北京邮电大学 离散数学下 群论 作业错题讲解(17)

北京邮电大学离散数学群论作业详解

群论9.5 Ex1: Let G be a group, and let N and H be subgroups of G such that N is normal in G. Prove that (1)HN is a subgroup of G.(2)N is normal subgroup of HN.

(1) first show HN is closure;

?x,y ∈HN, x=an 1,y=bn 2 for some a,b ∈H.

xy=an 1*bn 2, because bN=Nb => n 1*b=bn 3,

so xy=abn 3n 2∈HN.

e ∈H and e ∈N, so e*e=e ∈HN.

?x ∈HN, x=an 1, x -1=(an 1)-1= n1-1a -1 ,

because Nb=bN,so n1-1a -1 =a -1n 4 ∈HN.

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