北京邮电大学离散数学群论作业详解
群论9.4 (1)22,28@page 331;
22-Show S 1?S 2is a subsemigroup of S.
proof: closure; ?a,b ∈S 1?S 2, then a*b ∈S 1, a*b ∈S 2, so a*b ∈S 1?S 2.
if S 1?S 2 =?, still be subsemigroup.
28-If f:S 1→S 2and g :S 2→S 3 be isomorphisms, show g ?f :S 1→S 3is an isomorphism.
proof: (g ?f)(x *1y)= g(f(x *1y))=g(f(x) *2f(y))
=g(f(x))*3g(f(y))=(g ?f)(x) *3(g ?f)(y).
one-to-one: suppose (g ?f)(a)=(g ?f)(b), g(f(a))=g(f(b)), f(a)=f(b), a=b.
onto: ?z ∈S 3 有g(y)=z; ?y ∈S 2 有f(x)=y;