[FONT="]Let *n* be an even integer. Prove that D*n*/Z(D*n*)[/FONT] is isomorphic to [FONT="]D(*n*/2). [/FONT]

Hints: if \(\displaystyle D_{2n}=\left\{a,b\;;\;a^2=b^n=1\,,\,aba=b^{-1}=b^{n-1}\right\}\) , then:

1) \(\displaystyle Z\left(D_{2n}\right)=\{1,b^{n/2}\}\)

2)\(\displaystyle D_{2n}/Z\left(D_{2n}\right)=\left\{\overline{a}\,,\,\overline{b}\;;\;\overline{a}^2=\overline{b}^{n/2}=\overline{1}\,,\,\overline{a}\overline{b}\overline{a}=\overline{b}^{-1}\right\}\) , with \(\displaystyle \overline{x}:=xZ\left(D_{2n}\right)\in D_{2n}/Z\left(D_{2n}\right)\,,\,x\in D_{2n}\)

Tonio