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 $n \ge 8$ , 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 $n \ge 8 \implies n - 3 \ge 5$ (see alternating groups are simple). Thus, $N_G(\langle x \rangle)$ is not solvable, so $A_n$ is not a N-group.