课程名称︰代数导论一
课程性质︰必修
课程教师︰庄武谚
开课学院：理学院
开课系所︰数学系
考试日期（年月日）︰2013/12/19
考试时限（分钟）：30分钟
试题 :
代数导论第六次小考
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.)
注：H ╳ K表the semidirect product of H and K with respect to homomorphism ψ