175 8 I External Direct Products 19. If r is a divisor of m and s is a divisor of n, find a subgroup of Z Z, that is isomorphic to Z, Z 20. Find a subgroup of Z, Zg that is isomorphic to Z, Z. 21. Let G and H be finite groups and (g, h) E GO H. State a necessary and sufficient condition for ((g, h)) = (g) {h). 18 22. Determine the number of elements of order 15 and the number of cyclic subgroups of order 15 in Z30 Z20 23. What is the order of any nonidentity element of Z Z Z? Generalize. of 24. Let m > 2 be an even integer and let n > 2 be an odd integer. Find a formula for the number of elements of order 2 in D D n- т 25. Let M be the group of all real 2 X 2 matrices under addition. Let N = ROROROR under componentwise addition. Prove that M and N are isomorphic. What is the corresponding theorem for the group of m Xn matrices under addition? 26. The group S, Z, is isomorphic to one of the following groups: Z12, Z Z, A, D Determine which one by elimination. 27. Let G be a group, and let H = {(g, g) | g E G}. Show that H is a subgroup of G G. (This subgroup is called the diagonal of GG.) When G is describe G G and H geometrically. 28. Find a subgroup of Z, Z, that is not of the form H K, where H is a subgroup of Z, and K is a subgroup of Z,. 29. Find all subgroups of order 3 in Z, Za. 30. Find all subgroups of order 4 in Z Z. 31. What is the largest order of any element in Z Z? 32. What is the order of the largest cyclic subgroup of Z, Z10 Z,? What is the order of the largest cyclic subgroup of Z, Z, n. O- 4' ral the set of real numbers under addition, his 4 4 this me ral- V 30 are phic 33. Find three cyclic subgroups of maximum possible order in Z ZtZ, of the form (a) (b) (c), where a E Z, bEZ and z ro- iso- 10 15 cE Z1S 34. How many elements of order 2 are in Z200000 Z4000000? Generalize. 35. Find a subgroup of Z Z200 that is isomorphic to Z, Z 36. Find a subgroup of Z12 Z Zs that has order 9. 37. Prove that R* R* is not isomorphic to C. (Compare this with iso- Exercise 15.) 38. Let neral Н- a, b E z L0 0