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 $e y$ . 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 $\mathbb{R}^{\times}$ of reals is the semi-direct product of the subgroups $\{1,\,-1\}=\{\pm 1\}$ and $\mathbb{R}_{+}$ . The quotient group $\mathbb{R}^{\times}/\{\pm 1\}$ consists of all cosets

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

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

Title	semi-direct factor and quotient group
Canonical name	SemidirectFactorAndQuotientGroup
Date of creation	2013-03-22 15:10:22
Last modified on	2013-03-22 15:10:22
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	8
Author	yark (2760)
Entry type	Theorem
Classification	msc 20E22