More results on E loops (Q1061238)

The author establishes stringent conditions under which a finite Moufang loop is a group. These are as follows. 1. For any three elements of the loop, x, y, and z, the associator (x,y,z) is contained in the centre and nucleus of the subloop generated by x, y and z. 2. For each prime p dividing the order of the loop a Sylow-p-subloop exists. 3. If the order of the loop is \(2^{n_ 1}3^{n_ 2}p_ 1^{m_ 1}p_ 2^{m_ 2}...p_ k^{m_ k}\) where the \(p_ i\) are distinct primes \(\geq 5\), then \(n_ i<4\), \(m_ i<5\).