1. This is an exercise from math100a which gives us a characterization of cyelic groups. (a) Suppose C, := {1,a,a², .,a"-1} is a cyclic group of order n. Show that if d\n, then C, has exactly ø(d) elements that have order d. Use this to deduce that Eod) = n. dịn (b) Suppose G is a finite group and for every positive integer d, |{g € G|gª = 1}|< d. Prove that G is cyclic. (Hint. Let p(d) be the number of elements of G that have order d. Show that if o(g) = d, then 1, g,...,gª-1 are all the elements of G that satisfy rd = 1. Use this to deduce that if (d) # 0, then »(d) = ø(d). Argue why we have Edn v(d) = n where n = |G|. Use the first part to obtain that ý(d) = ¢(d) if d\n, and so G is cyclic.)

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