Classification of symmetric groups that are N-groups

From Groupprops
Jump to: navigation, search
This article classifies the members in a particular group family symmetric group that satisfy the group property N-group.

Statement

Suppose n is a positive integer. The symmetric group S_n is a N-group if and only if n \in \{ 0,1,2,3,4,5,6 \}.

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 n

Consider n \ge 7. Describe S_n concretely as the group of all permutations on the set \{ 1,2,\dots, n\}. Let x be the element (1,2) in S_n. The centralizer C_G(x) equals the normalizer N_G(\langle x \rangle) and it is the internal direct product \operatorname{Sym} \{ 1,2 \} \times \operatorname{Sym} \{ 3,4,\dots,n \}. This is isomorphic to S_2 \times S_{n-2}. For n \ge 7, n - 2 \ge 5, so S_{n-2} is non-solvable (follows from alternating groups are simple for degree \ge 5), hence S_2 \times S_{n-2} is non-solvable.

A similar example works for the symmetric group on an infinite set.

Proof of success

For n \in \{0,1,2,3,4 \}, the whole group is solvable, so it is a N-group.

The case n = 5 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 n = 6, 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.