北京邮电大学 离散数学下 群论 作业错题讲解(18)

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.

(2) first show N is subset of HN;

?n∈N, because e∈H, so e*n=n∈HN.

Since N is subgroup of G, so N is closed and have identity, and inverse of all elements, thus N is subgroup of HN.

Second, show aN=Na, for ?a∈HN;

because (1), a∈G,

N is mormal in G, so aN=Na,

Hence N is normal subgroup of HN.

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