Random generation of associative algebras

TL;DR

This paper investigates the probability that two randomly chosen elements generate finite and profinite associative algebras over finite fields, extending known results for simple algebras and exploring generation properties.

Contribution

It extends generation probability results to a broad class of finite associative algebras, providing bounds, growth estimates, and conditions for profinite algebra generation.

Findings

Probability approaches 1 for large simple algebras

Established optimal lower bounds for simple algebra generation

Linked profinite algebra generation to polynomial maximal subalgebra growth

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 $k=F_q$ be a finite field. Let $A$ be a finite dimensional, associative, unital algebra over $k$. Let $P(A)$ be the probability that two elements of $A$ chosen (uniformly and independently) at random will generate $A$ as a unital $k$-algebra. It is known that, if $A$ is simple, then $P(A) \to 1$ as $|A| \to \infty$. We extend this result to a large class of finite associative algebras. For $A$ simple, we find the optimal lower bound for $P(A)$ and we estimate the growth rate of $P(A)$ in terms of the minimal index $m(A)$ of any proper subalgebra of $A$. We also study the random generation of simple algebras $A$ 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 $A$ be a profinite algebra over $k$. We show that $A$ is positively finitely generated if and only if $A$ has polynomial maximal subalgebra growth. Related quantitative results are also established.