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?
OK so this idea is very neat and elegant. In order to deduce that p divides the index [G:N] I'm guessing that we show p does not divide |N| (since we know p divides |G| and then we'll be done). I can't see how we can use what we already have to see this though.
If x is not in N and m is the smallest integer for which is in N, then it looks as if and that are distinct cosets. Is this correct? And what is this supposed to say about G/N?? Thanks!