北京邮电大学离散数学群论作业详解
群论9.1-9.2 Ex3: Let G ={x∈R|x>1} be the set of all real numbers greater than 1. For x,y∈G, define
x*y=xy-x-y+2. Show that (G,*) is a group.
proof: (1)closure; because x>1, xy>x+y,
so x*y=xy-x-y+2>2, x*y∈G.
(2)associate; (x*y)*z=(xy-x-y+2)*z=xyz-xz-yz+2z-(xy-x-y+2)-z+2=xyz-xy-xz-yz+x+y+z .
x*(y*z)=x*(yz-y-z+2)=xyz-xy-xz+2x-x-yz+y+z-
2+2=xyz-xy-xz-yz+x+y+z .
(3)identity: e=2, 2*x=2x-x-2+2=x.
(4)reverse: x*x-1=xx-1-x-x-1+2=e=2, x-1=x/(x-1).