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:
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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.
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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 |
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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 |