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 $2n$, denoted $D_{2n}$, can be considered as the symmetries of a regular $n$-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 permutation on 6 points. So a rotation by $\pi /3$ can be encoded as the permutation $\rho =(123456)$ and the reflection fixing the axis through the vertices 1 and 4 can be encoded as $\varphi =(26)(35)$. Indeed these two permutations generate a permutation group isomorphic to $D_{12}$.

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 $D_4\cong {\mathbb{Z}}_2\oplus {\mathbb{Z}}_2$, see the remark below). Specifically, if $\rho$ is a rotation of order $n$, and $n=2m$, then $⟨{\rho }^m⟩$ is the center of $D_{2n}$. (Note this is the only rotation or order 2, and in particular it is always a rotation by $\pi$.)So when $n=6$, the center is $⟨(14)(25)(36)⟩$.

Now fix $n=6$ and note the following assignment of generators determines an endomorphism $f:D_{12}\to D_{12}$:

$$(123456)\mapsto (26)(35),(26)(35)\mapsto (14)(25)(36).$$

Note that image $K:=⟨(26)(35),(14)(25)(36)⟩\cong {\mathbb{Z}}_2\oplus {\mathbb{Z}}_2$, as $(14)(25)(36)$ is central in $D_{12}$ and the generators of$K$ are distinct elements of order $2$. [This can be proved with the relations of the dihedral group.]

Remark 1.

Geometrically we note that the kernel of the homomorphism is $\mathrm{⟨}{\rho }^{\mathrm{2}}\mathrm{⟩}$ – 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 ${\mathrm{Z}}_{\mathrm{2}}\mathrm{\oplus }{\mathrm{Z}}_{\mathrm{2}}$, sometimes called $D_{\mathrm{4}}$.

Now the center is mapped via $f$ to the subgroup $⟨(26)(35)⟩$ so$f(Z(D_{12}))$ is not contained in $Z(D_{12})$ proving $Z(D_{12})$ 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 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):

$$\rho \mapsto {\rho }^m\mapsto \varphi ,\varphi \mapsto \varphi \mapsto {\rho }^m.$$

As $m$ is odd and the center, $⟨{\rho }^m⟩$, has order 2, it follows $⟨{\rho }^m⟩$ maps to $⟨{\rho }^m⟩$ 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.