Classification of alternating groups that are N-groups

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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

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.

Proof of success

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