Exercise 2. This software computes the elements of the subgroup \(U(n)^k = \{x^k | x \in U(n)\}\) of \(U(n)\) and its order. Run the program for \((n,k) = (27,3), (27,5), (27,7)\), and \((27,11)\). Do you see a relationship connecting \(|U(n)|\) and \(|U(n)^k|\), \(\phi(n)\), and \(k\)? Make a conjecture. Run the program for \((n,k) = (25,3), (25,5), (25,7)\), and \((25,11)\). Do you see a relationship connecting \(|U(n)|\) and \(|U(n)^k|\), \(\phi(n)\), and \(k\)? Make a conjecture. Run the program for \((n,k) = (32,2), (32,4)\), and \((32,8)\). Is your conjecture valid for \(U(32,16)\)? If not, restrict your conjecture. Run the program for \((n,k) = (77,2), (77,3), (77,5), (77,6), (77,10)\), and \((77,15)\)? Do you see a relationship among \(U(77,6)\) and \(U(77,2)\), and \(U(77,3)\)? What about \(U(77,10), U(77,2)\), and \(U(77,5)\)? What about \(U(77,15)\), \(U(77,3)\), and \(U(77,5)\)? Make a conjecture. Use the theory developed in this chapter about expressing \(U(n)\) as external direct products of cyclic groups of the form \(Z_n\) to analyze these groups to verify your conjectures.

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