Explain

ments of nite order does it have? 20. Suppose that G is an Abelian group of order 35 and every element of G satises the equation x35 = e. Prove that G is cyclic. Does your argument work if 35 is replaced with 33? 21. Let G be a group and let a be an element of G. a. If a\" = 6, what can we say about the order of a? b. If 1\" = 8, what can we say about the order of a? c. Suppose that IGI = 24 and that G is cyclic. If a3 #= e and a'2 a": 6, show that (a) = G. 22. Prove that a group of order 3 must be cyclic. 23. Let Z denote the group of integers under addition. Is every subgroup of Z cyclic? Why? Describe all the subgroups of Z. Let a be a group element with innite order. Describe all subgroups of (a). 24. For any element a in any group G, prove that (a) is a subgroup of (3(a) (the centralizer of a). 25. If :1 is a positive integer, d i 2, and d divides n, show that the num- ber of elements of order if in DM is (Md ). How many elements of order 2 does it)!1 have? 26. Find all generators of 2'. Let a be a group element that has innite order. Find all generators of (a). 27. Prove that C *, the group of nonzero complex numbers under multipli- cation, has a cyclic subgroup of order n for every positive integer n. 28. Let a be a group element that has innite order. Prove that (a') = (at) if and only ifi = ij. 29. List all the elements of order 8 in Zsoooooo' How do you know your list is complete? Let a be a group element such that In! = 8000000. List all elements of order 8 in (a). How do you know your list is complete? 30. Suppose that G is a group with more than one element. If the only subgroups of G are {e} and G, prove that G is cyclic and has prime order. 31. Let G be a nite group. Show that there exists a xed positive integer n such that a" = e for all a in G. (Note that n is independent of a.) 32. Determine the subgroup lattice for 212- Generalize to Z 2 F q, where p and q are distinct primes. 33. Determine the subgroup lattice for 23' Generalize to 2p, where p is a prime and n is some positive integer. 34. Prove that a nite group is the union of nroner suborouns if and