北京邮电大学 离散数学下 群论 作业错题讲解(15)

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.

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