Isomorphism Question

• May 17th 2010, 03:21 PM
wutang
Isomorphism Question
Let n be an even integer. Prove that Dn/Z(Dn) is isomorphic to
D(n/2).

• May 17th 2010, 07:18 PM
tonio
Quote:

Originally Posted by wutang
Let n be an even integer. Prove that Dn/Z(Dn) is isomorphic to D(n/2).

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

Tonio
• May 19th 2010, 06:27 PM
wutang
It should say that Dn/Z(Dn) is isomorphic to D(n/2). I understand your definition for Dn/Z(Dn), but I don't get how to set up the isomorphism. I think I should use the fact that any group generated by a pair of elements of order 2 is dihedral to get the isomorphism from Dn/Z(Dn) to D(n/2) ?
• May 19th 2010, 06:41 PM
wutang
never mind, I think I just have to play with the elements in Dn/Z(Dn) until I get it to look that D(n/2).