# Irreducible linear subgroups generated by pairs of matrices with large irreducible submodules

## TL;DR

This paper investigates the properties of pairs of 'fat' elements in finite general linear groups, showing that most such pairs generate irreducible subgroups and analyzing their structural characteristics.

## Contribution

It provides probabilistic bounds on the likelihood that pairs of fat elements generate reducible subgroups and explores their action on large invariant subspaces.

## Key findings

- Most pairs of fat elements generate irreducible subgroups.
- The probability of generating a reducible subgroup is less than q^{-d+1}.
- Reducible subgroups act irreducibly on large subspaces and preserve fatness.

## Abstract

We call an element of a finite general linear group $ \textrm{GL}(d,q) $ \emph{fat} if it leaves invariant, and acts irreducibly on, a subspace of dimension greater than $d/2$. Fatness of an element can be decided efficiently in practice by testing whether its characteristic polynomial has an irreducible factor of degree greater than $d/2$. We show that for groups $G$ with $ \textrm{SL}(d,q) \leq G \leq \textrm{GL}(d,q) $ most pairs of fat elements from $G$ generate irreducible subgroups, namely we prove that the proportion of pairs of fat elements generating a reducible subgroup, in the set of all pairs in $ G \times G $, is less than $q^{-d+1}$. We also prove that the conditional probability to obtain a pair $(g_1,g_2)$ in $G \times G$ which generates a reducible subgroup, given that $g_1, g_2$ are fat elements, is less than $2q^{-d+1}$. Further, we show that any reducible subgroup generated by a pair of fat elements acts irreducibly on a subspace of dimension greater than $ d/2 $, and in the induced action the generating pair corresponds to a pair of fat elements.