In his Notes on Group Theory, 2019 edition (http://pdvpmtasgaon.edu.in/uploads/dptmaths/AnotesofGroupTheoryByMarkReeder.pdf p. 83 and ff.) Mark Reeder gives a proof of the non-existence of simple groups of order 720.
Here : (No simple group of order 720 ) I asked for a clarification about a statement at the beginning of the proof and I got it. (Thanks again to David A. Craven.)
I have another problem, perhaps more serious, at the end of the proof. M. Reeder proves that if $G$ is a simple group of order 720, if $s$ is an involution (element of order 2) of $G$, then $C_{G}(s)$ is a Sylow 2-subgroup of $G$ and that if $s$ and $t$ are two differents involutions of $G$, then $C_{G}(s)$ and $C_{G}(t)$ are different. So far, so good. A little further, M. Reeder says : "Since $s$ is contained in just one Sylow 2-subgroup, namely $C_{G}(s)$ (...)". I don't understand how that can be deduced from what precedes.
I must say that I am skeptical about the possibility of proving this in the frame of M. Reeder's proof, for the following reason. After proving that (for a simple group $G$ of order 720), $G$ has exactly ten Sylow 3-subgroups, that these Sylow 3-subgroups are non-cyclic and intersect pairwise trivially, M. Reeder seems to derive a contradiction from only the following properties :
s1 : $G$ is a group (I don't say "simple group") of order 720 ;
s2 : $G$ has exactly ten Sylow 3-subgroups ;
s3 : these Sylow 3-subgroups are non-cyclic ;
s4 : these Sylow 3-subgroups intersect pairwise trivially ;
s5 : $G$ has no element of order 6 ;
s6 : every involution of $G$ normalizes at least two Sylow 3-subgroups of $G$ ;
s7 : $G$ has several Sylow 2-subgroups.
(M. Reeder uses another property to prove that some given subgroups of $G$ are isomorphic to $Q_{8}$, but he notes that this fact plays no real role.)
Now, properties s1 to s7 are not contradictory, because the group $M_{10}$ possesses them. (By the way, properties s1 to s7 are not independent : s3 can be deduced from s1, s2 and s5, using the N/C lemma.)
My question is : do you see how the statement "$s$ is contained in just one Sylow 2-subgroup, namely $C_{G}(s)$" can be proved in the frame of Mark Reeder's proof ? Thanks in advance.
Note : I wrote up M. Reeder's proof by breaking it down into simple elements and indicating, for each element, the minimum hypotheses. If anyone is interested, I can post this work here. I can also post a proof that $M_{10}$ possesses properties s1 to s7.