# 11Fundamental Theorem of Finite Abelian Groups23515. How many Abeliangroups (up to isomorphism) are therea. of order 6?b. of order 15?NEY,c. of order 42?ko"*d. of order pq, where p andof order par, wheref. Generalize parts d andq are distinct primes?P, q, and r are distinct primes?e.oni-16. How does the number (up to isomorphism) of Abelianofreegroupsorder n compare with the number (up to isomorphism) of Abelianups.groups of order m where32 and m 52?24 and m 54?ctlyа. п —fourb. nc. 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 16have the property that x + x +x + x = 0 for all x in the group?er 4.- 15.9?ave18. Let pi, P2 .. , be distinct primes. Up to isomorphism, how manyAbelian groups are there of order p p...p?ave19. The symmetry group of a nonsquare rectangle is an Abelian groupof order 4. Is it isomorphic to Zg or Z, Z,?420. Verify the corollary to the Fundamental Theorem of FiniteAbelian Groups in the case that the group has order 1080 and theavedivisor 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.:lass21. The set22. 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 oforder n are cyclic.the, npre-24. Characterize those integersn such that any Abelian group of ordern 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 modulo96. Express G as an external and an internal direct product of cyclice bydi-r4sionр.groups.d byights

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Step 1

From the given information let G ={1,7,17,23,49,55,65,71} be a group under multiplication modulo 96.

It is needed to express G as an external and internal direct product of cyclic groups.

Since the number of elements in G is 8 so this will be either isomorphic to:

Step 2

Since the order of element 1 is 1 and there are four elements whose order is 4 that is {7,23,55,71}

And the rest of elements {17,4...

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