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

Original language | American English |
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Journal | Journal of the London Mathematical Society |

DOIs | |

State | Accepted/In press - 2023 |

### Bibliographical note

Funding Information:DS was supported by a Post‐Doctoral Fellowhip from ISF grant 686/17 of AS. AS was partially supported by ISF grant 686/17 and the Vinik Chair of mathematics which he holds.

Publisher Copyright:

© 2023 The Authors. Journal of the London Mathematical Society is copyright © London Mathematical Society.