center of the dihedral group

**kimberu** · March 25th 2010, 07:21 PM

Hi,
my question is, "for n = 3 or greater, show $Z(D_n)$ is trivial if n is odd and is {1, $a^{n/2}$ } 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 $b^ja^i$ is in $Z(D_n)$ , then $g(b^ja^i)g^{-1} = b^ja^i$ for all g.
if g=a then $a(b^ja^i)a^{-1} = b^ja^i$ and if g=b $b(b^ja^i)b^{-1} = b^ja^i$
but from here I'm quite stuck.

Thanks!!

**Haven** · March 25th 2010, 10:45 PM

I assume $a$ denotes a rotation and $b$ denotes a flip

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

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

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

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

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