Exercise 17, § 4

Suppose that G is nilpotent. Let P be a set of prime numbers and let $\mathfrak{V}_p(G)$ be the set of normal subgroups of G of finite index whose index is a P-integer (cf. Exercise 14). Let $\mathscr{T}_P(G)$ be the topology defined on G by the filter base $\mathfrak{V}_p(G)$ (General Topology, Chapter III, § 1, no. 2, Example). (a) Let N be a subgroup of G of finite index such that $(G:N)$ is a P-integer and let N' be the intersection of the conjugates of N. Show that $(G:N')$ is a P-integer (use the fact that the group G/N' is the product of its Sylow groups); deduce that N is open with respect to $\mathscr{T}_P(G)$. (b) Let H be a subgroup of G of finite index. Show that $\mathscr{T}_P(H)$ coincides with the topology induced on H by $\mathscr{T}_P(G)$. (Same method.) (c) Let H be a normal subgroup of G such that G/H is isomorphic to $\mathbf{Z}$ and let x be a representative in G of a generator of G/H. Let $N \in \mathfrak{V}_P(H)$ and let $N' = \bigcap_{n \in \mathbf{Z}} x^n N x^{-1}$. Show that $N' \in \mathfrak{V}_P(H)$ and that $x N' x^{-1} = N'$. If $N''$ is the subgroup generated by N' and x, show that $(G:N'') = (H:N')$. Deduce that $\mathscr{T}_P(H)$ is induced by $\mathscr{T}_P(G)$. (d) Suppose that G is finite. Show that, if H is a subgroup of G, there exists a sequence of subgroups

such that $H_i$ is normal in $H_{i+1}$ for $0 \leq i < k$ and that $H_{i+1}/H_i$ is finite or isomorphic to $\mathbf{Z}$. Deduce, using (b) and (c), that $\mathscr{T}_P(H)$ is induced by $\mathscr{T}_P(G)$.

(e) With the hypotheses of (d), let $P'$ denote the complement of $P$ in the set of prime numbers. Show that the closure of $H$ under $\mathscr{T}_P(G)$ coincides with the $P'$-saturation of $H$ in $G$ (Exercise 14); in particular, $G$ is Hausdorff if and only if it is $P'$-torsion-free.