Problem: Intersection of 2-Sylow-groups

Hi,

I am stuck proving that a simple group of order is isomorphic to .

In particular:

- 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!

Re: Problem: Intersection of 2-Sylow-groups

Could you possibly use the fact that all sylow p subgroups are conjugate to each other (with p=2)?

Re: Problem: Intersection of 2-Sylow-groups

I could not yet figure out how to use that, but I will try for some hours. Thanks for the input!

Re: Problem: Intersection of 2-Sylow-groups

Had vague thoughts about the conjugacy class of 2-Sylow-subgroups generating a normal subgroup in G, but I couldn't see how that would lead me further. Can you help me a little?

1 Attachment(s)

Re: Problem: Intersection of 2-Sylow-groups

Hi,

I believe this is what you need to finish the proof that G is isomorphic to A_{5}; namely that there are 5 Sylow 2 subgroups of G. After this determination, I left the rest of the proof to you:

Attachment 27950

Re: Problem: Intersection of 2-Sylow-groups

Thank you!

That was all I need to conclude my proof.