Abstract Algebra Answers

Use the fundamental theorem ofhomomorphism for groups to prove

check whether or not q[x]/<8x³+6x²-9x+24> is a field.

If (a,b) E AxA, where A is a group, then o((a,b))=o(a)o(b)

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

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)

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