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. 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.