Counting element

jackie · #1 $Old$ November 3rd, 2009, 10:19 PM

Can someone help me with counting the number of elements of order 2 in the dihedral group $D_{10}$ , $ord(D_{10})=20$ . I have no idea how to do this. How can I count the number of elements of order 2 of finite dihedral group in general? Really appreciate any help
I only know $D_{10}=\{x,y:x^2=1, y^{10}=1, (xy)^2=1\}$
From this, I think I say that $x$ and $xy$ have order 2. But how can I find others?

NonCommAlg · #2 $Old$ November 3rd, 2009, 10:43 PM

Quote:

Originally Posted by jackie $View Post$

Can someone help me with counting the number of elements of order 2 in the dihedral group $D_{10}$ , $ord(D_{10})=20$ . I have no idea how to do this. How can I count the number of elements of order 2 of finite dihedral group in general? Really appreciate any help
I only know $D_{10}=\{x,y:x^2=1, y^{10}=1, (xy)^2=1\}$
From this, I think I say that $x$ and $xy$ have order 2. But how can I find others?

if $x,y$ are the generators of the dihedral group of order 2n with the relations $x^2=y^n=(xy)^2=1,$ then the elements of order 2 are $xy^j, \ 0 \leq j \leq n-1,$ and if $2 \mid n,$ there will be one more

element: $y^{\frac{n}{2}}.$ so the number of elements of order 2 in the dihedral group of order 2n is: $n + \frac{1 + (-1)^n}{2}.$

jackie · #3 $Old$ November 3rd, 2009, 11:02 PM

Thanks a lot for your help, NonComm. This is a very useful formula, which I have not seen before.