examples of semidirect products of groups

Suppose H=/n and let r be a generatorPlanetmathPlanetmathPlanetmath for H. Let Q=/2=<s>. Define θ:QAut(H) by θ(s)(r)=r-1. Let G=HθQ. Then in G,


by the canonical equivalence of inner and outer semidirect productsMathworldPlanetmath. So G has 2n elements, two generators r,s satisfying


and thus G=𝒟2n, the nth dihedral groupMathworldPlanetmath.

If instead H=, the result is the infinite dihedral group.

As another example, if G is a group, then the holomorph of G is GAut(G) under the identity map from Aut(G) to itself.

Title examples of semidirect products of groups
Canonical name ExamplesOfSemidirectProductsOfGroups
Date of creation 2013-03-22 17:22:52
Last modified on 2013-03-22 17:22:52
Owner rm50 (10146)
Last modified by rm50 (10146)
Numerical id 4
Author rm50 (10146)
Entry type Example
Classification msc 20E22