normality of subgroups is not transitive
Let G be a group.
A subgroup K of a subgroup H of G is obviously a subgroup of G.
It seems plausible that a similar
situation would also hold for normal subgroups
, but in fact it does not:
even when K⊴H and H⊴G, it is possible that K⋬. Here are two examples:
-
1.
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.
-
2.
A related example uses finite subgroups. Let be the dihedral group
with eight elements (the group of automorphisms
of the graph of the square). Then
is generated by , rotation, and , flipping.
The subgroup
is isomorphic to the Klein 4-group – an identity
and 3 elements of order 2. since . Finally, take
We claim that . 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 |