The problem statement, all variables and given/known data Let G be a group and let g be an element G such that |g| = mn, where gcd(m, n) = 1. Show that there are unique elements a, b in G such that g = ab = ba and |a| = m and |b| = n. The attempt at a solution Since gcd(m, n) = 1, there are integers s and t such that ms + nt = 1. Hence g = gms+nt = gmsgnt. Let a = gnt and b = gms. Then g = ab = ba. We also have that |a| ≤ m and |b| ≤ n. I'm having trouble showing that |a| = m and |b| = n. Also, the fact that s and t are not unique proves that a and b are not unique right? This is as my thought process has taken me. Any tips?