Answer to Question #105103 in Abstract Algebra for Sourav Mondal

Question

Find Z (D8) the centre of . D8 Also give the algebraic structure of D8/Z (D8.)

Answer 1

Notation:

$R_0$$=$ Rotation of $0^°$ , $R_{90}$ =Rotation of ${90}^°$ , $R_{180}=$ Rotation of${180}^°$,$R_{270}=$ Rotation of ${270}^°$ ,$H=$ Flip about horizontal axis ,$V=$

Flip about vertical axis, $D=$ Flip about main diagonal ,$D'$$=$ Flip about the other diagonal.

$D_8=$ { $R_0,R_{90},R_{180},R_{270},H,V,D,D'$ }

$Z(D_8)=$ { $a\in D_8:ax=xa \space \forall \space x\in D_8$ }

First observe that since every rotation in $D_8$ is a power of $R_{90}$ ,rotation commute with rotation. We now investigate when a rotation commutes with a reflection. Let $R$ be any rotation and $F$ be any reflection in $D_8$ .Observe that since $RF$ is a reflection we have $RF=(RF)^{-1}=F^{-1}R^{-1}=FR^{-1}.$ Thus it follows that $R\space and \space F$ commute if and only if $FR=RF=FR^{-1}.$ By cancellation ,this hold if and only if $R=R^{-1}$ .But $R=R^{-1}$ only when $R=R_0 \space and \space R=R_{180}$ .

Thus $Z(D_8)=$ {$R_0,R_{180}$ }=$K$ (say).

Now,

$\frac{D_8}{Z(D_8)}=$ { $R_0K,R_{90}K,R_{180}K,VK,DK$ }.