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.
(5) ?x ∈H, x(H ?N) = {xn | n ∈H ?N}, let any n 1∈H ?N,
because aN=Na, x*n 1=n 2*x for some n 2∈N,
n 2= x*n 1*x -1,
since x -1 ∈H, n 1∈H, then n 2∈H.
n 2∈H ?N,
Thus x(H ?N) = (H ?N)x.
Hence, H ?N is a normal subgroup of H.