1. The problem statement, all variables and given/known data If N is a normal subgroup in the finite group such that number of cosets of N in G [G:N] and o(N) are relatively prime, then show that any element x in G satisfying x^o(N) = e must be in N? 2. Relevant equations 3. The attempt at a solution For any x in G, Nx will be an element in G/N . As N is normal, G/N is a group. By Lagrangian Theorem, we will have x^o(G/N) belongs to N. I am not able to get any clue after making lot of attempts beyond this point. Can you please throw some light regarding this? Regards, Henry.