15. How а. of Groups b. of 234 C. of d. of Exercises JOHN LENNON AND PAUL MCCARTNEY, "The Ballad of John and Yoko" e. of f. G You know it ain't easy, you know how hard it can be. 16. How orde 1. What is the smallest positive integer n such that there are two noni- of order n? Name the two groups. somorphic groups 2. What is the smallest positive integer n such that there are three nonisomorphic Abelian 3. What is the smallest positive integer n such that there are exactly grou of order n? Name the three groups. а. п groups b. n of order n? Name the four groups с. т four nonisomorphic Abelian d. n groups. 4. Calculate the number of elements of order 2 in each of Z, ZgZ Z, and Z, Z, Z,. Do the same for the elements of order 4. of order 45 has an element of order 15 of order 45 have an element of order 9? е. r 17. Up hav 5. Prove that any Abelian Does every Abelian 6. Show that there are two Abelian groups of order 108 that have exactly one subgroup of order 3. 7. Show that there are two Abelian groups of order 108 that have exactly four subgroups of order 3. 8. Show that there are two Abelian groups of order 108 that have exactly 13 subgroups of order 3. 9. Suppose that G is an Abelian group of order 120 and that G has exactly three elements of order 2. Determine the isomorphism class of G 10. Find all Abelian groups (up to isomorphism) of order 360. 11. Prove that every finite Abelian group can be expressed as the (external) direct product of cyclic groups of orders n,, n,,.n where group group 18. Let Ab 19. Th of 20. Ve At div 21. Th ca 22. Su divides n, for i nit1 ferred to in this chapter.) gr 23. C 1, 2, .. . , t - 1. (This exercise is re- 12. Suppose that the order of some finite Abelian group is divisible by 10. Prove that the group has a cyclic subgroup of order 10. 0F 13. Show, by example, that if the order of a finite Abelian group is di- 24. C visible by 4, the n need not have a cyclic subgroup of order 4. group 14. On the basis of Exercises 12 and 13, draw a general conclusion about the existence of cyclic subgroups of a finite Abelian group. 25. R *Copyright 1969 (Renewed) Stony/ATV Tunes LLC. All rights administered by Sony/ATV Music Publishing, 8 Music Square West, Nashville, TN 37203. All rights 26. reserved. Used by permission.