Sylow subloops exist for Sylow primes in finite Moufang loop

Definition
Suppose $$M$$ is a finite Moufang loop (i.e., a Moufang loop whose underlying set is finite) and $$p$$ is a prime number dividing the order of $$M$$.

Call $$p$$ a Sylow prime for $$M$$ if the following is true: there is a composition series of $$M$$ that does not contain any simple composition factors that are isomorphic to a Paige loop over a field of size $$q$$ for which $$p$$ divides $$(q^2 + 1)/\operatorname{gcd}(2,q - 1)$$.

Then:

$$p$$ is a Sylow prime for $$M$$ $$\iff$$ $$M$$ has a $$p$$-Sylow subloop, i.e., a subloop whose order is the largest power of $$p$$ dividing the order of $$M$$.

Analogues in other algebraic structures

 * Sylow subgroups exist (analogue for finite groups)

Corollaries

 * 2-Sylow subloops exist in finite Moufang loops
 * 3-Sylow subloops exist in finite Moufang loops
 * Sylow subloops exist in finite Moufang loops of group type
 * Hall subloops exist in finite solvable Moufang loops