Let G be a finite group and N a normal subgroup. Suppose that x is an element of G of prime order p with x not in N. Show that $\displaystyle <x> \cap N = 1$ and that the cosets $\displaystyle N, Nx,..., N x^{p-1}$ are distinct. Deduce that p divides the index [G:N].

Now I have the ideas for the first two parts although I'm having trouble writing this concisely and properly. Also, I'm not sure where the last deduction comes into this - what am I missing?

Finally (sketch thoughts for this bit will be fine!) - if simply x is not in N and m is the smallest integer for which $\displaystyle x^m$ is in N, generalise the above result. What does this say about the quotient group G/N?