Exercise 29, § 6

Let G be a finite group of even order and let H be a Sylow 2-subgroup of G. For all $s \in G$, let $\varepsilon(s)$ be the signature of the permutation $x \mapsto sx$ of G.

(a) Let $s \in H$. Show that $\varepsilon(s) = -1$ if and only if s generates H.

(b) Show that $\varepsilon$ is surjective if and only if H is cyclic.

(c) Suppose that H is cyclic. Show that there exists one and only one normal subgroup D of G such that G is the semi-direct product of H and D. (Argue by induction on the order of H.) Show that the normalizer N of H in G is the direct product of H and $N \cap D$.

(d) Show that the order of a non-commutative simple group is either odd† or divisible by 4.