Groups TT23, Class equation
Flashcards
Given that the orbits of $G$ acting on $S$ partition the set, what expression can you give for the size of $S$?
\[ \vert S \vert = \sum _ {g \text{ representative}\,} \vert \mathrm{Orb}(g) \vert \]
Given that the orbits of $G$ acting on $S$ partition the set, and that conjugacy classes are orbits under conjugation, @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.