# Classification of symmetric groups that are N-groups

This article classifies the members in a particular group family symmetric group that satisfy the group property N-group.

## Contents

## Statement

Suppose is a positive integer. The symmetric group is a N-group if and only if .

Also, the symmetric group on an infinite set is never a N-group. Similarly, the finitary symmetric group on an infinite set is never a N-group.

## Related facts

## Proof

### Proof of failure for larger

Consider . Describe concretely as the group of all permutations on the set . Let be the element in . The centralizer equals the normalizer and it is the internal direct product . This is isomorphic to . For , , so is non-solvable (follows from alternating groups are simple for degree ), hence is non-solvable.

A similar example works for the symmetric group on an infinite set.

### Proof of success

For , the whole group is solvable, so it is a N-group.

The case is also direct: the only non-solvable subgroups are the whole group symmetric group:S5 and A5 in S5 (isomorphic to alternating group:A5) and neither of them has a nontrivial solvable normal subgroup.

For the case , the only non-solvable subgroups are the whole group symmetric group:S6, the subgroup A6 in S6 (isomorphic to alternating group:A6), and subgroups isomorphic to alternating group:A5 and symmetric group:S5. None of them have a nontrivial solvable normal subgroup. Thus, the group is a N-group.