Groups, conjugacy classes, and inverses

Factual Questions
1

This may be an over-specialized question for the SDMB, but I figured I’d post it anyway and see if any mathematically-oriented folks knew an answer.

The symmetric group S[sub]n[/sub] (for any value of n) has the property that every element is in the same conjugacy class as its inverse; in other words, for every element a in the group, there exists another element b such that b[sup]-1[/sup]ab = a[sup]-1[/sup]. (This can be seen by remembering that the conjugacy classes of S[sub]n[/sub] are all elements with a given cycle structure.) Does this property have a name, or does it follow from another well-known group property? Do any finite (non-Abelian) groups other than S[sub]n[/sub] share it?

2

It is easy to prove that all of the dihedral groups D[sub]n[/sub] have this property. Most of the alternating groups A[sub]n[/sub] do not, though a few do. You may be interested in this paper (link to abstract; PDF at the link), which finds the A[sub]n[/sub] with this property (proof is easy).

3

I forgot to mention: that paper also provides another characterization (no pun intended) of these groups, in terms of their irreducible characters.

4

Thanks for the info — as a physicist, I usually don’t have much call for this stuff, but I still find it fascinating.

5

Yes, same here (on all counts).