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）︰2013/10/17
考试时限（分钟）：30分钟
试题 :
代数导论第二次小考
INTRODUCTION TO ALGEBRA I - QUIZ II
OCT 17 2013
(1) (10 points) Let g be a finite group and let x ∈ G. Prove that
if g ∈ N_G((x)), then gx(g^-1) = x^a for some a ∈ Z.
(2) (10 points) Prove that the subgroup generated by any two distinct elements
of order 2 in S_3 is exactly S_3.
(3) (10 points) Please find all possible normal subgroups in S_3.
(4) (10 points) Let G be a group. Prove that N = (x^-1y^-1xy | x,y ∈ G) is a
normal subgroup of G and G/N is abelian.
(5) (10 points) Prove if N ≦ G and H is a subgroup of G, then H ∩ N ≦ H.