(a) Let $t$ be the maximal power of the prime $p$ dividing the order of the finite group $G$, and let $N\left(p^{t}\right)$ denote the number of subgroups of $G$ of order $p^{t}$. State clearly the numerical restrictions on $N\left(p^{t}\right)$ given by the Sylow theorems.

If $H$ and $K$ are subgroups of $G$ of orders $r$ and $s$ respectively, and their intersection $H \cap K$ has order $t$, show the set $H K=\{h k: h \in H, k \in K\}$ contains $r s / t$ elements.

(b) The finite group $G$ has 48 elements. By computing the possible values of $N(16)$, show that $G$ cannot be simple.