课程名称︰代数导论 课程性质︰数学系必修 课程教师︰朱桦 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰98/11/19 考试时限（分钟）：150min 是否需发放奖励金：yes （如未明确表示，则不予发放） 试题 : (1)(10%)Solve the follow system of equations ┌5x^2 + 4x ≡5 (mod7) │ └4x ≡6 (mod22) (2)(10%)Let m >= 2 be an integer and S = { a:gcd(a,m) = 1 , 1 <= a <= m }. Let n = (a的连乘 其中a属於S) show that n≡ +-1(mod m). (3)(10%)(a) Determine the number of elements of order 6 in S7. (b)Determine the number of odd permutations of order 6 in S7. (4)(10%)Show that the symmetric group Sn, n>=3, can be generated by the permutations (1 2) and (1 2....n). (5)(30%)Let G be a group. (a) Let |g| = m, |h| = n in G. Suppose that gh = hg and gcd(m,n) = 1. Show that |gh| = mn. (b) Let |g| = m, |h| = n in G. Suppose that gh = hg. Show that there exists a 属於 G such that |a| = lcm(m,n). (c) Let |a| = mn in G. Suppose that gcd(m,n) = 1. Show that there exist g,h 属於 G such that |g| = m, |h| = n and a = gh. (6)(30%)(a) Show that for any positive integer n, there is a group G such that it has exactly n subgroups. (b) Find a noncyclic group G which has exactly five subgroups. (c) Let G be a group. Show that G is a finite group if and only if G has only finitely many subgroups. (7)(10%) Find all group homomorphisms from the symmetric group S3 to the Klein 4-group K4 (8)(10%) Let H and K be subgroups of G. Show that Ha∩Ka = (H∩K)a. --

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