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 groupMathworldPlanetmath of order 2n, denoted D2n, can be considered as the symmetriesPlanetmathPlanetmath of a regularPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath n-gon. If we consider a regular hexagon, so n=6, and label the vertices counterclockwise from 1 to 6 we can then encode each symmetry as a permutationMathworldPlanetmath on 6 points. So a rotation by π/3 can be encoded as the permutation ρ=(123456) and the reflection fixing the axis through the vertices 1 and 4 can be encoded as ϕ=(26)(35). Indeed these two permutations generate a permutation groupMathworldPlanetmath isomorphicPlanetmathPlanetmathPlanetmathPlanetmath to D12.

The center of a dihedral group of order 2n is trivial if n is odd, and of order 2 if n>2 is even (if n=2 it is the entire group D422, see the remark below). Specifically, if ρ is a rotation of order n, and n=2m, then ρm is the center of D2n. (Note this is the only rotation or order 2, and in particular it is always a rotation by π.) So when n=6, the center is (14)(25)(36).

Now fix n=6 and note the following assignment of generatorsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath determines an endomorphismPlanetmathPlanetmath f:D12D12:


Note that image K:=(26)(35),(14)(25)(36)22, as (14)(25)(36) is central in D12 and the generators of K are distinct elements of order 2. [This can be proved with the relationsMathworldPlanetmathPlanetmath of the dihedral group.]

Remark 1.

Geometrically we note that the kernel of the homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath is ρ2 – 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 Z2Z2, sometimes called D4.

Now the center is mapped via f to the subgroupMathworldPlanetmathPlanetmath (26)(35) so f(Z(D12)) is not contained in Z(D12) proving Z(D12) is not fully-invariant.

Of course the example applies without serious modification to the dihedral groups on 2m-gons, where m>1 is odd. Here a generally offending endomorphism may be described with a compositionMathworldPlanetmathPlanetmath 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 m is odd and the center, ρm, has order 2, it follows ρm maps to ρm under the first map, and then can be interchanged with a reflection to violate the condition of full invariance. If m 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
Canonical name ExampleOfANonfullyInvariantSubgroup
Date of creation 2013-03-22 16:06:26
Last modified on 2013-03-22 16:06:26
Owner Algeboy (12884)
Last modified by Algeboy (12884)
Numerical id 7
Author Algeboy (12884)
Entry type Example
Classification msc 20D99