课程名称︰代数导论一 课程性质︰系必修 课程教师︰于靖 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2010年10月28日 考试时限（分钟）：三小时 是否需发放奖励金：是 试题 : In answering the following problems, you may use freely any Theorem already proved (or Lemmas, Propositions) from the textbook(Section 2.2-2.6), my class- room lectures, or previous course on Linear Algebra. You don't need to give proofs of the theorems you are using, but you MUST write down complete statem- ent of the theorem which you use as a second part of your answer to that prob- lem. NOTATIONS： Ⅱm ,the cyclic subgroup of order m>=1, known also as the additive group of the integers modulo m. D2n ,the dihedral group of order 2n, known also as the group symmetries of the regular n-gon, n>=3. Sn , the symmetric group of n>=1 letters. An , the subgroup of Sn consists of even permutations. V = {1,(12)(34),(13)(24),(14)(23)} <= S4. Q, the quaternion group generated by A = [ 0 1 ] and B = [ 0 i ] [ -1 0 ] [ i 0 ] under the matrix multipication. GL(2,R), the group of all real 2 ×2 matrices with non-zero determinant. SL(2,R), the subgroup of GL(2,R) consisting of matrices having determinant 1. I, the identity square matrix. (1)(a) Prove that Ⅱ3 ×Ⅱ9 is not isomorphic to Ⅱ3 ×Ⅱ3 ×Ⅱ3. (b) Suppose gcd(m,n) = 1, prove that Ⅱm ×Ⅱmn　~ Ⅱn ×Ⅱm ×Ⅱm. = (2)(a)Write down all subgroups of D10. Which ones are normal subgroups? Which ones are non-normal subgroups? (b)Let n>=2 be integer. Prove that there exist two elements in the group D2n, both of order 2, such that any subgroup of D2n containing these two ele- ments must be the whole group D2n (in order words D2n can be generated by these two elements). (c)Prve that D12 ~ S3 ×Ⅱ2. = (3)(a)Show that S4 has three distinct subgroups of order 8, and they are all isomorphic_to D8. (b)Show that V <| S4, and prove that S4/V ~ S3. = (4)(a)Let H,K <= G be subgroups of G. Prove that HK is a subgroup if and only if HK = KH holds. (b)The following statement is NOT true: let K, H1 ,H2 be subgroups of group G, with K <| G satisfyinh H1K = H2K , then H1 = H2. Give an example of a group G, together with subgroups K,H1,H2 satisfying the hypothesis of this statement, but H1 ≠ H2. (5)(a)Show that every element x ≠ ±I. Then prove that every subgroup of Ⅱ3 ×Q is normal. (b)Suppose G is finite abelian group with |G| odd. Show that for every divior d of 8|G|, there exists subgroup in G ×Q having order d. (6)(a)Given x,y in group G, the element xy x^(-1) y^(-1) is called the commut- ator of x and y. The set of all commutators of pairs of elements form G generate a subgroup, called the commutator subgroup of G. Show that the commutator subgroup of GL(2,R) is contained in SL(2,R). (b)Prove that the commutator subgroup of Sn is contained in An. Show also that the commutator subgroup of S3 is A3. 以上试题结束。 附注： (3)(a)只需要找到三个，不必证明只有三个。 另外提供得分分布： A 12 , A- 6 B+ 1 , B 4 ,B- 3 C+ 2 , C 4 ,C- 1 F 70.　　　　　　　　　（如果你作不太出来并不用太难过，但还是请好好加油吧!) --

※ 发信站: 批踢踢实业坊(ptt.cc) ◆ From: 140.112.251.220