Difference between revisions of "The Class Equation"
(Created page with "The '''Class Equation''' is a result in group theory. It states that <center><math>|G|=|Z(G)|+\sum_{i=1}^r|G : C_{G}(g_i)|</math></center> where <math>g_i</math> are rep...") |
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Latest revision as of 22:26, 12 May 2022
The Class Equation is a result in group theory. It states that
where are representatives of the conjugacy classes not in the center of .
Proof: Let . We define for all . The number of conjugates of an element is the index of the centralizer of , denoted as . The element is a conjugacy class of order as long as , since then for all . Let and denote the conjugacy classes not in the center. Let be a representative of each conjugacy class (which is just an element in the conjugacy class). Since these all partition , we have which is exactly what we wanted to show