Exercise 18, § 21

Suppose that the group G is simple but not cyclic. a) Prove that there is no conjugacy class different from $\{1\}$ whose cardinal is a power of a prime number. (Let C be a class of cardinal $p^r$ with $r\geqslant 1$. Deduce from Exercise 17, b) that $p^{-1}\chi_{\lambda}(1)\chi_{\lambda}(C)$ is integral over $\mathbf{Z}$ when $\lambda$ is different from the unit representation, and deduce a contradiction using the second orthogonality relation for characters).

b) Prove that the order of G is divisible by at least three distinct prime numbers (apply a) to the conjugacy class of a central element $x\not= 1$ in a Sylow subgroup of G).

c) Prove that a finite group whose order is divisible by at most two distinct prime numbers is solvable (“Burnside’s theorem”: reason by induction on the cardinal of the group, using b)).