Random generation of associative algebras

Abstract

There has been considerable interest in recent decades in questions of random generation of finite and profinite groups and finite simple groups in particular. In this paper, we study similar notions for finite and profinite associative algebras. Let be a finite field. Let be a finite‐dimensional, associative, unital algebra over . Let be the probability that two elements of chosen (uniformly and independently) at random will generate as a unital ‐algebra. It is known that if is simple, then as . We extend this result to a large class of finite associative algebras. For simple, we find the optimal lower bound for and we estimate the growth rate of in terms of the minimal index of any proper subalgebra of . We also study the random generation of simple algebras by two elements that have a given characteristic polynomial (resp. a given rank). In addition, we bound above and below the minimal number of generators of general finite algebras. Finally, we let be a profinite algebra over . We show that is positively finitely generated if and only if has polynomial maximal subalgebra growth. Related quantitative results are also established.