代数代写 Algebra代写 数学考试代写 MATH exam代写

Algebra MATH-UA.0343-003

Final exam

代数代写 • Make sure that this file has your last name in its name! • This is a timed exam. You need to finish the exam by 2:00pm and submit you solutions to

• Make sure that this file has your last name in its name!

• This is a timed exam. You need to finish the exam by 2:00pm and submit you solutions to NYUClasses (not by email!) by 2:10pm. No late submission will be accepted.

• This is an open book exam: you can use lecture notes and the textbook.

• Only complete, detailed solutions will get the full credit. No credit will be given for an answer without a proof.

• Include the following pledge on the top of your first page

“I shall perform honestly all my academic obligations. I will not represent the words, works, or ideas of others as my own; will not cheat; and will not seek to mislead faculty or other academic officers in their evaluation of my course work or in any other academic affairs.”

• Write you NetID (e.g. yu3) in the top right corner of every page.

Problem 1. 代数代写

We know that there is an isomorphism of groups with respect to addition:

Z₅ × Z₇ ≃Z₃₅.

Describe all elements (a,b) ∈ Z₅ × Z₇ such that (a,b) has order 35.

Problem 2.

Let f : Z₁₅ → Z₁₅ be a group homomorphism given by f (x) = 3x. Find Ker(f ) and= (f ). Compute the index [Z₁₅ : Ker(f )].

Problem 3. 代数代写

A group G is called simple if it has no proper normal subgroups. Find, with a proof, all n ≧ 1 such that the permutation group S_n is simple?

Problem 4

(Prove or Disprove). Let G be a group and H a normal subgroup of G. If G is abelian, then G/H is abelian.

Problem 5. 代数代写

Give, with a proof, the list of all possible orders of elements in S₅.

Problem 6.

Define a homomorphsim of rings

f : Z₃₅ → Z₅ × Z₇, f (a) := (a mod 5,a mod 7).

Prove that Ker(f ) = {0}. Conclude that f is an isomorphism.

Problem 7. 代数代写

Let f : G → H be a homomorphism of finite groups G and H. Prove that |Ιm(f )| is a common divisor of |G| and |H|. Conclude that |Ιm(f )| divides gcd(|G|,|H|).

更多代写：CS北美高中代修网课案例  加拿大雅思代考  英国宏观经济网课托管  北美BIO生物学论文代写  北美Finance论文代写 护理报告代写

合作平台：essay代写 论文代写 写手招聘 英国留学生代写