(a) State Sylow's theorems and give the proof of the second theorem which concerns conjugate subgroups.

(b) Show that there is no simple group of order 351 .

(c) Let $k$ be the finite field $\mathbb{Z} /(31)$ and let $G L_{2}(k)$ be the multiplicative group of invertible $2 \times 2$ matrices over $k$. Show that every Sylow 3-subgroup of $G L_{2}(k)$ is abelian.