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2-Sylow subloops exist in finite Moufang loop

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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

Facts used

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.