课程名称︰代数导论一
课程性质︰必修
课程教师︰庄武谚
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2013/10/31
考试时限（分钟）：30分钟
试题 :
代数导论第三次小考
INTRODUCTION TO ALGEBRA I - QUIZ III
OCT 31 2013
(1) (10 points) Please state the first isomorphism theorem and then apply it
to prove the following corollary.
Corollary: Let ψ : G → H be a group homomorphism. Then ψ is injective
if and only if Ker(ψ) = 1.
(2) (10 points) Let G be a group with order being a prime p. Show that G is
isomorphic to a cyclic group.
(3) (10 points) Prove that the permutation group S_n is generated by the set of
transposition {(i i+1)|1≦i≦n-1}.
(4) (10 points) Let A,B be groups and C,D be normal subgroups of A,B
respectively. Show that (A ×B)/(C ×D) is isomorphic to A/C ×B/D.
(5) (10 points) Let G be a group and N is a normal subgroup of G such that
G/N is a finit group. Let H be a finit subgroup of G such that
(|H|,|G/N|)=1. Then we have H≦N.