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# 代数代写 Algebra代写 数学考试代写 MATH exam代写

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## Final exam

• 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 Sn 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|).

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MT3505 Algebra: Rings & Fields: Solutions for Chapter 1. 环与域代写 1. Let R be a ring and let Mn(R) denote the set of n × n matrices with entries in R. Prove that Mn(R) is a ring. 2. For a...

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