北京邮电大学离散数学群论作业详解
群论9.4 (2)28,32@page 349; 24@page338.
32-Show f:G→G by f(a)=a-1 is a isomorphism iff Abelian.
Proof: (1)Suppose f is isomorphism,
f(xy)=(xy)-1=f(x)f(y)=x-1y-1
so xy=((xy)-1)-1=(x-1y-1)-1=yx. G is Abelian. (2)Suppose G is Abelian.
f(xy)=(xy)-1=x-1y-1=f(x)f(y), homomorphism. onto: ?x∈G, f(x-1)=(x-1)-1=x.
one-to-one: suppose f(x)=f(y), x-1=y-1,
xx-1=e=yy-1, right cancelation x=y.