课程名称︰代数一
课程性质︰数学系选修(可抵必修代数导论一)
课程教师︰林惠雯教授
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰105.10.3
考试时限（分钟）：50 mins
试题 :
1.
Show that a subgroup of a cyclic group is cyclic.
2.
Show that if d|n, then there exist unique H≦Z_n such that |H| = d.
3.
(a)
Let H,K≦G. Show that H∩K≦G, and HK≦G if HK = KH.
(b)
Let H≦G and K is normal in G. Show that H∩K is normal in H and K is normal in HK with HK≦G.
(c)
Let H≦G and K is normal in G. Show that HK/K～H/H∩K.
4.
Show that N is normal in G and G/N is abelian if and only if the commutator [G,G]≦N.
5.
Let G be a finite group and n be a positive positive integer with n||G|. Whether does there exist a subgroup H of G such that |H| = n? Prove it or disprove it by giving a counterexample.