# Problem: Intersection of 2-Sylow-groups

• Apr 13th 2013, 04:50 AM
omlele
Problem: Intersection of 2-Sylow-groups
Hi,

I am stuck proving that a simple group $\displaystyle G$ of order $\displaystyle 60$ is isomorphic to $\displaystyle A_5$.

In particular:

- I have shown $\displaystyle |Syl_5(G)|=6$ and $\displaystyle |Syl_3(G)|=10$, so there must be $\displaystyle 6\cdot(5-1)=24$ elements of order $\displaystyle 5$ and $\displaystyle 10\cdot(3-1)=20$ elements of order $\displaystyle 3$.

- I could further show $\displaystyle |Syl_2(G)|\notin\{1,3\}$.

- Now, where I get stuck: I need to show $\displaystyle |Syl_2(G)|\neq15$ and therefore $\displaystyle |Syl_2(G)|=5$.
I know I have to use the fact that there are already $\displaystyle 20+24=44$ elements of order coprime to $\displaystyle 2$. That leaves only $\displaystyle 60-44-1=15$ elements to have order 2 or 4, which somehow has to lead to a contradiction to $\displaystyle |Syl_2(G)|=15$. But the 2-Sylow-groups have order $\displaystyle 2^2$ and can thus intersect non-trivially. So how can I argue from here?

Any help would be much appreciated!
• Apr 13th 2013, 05:14 AM
Gusbob
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)?
• Apr 13th 2013, 06:28 AM
omlele
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!
• Apr 13th 2013, 08:46 AM
omlele
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?
• Apr 14th 2013, 01:21 PM
johng
Re: Problem: Intersection of 2-Sylow-groups
Hi,
I believe this is what you need to finish the proof that G is isomorphic to A5; namely that there are 5 Sylow 2 subgroups of G. After this determination, I left the rest of the proof to you:

Attachment 27950
• Apr 14th 2013, 04:36 PM
omlele
Re: Problem: Intersection of 2-Sylow-groups
Thank you!
That was all I need to conclude my proof.