2-Sylow subloops exist in finite Moufang loop
Then, contains a 2-Sylow subloop, i.e., a subloop whose order is the largest power of 2 dividing the order of .
- 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
We combine Fact (1) and the observation that, from purely number-theoretic considerations, 2 can never divide for any prime power , so it is a Sylow prime for every finite Moufang loop.