Let G be a group of order pn, where p is a prime and n = 2. Suppose that |Z(G)| = p. Show that there exists an a ? G such that |C(a)| = pn-1. If G is a group of order pn, for some prime p and positive integer n, show that G has a subgroup of order pm for each positive integer m = n. For each of the following lists, determine if it is the list of sizes of the conjugacy classes of some finite group. If it is, provide such a group. If not, explain why not. 1. 1, 1, 1, 1, 1, 5, 5, 5, 5 2. 1, 2, 3 3. 2, 4, 6

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