We prove the `Frobenius divisibility theorem' first. The main idea is to show that the quotient |G|/ dim V, where G is the given finite group and (V, rho) is a finite dimensional irreducible complex representation of G, is an algebraic integer and then use the fact that Z is integrally closed.
After that, we talk about Burnside's theorem, which says that, if p and q are primes and a and b are nonnegative integers with a+b is at least 2, then G cannot be simple. This theorem implies that every group of order p^aq^b, where p and q are primes and a and b are nonnegative integers is solvable. Thus we are provided with a large class of finite solvable groups. The Burnside's theorem stated above actually follows from another theorem of Burnside that says if a finite group has a nontrivial conjugacy class of prime power order then it cannot be simple. We show how the former follows from the latter. Then we prove a lemma which will be needed to prove the second theorem of Burnside, and with this we finish this lecture.
The proof of the second theorem of Burnside stated above will be provided in the next lecture.
