In $D_{33}$ how do I find out number of elements of each order? In $D_{33}$ i.e diehedral group of order 66.
How do I find out number of elements of each order?
The only idea I have is that possible order of any element can be 1,2,3,6,11,33,66. Now 1 is only for identity and since group is not cyclic 66 can not be order of any group. Thanks in advance to all.
 A: $D_{33}$ is the symmetry group of the regular polygon $A_1A_2\cdots A_{33}$ with 33 sides. If $\sigma$ is the generator of rotation cyclic group of the polygon and $\tau$ a reflection of the polygon on the altitude passing through $A_1$ (passing through $A_1$ and the mid point of the opposite side), then $\sigma^{33} = e$ and $\tau^2 =e$ where $e$ is the identity element of the group. Also $\tau \sigma = \sigma^{32}\tau$. From these relations, one can calculate the order of any element. Note that the elements of the group can be written as $\tau^i\sigma^j$, where $i = 0,1$ and $j = 0,1,\ldots, 32$.
A: There is an index $2$ subgroup $C$ which is cyclic of order $33$, generated by some element $\sigma$, so $C=\langle \sigma\rangle$. There is another element $\tau$ which has order $2$. Clearly $\tau\not\in C$, and also clearly the two cosets are $C$ and $\tau C$. Indeed, we can take $\tau$ to correspond to reflection and $\sigma$ to rotation, so we also have that $\tau\sigma\tau=\sigma^{-1}$.
Question 1. How many elements are there of orders respectively $1$, $3$, $11$ and $33$ in $C$?
Question 2. Suppose $x\in C$ (possibly trivial). What is the order of $\tau x$?
Question 3. Conclude. (That is, work out why answering Questions 1 and 2 answers your question.)
A: Every reflection is by definition self-inverse, order 2.
The smallest rotation $r_1$ (other than the identity) generates all other rotations in a cycle, including reaching the identity after $33$ steps. Some of the rotations will have shorter cycles, though, in the classic pattern of cyclic groups - specifically, for a rotation generated by $r_1^k$, it will be an $11$-cycle if $3 \mid k$, a $3$-cycle if $11\mid k$ and a $33$-cycle otherwise.
