习题 19, § 5

令 G 为 G 的群、H 子群以及 G 中左陪集 (mod. H) 的 G/H 集合。令 $r : G/H \to G$ 为与规范投影 $G \to G/H$ 相关的截面。那么对于所有$X \in G/H$，$X = r(X).H$。 合成的内部律在 G/H 上由设定 $X \circ Y = r(X)r(Y).H$ 定义。

(a) 证明 $X \uparrow H = X$ 对于所有 $X$ 并且在律 $\uparrow$ 下每个左平移都是双射的。如果$G'$是由元素$r(X)$和$H' = H \cap G'$的集合生成的$G$的子群，则内部律类似地由映射$r$确定定义在$G'/H'$上在此集合结构同构于 $G/H$ 由律 $\uparrow$ 确定的结果。

(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 集合 $E$ an internal 律 of 合成 $\uparrow$ 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 群 of 置换 of $E$ generated by the left translations $\gamma_x$ and $\Delta$ 子群 of 置换 in $\Gamma$ leaving $e$ invariant;证明对于每个左陪集 $X$ 模 $\Delta$ 都有对应且只有一个元素 $x \in E$ 使得 $\gamma_x \in X$；如果我们将$r(X) = \gamma_x$、映射$x \mapsto \gamma_x$写成集合$E$和律$\uparrow$的同构和集合$\Gamma/\Delta$和律 $(X, Y) \mapsto r(X)r(Y).\Delta$。