11 Fundamental Theorem of Finite Abelian Groups 235 15. How many Abelian groups (up to isomorphism) are there a. of order 6? b. of order 15? NEY, c. of order 42? ko"* d. of order pq, where p and of order par, where f. Generalize parts d and q are distinct primes? P, q, and r are distinct primes? e. oni- 16. How does the number (up to isomorphism) of Abelian of ree groups order n compare with the number (up to isomorphism) of Abelian ups. groups of order m where 32 and m 52? 24 and m 54? ctly а. п — four b. n c. n p' and m = q', where p and q are prime? d. n p andm = p'q, where p and q are distinct primes? e. n p' and m = p'q', where p and q are distinct primes? 17. Up to isomorphism, how many additive Abelian groups of order 16 have the property that x + x +x + x = 0 for all x in the group? er 4. - 15. 9? ave 18. Let pi, P2 .. , be distinct primes. Up to isomorphism, how many Abelian groups are there of order p p...p? ave 19. The symmetry group of a nonsquare rectangle is an Abelian group of order 4. Is it isomorphic to Zg or Z, Z,? 4 20. Verify the corollary to the Fundamental Theorem of Finite Abelian Groups in the case that the group has order 1080 and the ave divisor is 180. has {1,9, 16, 22, 29, 53, 74, 79, 81 is a group under multipli- cation modulo 91. Determine the isomorphism class of this group. :lass 21. The set 22. Suppose that G is a finite Abelian group that has exactly one sub- group for each divisor of IGl. Show that G is cyclic. 23. Characterize those integers n such that the only Abelian groups of order n are cyclic. the , np re- 24. Characterize those integersn such that any Abelian group of order n belongs to one of exactly four isomorphism classes. 25. Refer to Example 1 in this chapter and explain why it is unneces- sary to compute the orders of the last five elements listed to deter- mine the isomorphism class of G. 26. Let G {1,7, 17, 23, 49, 55, 65, 71} under multiplication modulo 96. Express G as an external and an internal direct product of cyclic e by di- r4 sion р. groups. d by ights