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.