Classification of alternating groups that are N-groups

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

Statement

Suppose $n$ is a positive integer greater than or equal to 3. The alternating group $A_{n}$ is a N-group if and only if $n\in \{3,4,5,6,7\}$ .

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

Related facts

Classification of symmetric groups that are N-groups

Proof

Proof of failure for larger $n$

For $n\geq 8$ , denote by $A_{n}$ the alternating group on the set $\{1,2,3,\dots ,n\}$ . Let $x=(1,2,3)$ . The normalizer $N_{G}(\langle x\rangle )$ contains the centralizer of $x$ , which in turn contains the alternating group on the set $\{4,5,\dots ,n\}$ , which is isomorphic to $A_{n-3}$ , which is simple non-abelian since $n\geq 8\implies n-3\geq 5$ (see alternating groups are simple). Thus, $N_{G}(\langle x\rangle )$ is not solvable, so $A_{n}$ is not a N-group.