北京邮电大学离散数学群论作业详解
群论9.5 (1)4、18、29、30@353-354,
4: Show f:G 1?G 2→G 1by f(a,b)=a is a homomorphism. f((a,b )*‘(c,d))=f(a*c,b*d)=a*c=f(a,b)*f(c,d)
18: Prove N is normal subgroup iff a -1Na=N for all a ∈G.
If N is normal subgroup, then ?a ∈G, aN=Na, and
e ∈N. so a -1Na=Na -1a=Ne=N.
If a -1Na=N for all a ∈G, then N=a -1aN=a -1Na,
left cancelation, aN=Na.