The Number of Conjugacy Classes of a Finite Group of Odd Order is Equivalent to The Order of The Group Modulo Sixten
Point of Post: In this post we prove a remarkable fact about finite groups of odd order which isn’t easily proven without using representation theory. Namely, we prove that if is a finite group of odd order and is the number of conjugacy classes of then .
So far we’ve seen a few theorems in pure finite group theory which are very difficult (if not impossible) to prove without using representation theory– of course Burnside’s Theorem is the quintessential example. In this post we prove another such theorem, namely that for a finite group of odd order that the number of conjugacy classes is equivalent to the order of mod sixteen. This is really a remarkable and unexpected theorem. And at first it may seem quite useless, but think about its implications for finite group theory. Namely, suppose you had a group of odd order which suspected is abelian. Of course all you need to do is show that the number of conjugacy classes is . Thus, one is reduced to disproving the other possible number of conjugacy classes, but our little theorem will erase (in the case of most finite groups of small order) all but one or two possible cases. As a less impressive application it clearly implies that all groups of odd order less than or equal to sixteen are abelian (i.e. the only remotely non-trivial one of these is and since this has been previously established). Probably the most fantastic part of this theorem is how incredibly easy it is to prove using representation theory. Namely, it almost falls out from our previous theorem that every non-trivial irrep of a group of finite order is complex.
Without further ado we prove our very interesting theorem:
Theorem: Let be a finite group of odd order and the number of conjugacy classes of . Then, .
Proof: Since is odd we know that every non-trivial irrep of is complex. Thus, for every in one has that and . But, since is odd and we know that we may conclude that for some . Thus, since we know that the cardinality of is (where ) we enumerate the elements of as . Thus, we have by prior theorem that
from where the conclusion follows.
1. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Math. Soc., 1996. Print.