# Sylow subloops exist for Sylow primes in finite Moufang loop

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## Definition

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

Call  a Sylow prime for  if the following is true: there is a composition series of  that does not contain any simple composition factors that are isomorphic to a Paige loop over a field of size  for which  divides .

Then:

 is a Sylow prime for    has a -Sylow subloop, i.e., a subloop whose order is the largest power of  dividing the order of .