Prove that, in the category of groups, $G \stackrel{\varphi}{\rightarrow} H$ is an epimorphism if and only if $\varphi$ is onto. (Hints: The proof that $\varphi$ onto implies $\varphi$ an epimorphism is the same for groups as for sets. For the converse, consider $G \xrightarrow{\varphi} H \xrightarrow[\alpha^{\prime}]{\alpha} X$, where $X$ is the permutation group on the set $H$. Let $\alpha$ be the natural homomorphism from $H$ to $\operatorname{Perm}(H)$. Let $\alpha^{\prime}$ be defined by $\alpha^{\prime}(h)= x \alpha(h) x^{-1}$, where $x$ is an element of $X$. One must now choose $x \neq e$ so that $\alpha \circ \varphi=\alpha^{\prime} \circ \varphi$. Denote by $Y$ the subgroup of $X$ consisting of elements of the form $\alpha \circ \varphi(g)$ for $g$ in $G$. Choose $x$ so that it "rearranges the cosets of $Y$ in $\left.X .{ }^{\prime \prime}\right)$