# Random bases for coprime linear groups

## TL;DR

This paper investigates the base size of coprime primitive linear groups, showing that almost all 11-tuples form bases, extending understanding of probabilistic base properties in permutation group theory.

## Contribution

It proves that for coprime primitive linear groups, almost all 11-tuples are bases, providing a probabilistic perspective on base sizes in this class of groups.

## Key findings

- Almost all 11-tuples are bases for coprime primitive linear groups.
- The probability that a random 11-tuple is a base tends to 1 as the dimension grows.
- The result extends previous bounds on base sizes for these groups.

## Abstract

The minimal base size $b(G)$ for a permutation group $G$, is a widely studied topic in the permutation group theory. Z. Halasi and K. Podoski proved that $b(G)\leq 2$ for coprime linear groups. Motivated by this result and the probabilistic method used by T. C. Burness, M. W. Liebeck and A. Shalev, it was asked by L. Pyber that for coprime linear groups $G\leq GL(V)$, whether there exists a constant $c$ such that the probability of that a random $c$-tuple is a base for $G$ tends to 1 as $|V|\to\infty$. While the answer to this question is negative in general, it is positive under the additional assumption that $G$ is even primitive as a linear group. In this paper, we show that almost all $11$-tuples are bases for coprime primitive linear groups.