Classification of symmetric groups that are N-groups
This article classifies the members in a particular group family symmetric group that satisfy the group property N-group.
Statement
Suppose is a positive integer. The symmetric group
is a N-group if and only if
.
Also, the symmetric group on an infinite set is never a N-group. Similarly, the finitary symmetric group on an infinite set is never a N-group.
Related facts
Proof
Proof of failure for larger 
Consider . Describe
concretely as the group of all permutations on the set
. Let
be the element
in
. The centralizer
equals the normalizer
and it is the internal direct product
. This is isomorphic to
. For
,
, so
is non-solvable (follows from alternating groups are simple for degree
), hence
is non-solvable.
A similar example works for the symmetric group on an infinite set.
Proof of success
For , the whole group is solvable, so it is a N-group.
The case is also direct: the only non-solvable subgroups are the whole group symmetric group:S5 and A5 in S5 (isomorphic to alternating group:A5) and neither of them has a nontrivial solvable normal subgroup.
For the case , the only non-solvable subgroups are the whole group symmetric group:S6, the subgroup A6 in S6 (isomorphic to alternating group:A6), and subgroups isomorphic to alternating group:A5 and symmetric group:S5. None of them have a nontrivial solvable normal subgroup. Thus, the group is a N-group.