# 1577 I Cosets and Lagrange's Theorem(15 Let G be a group of order 60. What are the possible orders for thesubgroups of G?16.)Suppose that K is a proper subgroup of H and H is a proper sub-group of G. If IKI 42 and IGI 420, what are the possible ordersof H?17. Let G be a group with IGI pq, where p and q are prime. Provethat every proper subgroup of G is cyclic.18. Recall that, for any integer n greater than 1, (n) denotes the num-ber of positive integers less than n and relatively prime to n. Provethat if a is any integer relatively prime to n, then abin) mod n = 1.19. Compute 515 mod 7 and 713 mod 11.20. Use Corollary 2 of Lagrange's Theorem (Theorem 7.1) to provethat the order of U(n) is even when n > 2.21. Suppose G is a finite group of order n and m is relatively prime to n.If g E G and g"m22. Suppose H and K are subgroups of a group G. If IHI = 12 andKI= 35, find H KI. Generalize.23. Suppose that H is a subgroup of S and that H contains (12) and(234). Prove that H S.=e, prove that ge.24. Suppose that H and K are subgroups of G and there are elementsa and b in G such that aHC bK. Prove that HC K.25. Suppose that Gis an Abelian group with an odd number of elements.Show that the product of all of the elements of G is the identity.26. Suppose that G is a group with more than one element and G hasno proper, nontrivial subgroups. Prove that IGI is prime. (Do notassume at the outset that G is finite.)27. Let IGI = 15. If G has onlyof order 5, prove that G is cyclic. Generalize to IGI = pq, whereand q are prime.28. Let G be a group of order 25. Prove that G is cyclicall g in G. Generalize to any group of order p2 where p is prime.Does your proof work for this generalization?29. Let 1Gl = 33. What are the possible orders for the elements of G?one subgroup of order 3 and only onepg5 = e forShow that G must have an element of order 3.30. Let IGl = 8. Show that G must have an element of order 2.31. Can a group of order 55 have exactly 20 elements of order 11?Give a reason for your answer.32. Determine all finite subgroups of C*, the group of nonzero com-plex numbers under multiplication.

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Step 1

To establish the stated property of the group of order p^2 , for any prime (in particular for p=5)

Step 2

First part of the...

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