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.