Notation:
R0= Rotation of 0° , R90 =Rotation of 90° , R180= Rotation of180°,R270= Rotation of 270° ,H= Flip about horizontal axis ,V=
Flip about vertical axis, D= Flip about main diagonal ,D′= Flip about the other diagonal.
D8= { R0,R90,R180,R270,H,V,D,D′ }
Z(D8)= { a∈D8:ax=xa ∀ x∈D8 }
First observe that since every rotation in D8 is a power of R90 ,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 D8 .Observe that since RF is a reflection we have RF=(RF)−1=F−1R−1=FR−1. Thus it follows that R and 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=R0 and R=R180 .
Thus Z(D8)= {R0,R180 }=K (say).
Now,
Z(D8)D8= { R0K,R90K,R180K,VK,DK }.
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