Abstract Algebra Answers

prove that every non trival subgroup of a cyclic group has finite index hence prove that (Q,+) is non cyclic

 a) Using Cayley’s theorem, find the permutation group to which a cyclic group of 

order 12 is isomorphic. (4) 

 b) Let τ be a fixed odd permutation in .

S10 Show that every odd permutation in S is 

10

a product of τ and some permutation in .

A10 (2) 

 c) List two distinct cosets of < r > in ,

D10 where r is a reflection in .

D10 (2) 

 d) Give the smallest n ∈ N for which An is non-abelian. Justify your answer. 

What is the cardinality of the ff. sets? 

1. Z2 5. 2Z 

2. Z*4 6. R 

3. N 

4. Z

give an example of a non–trivial homomorphism or explain why none exists

φ:S₄ → S₃

give an example of a nontrivial homomorphism or explain why none exists. φ:S₃ → S₄

give an example of a non-trivial homorphiem or explain why none exists.  φ : Z₁₂ → Z₄

Use Cauchy’s mean value theorem to prove that:

{Cos(alpha)- cos(beta) }/{sin(alpha) -sin(beta) }=tan(theta)

a) Find all the units of Z[ √− 7].

b) Check whether or not Q[x]/< 8x + 6x − 9x + 24 >

Is a field.

c) Construct a field with 125 elements.

Prove that every non-trivial subgroup of a cyclic group has finite index. Hence

prove that (Q, +) is not cyclic. (7)

b) Let G be an infinite group such that for any non-trivial subgroup H of

G, G : H < ∞. Then prove that

i) H ≤ G ⇒ H = {e} or H is infinite;

ii) If g ∈G, g ≠ e, then o(g) is infinite

For an ideal I of a commutative ring R, define

√I {x belongs to R |x^n belongs to I |for some n ∈N}. Show that

i) √I is an ideal of R.

ii) I ⊆ √I.

iii) I ≠ √I in some cases.

b) Is R/I×R/J=R×R/I×J ,for any two ideals I and J of a ring R ? Give reasons for your

answer