Subsets of groups closed under conjugation

I shall notate conjugation like so: C(a, b) = b^-1ab.

A subset of a group closed under conjugation is a subset X of a group such that for any a and b in X, C(a,b) and C(a,b^-1) are also in X.

For abelian groups, any subset will work as every element is its own conjugacy class. So what about the subsets of S3? It’s tempting to think that it’d just be any set of conjugacy classes, however these aren’t the only ones. Any single-element set (plus, optionally, the identity) would also work, as for any a, C(a, a) = a. It can be proven that these are all of them in this case.

Another example of one of these is all the 72 degree rotations inside the icosahedral group inside the sphere group SO(3). If a set is closed under conjugation in SO(3), it can be used as the allowed turns in a non-jumbling twisty puzzle, in this case the Megaminx. (All that’s needed in that case is that for any two turns a and b that overlap, the conjugations C(a, b) and C(a, b^-1) are also valid turns that overlap with b.)

Consider any finite set closed under conjugation in SO(3). Any element of the subgroup it generates will permute the finite set in some way when conjugating. If any two elements have distinct rotation axes, then those two elements’ destinations after conjugation will determine the entire rotation. There are only finitely many permutations of a finite set, so it generates a finite subgroup. The other case is that all its elements have one common rotation axis, which makes it a subset of SO(2).

The above section proved that every finite set closed under conjugation in SO(3) is either a subset of an SO(2) subgroup or is a generating set of a finite subgroup of SO(3).

Because C(C(a, b), c) = C(a, bc), any of these sets is closed under conjugation by any element in the subgroup that they all generate, and therefore any of these sets is just a union of conjugacy classes of a subgroup.

A simple corollary that would seem nontrivial without having done all this: the subset generated by conjugation starting with a set of elements {a, b, …, n} is precisely the union of conjugacy classes of a, b, …, and n in the subgroup generated by a, b, …, and n.