Exercise 5, § 5

Let G be a group. (a) If H is a subgroup of G of finite index n, show that the intersection N of the conjugates of H is of index a divisor of n! (note that $G/N$ is isomorphic to a subgroup of $\mathfrak{S}_n$). (b) G is called residually finite, if the intersection of its subgroups of finite index is $\{e\}$. Show that this is equivalent to saying that G is isomorphic to a subgroup of a product of finite groups.

(c) Suppose that G can be generated by a finite set. Show that, for every integer n, the set P_n of subgroups of G of index n is finite.

(d) Under the hypothesis of (c), consider an endomorphism f of G which is surjective. Show that, for every integer n, the mapping H $\mapsto f^{-1}(H)$ is a bijection of P_n onto itself (observe that it is an injection). Deduce that the kernel of f is contained in every subgroup of G of finite index. In particular, if G is residually finite, f is bijective.