semi-direct factor and quotient group

Theorem.

If the group $G$ is a semi-direct product of its subgroups $H$ and $Q$,then the semi-direct $Q$is isomorphic to the quotient group $G/H$.

Proof. Every element $g$ of $G$ has the unique representation $g=hq$with $h\in H$ and $q\in Q$.We therefore can define the mapping

$$g\mapsto q$$

from $G$ to $Q$.The mapping is surjective since any element $y$ of $Q$ is the image of $ey$.The mapping is also a homomorphism since if $g_1=h_1{q}_1$ and $g_2=h_2{q}_2$, then we obtain

$$f(g_1{g}_2)=f(h_1{q}_1{h}_2{q}_2)=f(h_1{h}_2{q}_1{q}_2)=q_1{q}_2=f(g_1)f(g_2).$$

Then we see that $\ker f=H$ because all elements $h=he$ of $H$are mapped to the identity element $e$ of $Q$.Consequently we get, according to the first isomorphism theorem, the result

$$G/H\cong Q.$$

Example.The multiplicative group ${ℝ}^{\times }$ of realsis the semi-direct product of the subgroups$\{1,-1\}=\{\pm 1\}$ and ${ℝ}_{+}$.The quotient group ${ℝ}^{\times }/\{\pm 1\}$ consists of all cosets

$$x\{\pm 1\}=\{x,-x\}$$

where $x\ne 0$, and is obviously isomorphic with${ℝ}_{+}=\{x\mid x>0\}$.