Editing Conjugacy class (section)

==Conjugacy of subgroups and general subsets==<!-- This section is linked from [[Symmetry group]] -->

More generally, given any [[subset]] <math>S \subseteq G</math> (<math>S</math> not necessarily a subgroup), define a subset <math>T \subseteq G</math> to be conjugate to <math>S</math> if there exists some <math>g \in G</math> such that <math>T = gSg^{-1}.</math> Let <math>\operatorname{Cl}(S)</math> be the set of all subsets <math>T \subseteq G</math> such that <math>T</math> is conjugate to <math>S.</math>

A frequently used theorem is that, given any subset <math>S \subseteq G,</math> the [[index of a subgroup|index]] of <math>\operatorname{N}(S)</math> (the [[normalizer]] of <math>S</math>) in <math>G</math> equals the cardinality of <math>\operatorname{Cl}(S)</math>:
<math display=block>|{\operatorname{Cl}(S)}| = [G : N(S)].</math>

This follows since, if <math>g, h \in G,</math> then <math>gSg^{-1} = hSh^{-1}</math> if and only if <math>g^{-1}h \in \operatorname{N}(S),</math> in other words, if and only if <math>g \text{ and } h</math> are in the same [[coset]] of <math>\operatorname{N}(S).</math>

By using <math>S = \{ a \},</math> this formula generalizes the one given earlier for the number of elements in a conjugacy class.

The above is particularly useful when talking about subgroups of <math>G.</math> 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 [[Group isomorphism|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.