Groups TT23, Class equation
Flashcards
Given that the orbits of $G$ acting on $S$ partition the group, what expression can you give for the size of $S$?
\[\vert S \vert = \sum _ {g \text{ representative}\,} \vert \text{Orb}(g) \vert\]
Given that the orbits of $G$ acting on $S$ partition the group, and that conjugacy classes are orbits under conjugation, can you state the class equation?
\[\vert G \vert = \vert Z(G) \vert + \sum _ {g \text{ nontrivial repr.}\,} \vert G/C(g) \vert\]
where here $g$ is a “nontrivial representative”, i.e. a representative of some non-trivial conjugacy class.