Does the dihedral group $D_{12}$ have a subgroup of order $4$?

Question

Let the dihedral group $D_{12}=\{xy\mid x^6=y^2=1,xy=yx^{-1}=yx^5\}$

Order 2 subgroups are: $\{1, x^3\}$, $\{1, y\}$, $\{1, xy\}$, $\{1, x^2 y\}$,$\{1, x^3 y\}$, $\{1, x^4 y\}$,$\{1, x^5y\}$.

Order 3 subgroups are: $\{1, x, x^5\}$, $\{1, x^2, x^4\}$.

Order 6 subgroups are: $\{1, x, x^2, x^3, x^4, x^5\}$.

Does $D_{12}$ have a subgroup of order 4?

@DietrichBurde That question is about the subgroups of $D_4$. — , Aug 05 '15 at 22:00
@NormalHuman The question has beed answered several times at MSE. So it is definitely a duplicate. Perhaps a better link is http://math.stackexchange.com/questions/1327191/is-there-a-general-formula-for-finding-all-subgroups-of-dihedral-groups. — Dietrich Burde, Aug 06 '15 at 08:16

score 1 · Answer 1 · answered Aug 05 '15 at 16:09

1

Yes. $\{1,x^3,y,x^3y\}$, i.e. the group generated by a reflection and a rotation by 180 degrees.

answered Aug 05 '15 at 16:09

Hagen von Eitzen

2

(Which is a $2$-Sylow so it must exist, and all subgroups of order four have this form) – Pedro Aug 05 '15 at 16:10
@Hagen von Eitzen ${1,x^3, y, x^3y}$ is the unique subgroup of orden 4? – Roiner Segura Cubero Aug 05 '15 at 16:12
@RoinerSeguraCubero That is exactly what Pedro Tamaroff said. – wythagoras Aug 05 '15 at 17:17

score 0 · Answer 2 · answered Aug 05 '15 at 17:25

0

$D_{12} \cong S_3 \times C_2$. Hence $C_2 \times C_2$, the Klein $4$-group, is a subgroup of order $4$.

answered Aug 05 '15 at 17:25

Nicky Hekster

2 Answers2