Sylow subloops exist for Sylow primes in finite Moufang loop

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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).


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.

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