北京邮电大学 离散数学下 群论 作业错题讲解(11)

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.

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