Group theory/Isomorphism theorem for residue class groups/Fact/Proof

Proof

For the first statement, see exercise. Therefore, the residue class group $Q/\bar{H}$ is well-defined. We consider the composition

$p\circ q:G⟶Q⟶Q/\bar{H}.$

Because of

$\begin{array}{cc}\mathrm{kern}(p\circ q)& =\{x\in G\mid (p\circ q)(x)=e\}\\ & =\{x\in G\mid q(x)\in \mathrm{kern}p\}\\ & =\{x\in G\mid q(x)\in \bar{H}\}\\ & =H,\end{array}$

we have $\mathrm{kern}(p\circ q)=H$. Hence, according to fact, we get the canonical isomorphism

$G/H⟶Q/\bar{H}.$

To fact