Solve Henry's Normal Subgroup Problem

[seshikanth]

Homework Statement

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?

Homework Equations

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.

 

[Office_Shredder]

Consider the coset xN