Exercise 19, § 5

Let G be a group, H a subgroup of G and G/H the set of left cosets (mod. H) in G. Let $r : G/H \to G$ be a section associated with the canonical projection $G \to G/H$. Then for all $X \in G/H$, $X = r(X).H$. An internal law of composition is defined on G/H by setting $X \circ Y = r(X)r(Y).H$.

(a) Show that $X \uparrow H = X$ for all $X$ and that under the law $\uparrow$ every left translation is bijective. If $G'$ is the subgroup of $G$ generated by the set of elements $r(X)$ and $H' = H \cap G'$, the internal law defined analogously on $G'/H'$ by the mapping $r$ determines on this set a structure isomorphic to that determined on $G/H$ by the law $\uparrow$.

(b) For the law $\uparrow$ to be associative, it is necessary and sufficient that $H'$ be a normal subgroup of $G'$ in which case the structure determined by $\uparrow$ is isomorphic to the structure of the quotient group $G'/H'$ (to see that the condition is sufficient, show first, with the aid of § 4, Exercise 2(a), that if $\uparrow$ is associative it determines on $G/H$ a group structure; denoting by $K$ the largest normal subgroup of $G'$ contained in $H'$, show then, using the associativity condition on $\uparrow$, that $(r(X \uparrow Y))^{-1} r(X) r(Y) \in K$ for all $X, Y$; deduce that the mapping $X \mapsto r(X).K$ is an isomorphism of the group $G/H$ (with the law $\uparrow$) onto the quotient group $G'/K$; conclude that $H' = K$, noting that $H'$ is a union of cosets mod. $K$.

(c) Conversely, suppose given on a set $E$ an internal law of composition $\uparrow$ such that every left translation is bijective and there exists $e \in E$ such that $x \uparrow e = x$ for all $x \in E$. Let $\Gamma$ be the group of permutations of $E$ generated by the left translations $\gamma_x$ and $\Delta$ the subgroup of permutations in $\Gamma$ leaving $e$ invariant; show that to every left coset $X$ modulo $\Delta$ there corresponds one and only one element $x \in E$ such that $\gamma_x \in X$; if we write $r(X) = \gamma_x$, the mapping $x \mapsto \gamma_x$ is an isomorphism of the set $E$ with the law $\uparrow$ onto the set $\Gamma/\Delta$ with the law $(X, Y) \mapsto r(X)r(Y).\Delta$.