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!!