Exercise 7, § 3

Let $G$ be a non-discrete locally compact group with a countable base of open sets, and let $V_0$ be a compact neighbourhood of the identity element $e$ of $G$ which contains no subgroup of $G$ other than $\{e\}$.

a) Let $(x_n)$ be a sequence of points of $V_0$ which converges to $e$. For each $n$, there exists an integer $p(n) > 0$ such that $x_n^k \in V_0$ whenever $k \leq p(n)$ and $x_n^{p(n)+1} \notin V_0$, and we have $\lim_{n \to \infty} p(n) = +\infty$. Show that there exists a subsequence $(x_{k_n})$ of $(x_n)$ such that, for every rational number $r$ with $|r| \leq 1$, the sequence $(x_{k_n}^{\lfloor rp(k_n)\rfloor})$ has a limit $f(r)$ in $V_0$ (use the diagonal technique); and that if $r, r'$ are rational numbers such that $|r| \leq 1, |r'| \leq 1$ and $|r + r'| \leq 1$ we have

b) Show that $f$ is continuous in a neighbourhood of $0$ in $\mathbf{Q}$. [Argue by contradiction: if there were a sequence $(r_n)$ of rational numbers tending to $0$ and such that $f(r_n)$ tended to an element $y \neq e$ in $V_0$, show that we should have $y^m \in V_0$ for all integers $m$.]

c) Deduce from b) that there exists a non-trivial continuous homomorphism of $\mathbf{R}$ into $G$ (use Proposition 6 of § 1).

d) Let $H$ be a closed normal subgroup of $G$ other than $G$ itself, and suppose that there is a neighbourhood of the identity element in $G/H$ which contains no non-trivial subgroup. Show that there exists a continuous homomorphism $f : \mathbf{R} \to G$ such that the composition

is non-trivial.