# 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 $\phi=(26)(35)$. 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 $\langle\rho^{m}\rangle$ 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 $\langle(14)(25)(36)\rangle$.

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

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

Note that image $K:=\langle(26)(35),(14)(25)(36)\rangle\cong\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$, 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 $\langle\rho^{2}\rangle$ – 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 $\mathbb{Z}_{2}\oplus\mathbb{Z}_{2}$, sometimes called $D_{4}$.

Now the center is mapped via $f$ to the subgroup $\langle(26)(35)\rangle$ 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):

 $\rho\mapsto\rho^{m}\mapsto\phi,\qquad\phi\mapsto\phi\mapsto\rho^{m}.$

As $m$ is odd and the center, $\langle\rho^{m}\rangle$, has order 2, it follows $\langle\rho^{m}\rangle$ maps to $\langle\rho^{m}\rangle$ 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 ExampleOfANonfullyInvariantSubgroup 2013-03-22 16:06:26 2013-03-22 16:06:26 Algeboy (12884) Algeboy (12884) 7 Algeboy (12884) Example msc 20D99