I need some direction with some of the following:
Let be a simple group
1. show that does not have sub-groups of index 3 or 4.
2. show that if are 2 different 2-sylow subgroups of such that then is a subgroup of index 5.
3. show that if for every 2-sylow subgroups the intersection is trivial, . then there are exactly five 2-sylow subgroups.
4. show that is isomorphic to .
I was able to show 1,2.
with 3, I am having trouble. using sylow's 3rd theorem I can show that the number of 2-sylow subgroups can be 1,3,5 or 15. I can eliminate the possibility of 1 (contradiction to simple group) and the possibility of 15 (too many elements considering the other p-sylow subgroups). but I can't eliminate the possibility of 3.
I need direction as weel on how to start 4.