Feit–Thompson theorem
It is quite mysterious that every finite group of odd order is solvable. You see that odd factorial order is even number. Prime number is included in this theory. 3!/2=3 5!/2=60 7!/2=2520 9!/2=181440 11!/2=19958400 13!/2=3113510400 15!/2=653837184000 17!/2=177843714048000 19!/2=60822550204416000 21!/2=25545471085854720000 3,5,7,11,13,17,19 are prime numbers. G is a group, and P and Q are odd primes. (P (1+KQ) is the subgroups of G, so it is a divisor of ｜G｜ . K is an integer. Therefore, P^2 is divided by (1+KQ). ∴ (1+KQ)=1,P or P^2