# Classification of alternating groups that are N-groups

From Groupprops

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

## Statement

Suppose is a positive integer greater than or equal to 3. The alternating group is a N-group if and only if .

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

## Related facts

## Proof

### Proof of failure for larger

For , denote by the alternating group on the set . Let . The normalizer contains the centralizer of , which in turn contains the alternating group on the set , which is isomorphic to , which is simple non-abelian since (see alternating groups are simple). Thus, is not solvable, so is not a N-group.

### Proof of success

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