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4. Show that G O H is Abelian if and only if G and H are Abelian. State the general case. 5. Prove or disprove that Z O Z is a cyclic group. 6. Prove, by comparing orders of elements, that Z Z, is not iso- morphic to Z, O ZA. 7. Prove that G O G, is isomorphic to G, D G. State the general case. 8. Is 23 9. Is Z, OZ, isomorphic to Z15? Why? 10. How many elements of order 9 does Z, D Z, have? (Do not do this exercise by brute force.) 11. How many elements of order 4 does Z, O Z, have? (Do not do this by examining each element.) Explain why Z, D Z, has the same number of elements of order 4 as does Z3000000 Z400000- General- ize to the case Z O Z,. 12. Give examples of four groups of order 12, no two of which are isomorphic. Give reasons why no two are isomorphic. 13. For each integer n > 1, give examples of two nonisomorphic groups of order n2. 14. The dihedral group D, of order 2n (n 2 3) has a subgroup of n ro- tations and a subgroup of order 2. Explain why D, cannot be iso- morphic to the external direct product of two such groups. 15. Prove that the group of complex numbers under addition is iso- morphic to R OR. 16. Suppose that G, ~ G, and H, ~ H2. Prove that G, OH, ~ GO H,. State the general case. 17. If GO H is cyclic, prove that G and H are cyclic. State the general case. 18. In Z O Z,, find two subgroups of order 12