---- > [!theorem] Theorem. ([[class equation]]) > Since the [[conjugate|conjugacy classes]] [[partition]] a (finite) [[group]] $G$, we have that $\begin{align} |G|= & (\text{sum of sizes of conjugacy classes}) \\ = & |Z(G)| + \sum_{[x], x \notin Z(G)}^{} |C_{x}| \\ = & |Z(G)| + \sum_{[x], x \notin Z(G)}^{} \frac{|G|}{|Z_{G}(x)|}, \end{align}$ \ Where we use [[conjugate#^a1150e|these properties]] along with facts such as [[order of quotient group is quotient of orders]]. The final equality follows from the content of the *proof* that [[size of conjugacy class divides order of finite group]]. ---- #### ----- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```