Conjugacy Class - Conjugacy of Subgroups and General Subsets

Conjugacy of Subgroups and General Subsets

More generally, given any subset S of G (S not necessarily a subgroup), we define a subset T of G to be conjugate to S if and only if there exists some g in G such that T = gSg−1. We can define Cl(S) as the set of all subsets T of G such that T is conjugate to S.

A frequently used theorem is that, given any subset S of G, the index of N(S) (the normalizer of S) in G equals the order of Cl(S):

|Cl(S)| =

This follows since, if g and h are in G, then gSg−1 = hSh−1 if and only if g−1h is in N(S), in other words, if and only if g and h are in the same coset of N(S).

Note that this formula generalizes the one given earlier for the number of elements in a conjugacy class (let S = {a}).

The above is particularly useful when talking about subgroups of G. The subgroups can thus be divided into conjugacy classes, with two subgroups belonging to the same class if and only if they are conjugate. Conjugate subgroups are isomorphic, but isomorphic subgroups need not be conjugate (for example, an abelian group may have two different subgroups which are isomorphic, but they are never conjugate).