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Random generation of associative algebras

Damian Sercombe, Aner Shalev

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.

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.

Paper Structure

This paper contains 10 sections, 37 theorems, 104 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['randomgeneration']} and Corollary \ref{['randomgenerationcor']}
  4. Proof of Theorem \ref{['minP']}
  5. Proof of Theorem \ref{['estimateprob']}
  6. Proof of Theorem \ref{['nilpotentrandomgeneration']} and Corollary \ref{['nilpotentrandomgenerationcor']}
  7. Proof of Theorem \ref{['polyrandomgeneration']}
  8. Proof of Theorem \ref{['rankrandomgeneration']} and Corollary \ref{['rankrandomgenerationcor']}
  9. The minimal number of generators
  10. Positively finitely generated algebras

Key Result

Theorem 1.1

Fix constants $1<c <q$ and $\lambda>0$. Let $A$ be a finite algebra, say $A = (\prod_{i=1}^r M_{n_i}(q^{m_i})) \oplus J(A)$, that is bounded by $(c,\lambda)$. Denote $n:=\min_{i=1,...,r}\{n_i\}$ and $m:=\min_{i=1,...,r}\{m_i\}$. Then $P(A) \rightarrow 1$ as $n \rightarrow \infty$, as $m \rightarrow

Theorems & Definitions (65)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Theorem 1.10
  • ...and 55 more