normality of subgroups is not transitive

Let G be a group. A subgroupMathworldPlanetmathPlanetmath K of a subgroup H of G is obviously a subgroup of G. It seems plausible that a similarMathworldPlanetmath situation would also hold for normal subgroupsMathworldPlanetmath, but in fact it does not: even when KH and HG, it is possible that KG. Here are two examples:

  1. 1.

    Let G be the subgroup of orientation-preserving isometries ( of the plane 2 (G is just all rotations and translationsPlanetmathPlanetmath), let H be the subgroup of G of translations, and let K be the subgroup of H of integer translations τi,j(x,y)=(x+i,y+j), where i,j.

    Any element gG may be represented as g=r1t1=t2r2, where r1,2 are rotations and t1,2 are translations. So for any translation tH we may write


    where tH is some other translation and r is some rotation. But this is an orientation-preserving isometry of the plane that does not rotate, so it too must be a translation. Thus G-1HG=H, and HG.

    H is an abelian groupMathworldPlanetmath, so all its subgroups, K included, are normal.

    We claim that KG. Indeed, if ρG is rotation by 45 about the origin, then ρ-1τ1,0ρ is not an integer translation.

  2. 2.

    A related example uses finite subgroups. Let G=D4 be the dihedral groupMathworldPlanetmath with eight elements (the group of automorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmath of the graph of the square). Then


    is generated by r, rotation, and f, flipping.

    The subgroup


    is isomorphic to the Klein 4-group – an identityPlanetmathPlanetmathPlanetmath and 3 elements of order 2. HG since [G:H]=2. Finally, take


    We claim that KG. And indeed,

Title normality of subgroups is not transitive
Canonical name NormalityOfSubgroupsIsNotTransitive
Date of creation 2013-03-22 12:49:27
Last modified on 2013-03-22 12:49:27
Owner yark (2760)
Last modified by yark (2760)
Numerical id 13
Author yark (2760)
Entry type Example
Classification msc 20A05
Related topic NormalIsNotTransitive