semi-direct factor and quotient group


If the group G is a semi-direct product of its subgroupsMathworldPlanetmathPlanetmath H and Q, then the semi-direct Q is isomorphicPlanetmathPlanetmathPlanetmath to the quotient groupMathworldPlanetmath G/H.

Proof. Every element g of G has the unique representation g=hq with hH and qQ. We therefore can define the mapping


from G to Q. The mapping is surjectivePlanetmathPlanetmath since any element y of Q is the image of ey. The mapping is also a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath since if g1=h1q1 and g2=h2q2, then we obtain


Then we see that kerf=H because all elements h=he of H are mapped to the identity elementMathworldPlanetmath e of Q. Consequently we get, according to the first isomorphism theoremPlanetmathPlanetmath, the result


Example. The multiplicative groupMathworldPlanetmath × of reals is the semi-direct product of the subgroups {1,-1}={±1} and +. The quotient group ×/{±1} consists of all cosets


where x0, and is obviously isomorphic with +={xx>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