# 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 .

## Related facts

### Analogues in other algebraic structures

- Sylow subgroups exist (analogue for finite groups)