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 $(q^2+1)/\mathrm{g}\mathrm{c}\mathrm{d}(2,q-1)$.

Then:

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

Related facts

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