example of a non-fully invariant subgroup
Every fully invariant subgroup is characteristic, but some characteristic subgroups need not be fully invariant. For example, the center of a group is characteristic but not always fully invariant. We pursue a single example.
Recall the dihedral group of order , denoted , can be considered as the symmetries of a regular -gon. If we consider a regular hexagon, so , and label the vertices counterclockwise from 1 to 6 we can then encode each symmetry as a permutation on 6 points. So a rotation by can be encoded as the permutation and the reflection fixing the axis through the vertices 1 and 4 can be encoded as . Indeed these two permutations generate a permutation group isomorphic to .
The center of a dihedral group of order is trivial if is odd, and of order 2 if is even (if it is the entire group , see the remark below). Specifically, if is a rotation of order , and , then is the center of . (Note this is the only rotation or order 2, and in particular it is always a rotation by .) So when , the center is .
Geometrically we note that the kernel of the homomorphism is – the group of rotations of order 3. So if we quotient by the kernel we are identifying the three inscribed (non-square) rectangles of the hexagon (1245, 2356 and 3461). The symmetry group of a non-square rectangle is none other than , sometimes called .
Of course the example applies without serious modification to the dihedral groups on -gons, where is odd. Here a generally offending endomorphism may be described with a composition of maps (the first leaves the center invariant, the second swaps the basis of the image of the first thus moving the image of the center):
As is odd and the center, , has order 2, it follows maps to under the first map, and then can be interchanged with a reflection to violate the condition of full invariance. If is even then the center lies in the kernel of the first map so no such trick can be played.
|Title||example of a non-fully invariant subgroup|
|Date of creation||2013-03-22 16:06:26|
|Last modified on||2013-03-22 16:06:26|
|Last modified by||Algeboy (12884)|