Exam 2

1. (a) Define ring.

(b) Give an example of a commutative ring with unity 1 ¹ 0 that is not an integral domain.

2. Find the characteristic of the ring Z₄×Z₆.

3. Compute Kerf and f(8), where f:Z₂₄®Z₄×Z₃ is the group homomorphism satisfying f(1)=(2,2).

4. Classify the group (Z₄×Z₆) / á(2,2)ñ according to the fundamental theorem of finitely generated abelian groups.

5. Give an example of a nontrivial homomorphism f:Z₁₀®Z₁₅, if an example exists. If no such homomorphism exists, explain why that is so.

6. List, up to isomorphism, all abelian groups of order 162.

7. Let H and K be subgroups of a group G. Define ~ on G by a ~ b if and only if a=hbk for some h Î H and some k Î K. Prove that ~ is an equivalence relation on G. Describe the elements in the equivalence class containing a Î G.

8. Prove that Q/Z under addition is an infinite group in which every element has finite order.