Generators of Nonassociative Simple Moufang Loops over Finite Prime Fields

The first class of nonassociative simple Moufang loops was discovered by L. Paige in 1956 [9], who investigated Zorn’s and Albert’s construction of simple alternative rings. M. Liebeck proved in 1987 [7] that there are no other finite nonassociative simple Moufang loops. We can briefly describe the class as follows: For every finite field F, there is exactly one simple Moufang loop. Recall Zorn’s multiplication ( a α β b )( c γ δ d ) = ( ac + α · δ aγ + αd− β × δ βc + bδ + α× γ β · γ + bd ) ,