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）︰2013/11/21
考試時限（分鐘）：30分鐘
試題 :
代數導論第四次小考
INTRODUCTION TO ALGEBRA I - QUIZ IV
NOV 21 2013
(1) (20 points) Let τ and σ be two elements of the permutation group S_n.
Suppose the cycle decomposition of σ is given by
(a_1 a_2 … a_k1)(b_1 b_2 … b_k2)…. Show that τσ(τ^-1) has the cycle
decomposition (τ(a_1) τ(a_2) … τ(a_k1))(τ(b_1) τ(b_2) … τ(b_k2))….
(2) (15 points) Let G be a group and H be a normal subgroup of G. Show that for
every conjugacy class Κ of G, we have either Κ ⊂ H or K ∩ H = ψ.
(ψ denotes empty set)
(3) (15 points) There are 7 conjugacy classes in S_5. Please write down a
representative for each conjugacy class and compute the number of elements
in each class.