175 I External Direct Products 8 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 GOH. 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 Za0 Z20 23. What is the order of any nonidentity element of Z2 Z, Z3? 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 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 G G.) 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 Zg Z. 30. Find all subgroups of order 4 in Z4 Z 31. What is the largest order of any element in Z Z? 32. What is the order of the largest cyclic subgroup of Zg Z10 Z5? What is the order of the largest cyclic subgroup of Z, Z - - т n. O- 65 4' ral the set of real numbers under addition, his 4 4 his ame ral- 30 are phic 33. Find three cyclic subgroups of maximum possible order in Z Z1g z, of the form (a) (b) (c), where a E Z bE Z and CEZIS 34. How many elements of order 2 are in Z200000 Z4000000? Generalize. 35. Find a subgroup of Zo Z200 that is isomorphic to Z, Z. 36. Find a subgroup of Z12 Z, Z15 that has order 9. 37. Prove that R* R* is not isomorphic to C. (Compare this with ro- 15 iso- iso- Exercise 15.) 38. Let neral [1 a Н- a, b E z Lo o