Exercise 29, § 6

Let G be a finite group of even order and let H be a 2-Sylow subgroup of G. For every $s \in G$, let $\varepsilon(s)$ be the signature of the permutation $x \mapsto sx$ of G. a) Let $s \in H$. Now prove that $\varepsilon(s) = -1$ if and only if s generates H. b) Now prove that $\varepsilon$ is surjective if and only if H is cyclic. c) Suppose H is cyclic. Now prove that there exists one and only one normal subgroup D of G such that G is semidirect product of H and D (reason by induction on the order of H). Now prove that the normalizer N of H in G is direct product of H and $N \cap D$. d) Now prove that the order of a noncommutative simple group is either odd¹ or divisible by 4.

¹ In fact, this case is impossible, as was shown by Feit and Thompson, Pac. J. of Maths., t. XIII (1963), p. 775–1029.