normality of subgroups is not transitive
Let be a group. A subgroup of a subgroup of is obviously a subgroup of . It seems plausible that a similar situation would also hold for normal subgroups, but in fact it does not: even when and , it is possible that . Here are two examples:
Let be the subgroup of orientation-preserving isometries (http://planetmath.org/Isometry) of the plane ( is just all rotations and translations), let be the subgroup of of translations, and let be the subgroup of of integer translations , where .
Any element may be represented as , where are rotations and are translations. So for any translation we may write
where is some other translation and 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 , and .
is an abelian group, so all its subgroups, included, are normal.
We claim that . Indeed, if is rotation by about the origin, then is not an integer translation.
is generated by , rotation, and , flipping.
We claim that . And indeed,
|Title||normality of subgroups is not transitive|
|Date of creation||2013-03-22 12:49:27|
|Last modified on||2013-03-22 12:49:27|
|Last modified by||yark (2760)|