Random generators of given orders and the smallest simple Moufang loop

Abstract

The probability that m randomly chosen elements of a finite power associative loop $$ \mathcal{C} $$ have prescribed orders and generate $$ \mathcal{C} $$ is calculated in terms of certain constants Γ related to the action of Aut( $$ \mathcal{C} $$ ) on the subloop lattice of $$ \mathcal{C} $$ . As an illustration, all meaningful probabilities of random generation by elements of given orders are found for the smallest nonassociative simple Moufang loop.