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）︰2013/12/19
考試時限（分鐘）：30分鐘
試題 :
代數導論第六次小考
INTRODUCTION TO ALGEBRA - QUIZ VI
DEC 19 2013
You could assume Recognition Theorem for the semidirect product. Let G be a
Group with subgroups H and K, such that (1) H≦G, (2) H ∩ K = 1,and
(3) G = HK. Then G is isomorphic to H ╳ K, where the group homomorphism
ψ : K → Aut(H) sends k ∈ K to the automorphism of conjugation by k on H.
The quiz starts from here.
(1) (20 points) Pleas apply Sylow theorem and the recognition theorem for
the semidirect product to classify the groups of order pq, where p,q are
primes, p<q, and p does not divide q - 1.
(2) (20 points) Let y be a nilpotent element of a commutative ring R. Show
that 1 + ry is a unit in R for all r ∈ R. (An element x in R is called
nilpotent if x^m = 0 for some positive integer m.)
(3) (20 points) Let G = {g_1,… ,g_n} be a finit group. Prove that the
element N = g_1 + … + g_n commutes with all elements in the group ring
RC multiplicatively. (i.e. N is in the center of RC.)
註：H ╳ K表the semidirect product of H and K with respect to homomorphism ψ