The existence of Sylow 2-subloops in finite Moufang loops

Here we prove that if L is a finite Moufang loop then every 2-subloop of L is contained in some Sylow 2-subloop of L. Moreover, the number of Sylow 2-subloops of L is congruent to one modulo two. We will also show that if p is any prime congruent to one modulo four then there exist finite Moufang loops that do not contain any Sylow p-subloops.