of the dihedral group

`Further information: dihedral group:D16`

Let be the dihedral group of order :

.

Let be the subgroup of generated by and :

.

- is a lattice-complemented subgroup of : Indeed, the subgroup is a lattice complement to in .
- is not a permutably complemented subgroup of : Any permutable complement to must have order four. Hence, the intersection of with must have order either two or four. In either case, that intersection must contain the cyclic subgroup . Hence, does not intersect trivially.

### A generic example: simple non-Abelian group

Every simple non-Abelian group is a K-group: every subgroup is lattice-complemented. On the other hand, by Hall's theorem on solvability, there does not exist a -Sylow complement for every , so not every -Sylow subgroup is permutably complemented. Thus, we can always find a lattice-complemented subgroup that is not permutably complemented.