課程名稱︰代數導論
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）︰2013/09/26
考試時限（分鐘）：30分鐘
試題 :
代數導論第一次小考
INTRODUCTION TO ALGEBRA I - QUIZ I
SEP 26 2013
(1) (10 points) Let σ be an element in the permutation group S_n, (n≧13),
given by the following expression,
σ = (1 12 8 10 4)(2 13)(6 9)。(2 3)
Please find the order of this element. Notice that this expression is still
not in the cycle decomposition form.
(2) (10 points) Show that the special linear group
def
SL_n(R) = {A ∈ GL_n(R) | det(A) = 1}
is a subgroup of the general linear group GL_n(R).
(Feel free to use the relation: det(Ab) = det(A)det(B).)
(3) (10 points) Show that N_G(Z(G)) = G and C_G(Z(G)) = G.
(4) (10 points) Let H be a subgroup of group G. Show that H≦N_G(H).
(5) (10 points) Consider the dihedral group D_2k. Use the generators and
relations of D_2k to show that if x is any element of D_2k which is not a
power of r, then we have rx = x(r^-1), where r is the clockwise 2π/k
rotation.