# Notes - 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 $S$?

\[|S| = \sum_{g \text{ representative}\,} |\text{Orb}(g)|\]

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?

\[|G| = |Z(G)| + \sum_{g \text{ nontrivial repr.}\,} |G/C(g)|\]

where here $g$ is a “nontrivial representative”, i.e. a representative of some non-trivial conjugacy class.