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,
30: Prove that if N is a normal subgroup of G, then H?N is a normal subgroup of H.
(1)show H?N is a subset of H;
?x∈H?N, x ∈H , H?N is a subset of H.
(2)show H?N is a closure;
?x,y∈H?N, x∈H and y∈H, x*y∈H; x∈N and y∈N, x*y∈N; so x*y∈H?N.
(3)Since H and N are subgroup of G, e∈N and e∈H,
so e∈H?N.
(4)?x∈H?N, x ∈H , x-1∈H, and x ∈N, x-1∈N;
so x-1∈H?N.