課程名稱︰代數導論一
課程性質︰必修
課程教師︰莊武諺
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2013/10/17
考試時限（分鐘）：30分鐘
試題 :
代數導論第二次小考
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.