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.

Please enter $$n$$ and $$k$$, the result will show below.