Let p and q be odd primes and let m and n be positive integers. Prove that U(p^m) + U(a^n) is not cyclic.
(+ is the external direct product (+ with a circle around it))
I think you mean to say $\displaystyle U(p^m) \times U(q^n)$. Since $\displaystyle p,q$ are odd it means $\displaystyle 2$ divides both $\displaystyle |U(p^m)|,|U(q^m)|$. And therefore $\displaystyle \gcd( |U(p^m)|,|U(q^m)|) \not = 1$. Thus, their product cannot be cyclic.