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 and that the cosets 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 is in N, generalise the above result. What does this say about the quotient group G/N?