# 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 $\u27e8{\rho}^{m}\u27e9$ 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 $\u27e8(14)(25)(36)\u27e9$.

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 $K:=\u27e8(26)(35),(14)(25)(36)\u27e9\cong {\mathbb{Z}}_{2}\oplus {\mathbb{Z}}_{2}$, 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 $\mathrm{\u27e8}{\rho}^{\mathrm{2}}\mathrm{\u27e9}$ – 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^{} $\u27e8(26)(35)\u27e9$ 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, $\u27e8{\rho}^{m}\u27e9$, has order 2, it follows $\u27e8{\rho}^{m}\u27e9$ maps to $\u27e8{\rho}^{m}\u27e9$ 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 |