2-Sylow subloops exist in finite Moufang loop

Statement
Suppose $$M$$ is a finite Moufang loop i.e., a Moufang loop whose order (the size of its underlying set) is finite.

Then, $$M$$ contains a 2-Sylow subloop, i.e., a subloop whose order is the largest power of 2 dividing the order of $$M$$.

Related facts

 * Sylow subloops exist for Sylow primes 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

Facts used

 * 1) uses::Sylow subloops exist for Sylow primes in finite Moufang loops

Proof
We combine Fact (1) and the observation that, from purely number-theoretic considerations, 2 can never divide $$(q^2 + 1)/\operatorname{gcd}(2,q - 1)$$ for any prime power $$q$$, so it is a Sylow prime for every finite Moufang loop.