D6

Aug 2009
639
2
find all the subgroups of D6 that have order 6.

my book says that there should be 6 but i can only think of 1, which is the one consisting of all the rotations.
 
Oct 2009
4,261
1,836
find all the subgroups of D6 that have order 6.

my book says that there should be 6 but i can only think of 1, which is the one consisting of all the rotations.

We have that \(\displaystyle D_6\cong C_2\times S_3\) , \(\displaystyle C_2:=\{c\;;\;c^2=1\}\) , so two of the subgroups of order 6 are \(\displaystyle 1\times S_3\cong S_3\,,\,\,<c>\times <(123)>\cong C_6 \) and, as far as I can tell, there's

only one subgroup more of order 6 which I leave to you to find.

Anyway, I can't see how that book says there are 6 sbgps. of order 6...what book is that?

Tonio