# 2-Sylow subloops exist in finite Moufang loop

From Groupprops

## Contents

## Statement

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

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

## 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

## Proof

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.