# Thread: center of the dihedral group

1. ## center of the dihedral group

Hi,
my question is, "for n = 3 or greater, show $\displaystyle Z(D_n)$is trivial if n is odd and is {1, $\displaystyle a^{n/2}$} if n is even."

Any help would be appreciated; we haven't learned anything advanced in class yet, just about cosets and lagrange's thm.

My initial attempt was to consider that if $\displaystyle b^ja^i$ is in $\displaystyle Z(D_n)$, then $\displaystyle g(b^ja^i)g^{-1} = b^ja^i$ for all g.
if g=a then $\displaystyle a(b^ja^i)a^{-1} = b^ja^i$and if g=b $\displaystyle b(b^ja^i)b^{-1} = b^ja^i$
but from here I'm quite stuck.

Thanks!!

2. I assume $\displaystyle a$ denotes a rotation and $\displaystyle b$ denotes a flip

note that $\displaystyle (a^ib^j)^{-1} = b^{-j}a^{-i} = b^{j}a^{-i}$
$\displaystyle \Rightarrow a^ib^j = b^ja^{-i}$

So, we want to find all values of i and j such that: $\displaystyle (a^ib^j)g = g(a^ib^j)$

if $\displaystyle g= a^s$
then $\displaystyle (a^ib^j)a^s = a^{i-s}b^j = a^{s+i}b^j$
$\displaystyle \Rightarrow j \in \{ 0,1 \}$ and $\displaystyle s+i \equiv i-s \bmod{n}$ $\displaystyle \Rightarrow 2s \equiv 0 \pmod{n}$
Which means the $\displaystyle s=\frac{n}{2}$, which is only an integer when n is even.

Now you can do the case if $\displaystyle g= a^sb^t$, You should get that $\displaystyle j=0.$