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.