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）︰2013/12/05
考試時限（分鐘）：30分鐘
試題 :
代數導論第五次小考
INTRODUCTION TO ALGEBRA I - QUIZ V
DEC 5 2013
You could assume the follow theorem.
Sylow theorem. Let G be a group of order p^a˙m, where p is a prime not
dividing m. Then we have (1) Sylow p-subgroup of G exist, (2) If P is a Sylow
p-subgroup of G and Q is any p-subgroup of G, then P is contained in a
conjugate of P, and (3) The number of Sylow p-subgroup of G, denote by n_p is
of the form 1 + kp. Moreover n_p = |G:N_G(P)|. Hence n_p divides m.
The quiz starts from here.
(1) (15points) Prove that if P is a Sylow p-subgroup of G and n_p = 1,then
P is normal in G.
(2) (20 points) Prove that if |G| = 1365 then G is not simple.
(3) (15 points) (Cauchy Theorem) Prove that if a prime p divides te order of
a finit group G, then there exists an element of order p in G.