I am stuck proving that a simple group of order is isomorphic to .
- I have shown and , so there must be elements of order and elements of order .
- I could further show .
- Now, where I get stuck: I need to show and therefore .
I know I have to use the fact that there are already elements of order coprime to . That leaves only elements to have order 2 or 4, which somehow has to lead to a contradiction to . But the 2-Sylow-groups have order and can thus intersect non-trivially. So how can I argue from here?
Any help would be much appreciated!