Exercise 14, § 4

Let $G$ be a group and $H$ a subgroup of $G$. Any subset $T$ of $G$ which meets every left coset mod. $H$ in one and only one point is called a representative system of the left cosets mod. $H$ (cf. Set Theory, II, § 6, no. 2); this condition is equivalent to the following: The mapping $(x, y) \mapsto xy$ is a bijection of $T \times H$ onto $G$. (a) Let $T$ be such a representative system; for all $g \in G$ and all $t \in T$, let $x(g, t) \in T$ and $y(g, t) \in H$ be such that $gt = x(g, t)y(g, t)$. Let $S$ be a subset of $G$ generating $G$. Show that the elements $y(g, t), g \in S, t \in T$ generate $H$. (If $H'$ denotes the subgroup of $H$ generated by these elements, show that $T.H'$ is stable under the $\gamma_g, g \in S$, hence also under the $\gamma_g, g \in G$, and deduce that $T.H' = G$, whence $H' = H$.) (b) Suppose that $(G:H)$ is finite. Show that $G$ can be generated by a finite subset if and only if $H$ can.