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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 ofdimension greater than $d/2$. Fatness of an element can be decided efficientlyin practice by testing whether its characteristic polynomial has an irreduciblefactor 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 ofpairs of fat elements generating a reducible subgroup, in the set of all pairsin $ G \times G $, is less than $q^{-d+1}$. We also prove that the conditionalprobability to obtain a pair $(g_1,g_2)$ in $G \times G$ which generates areducible 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 pairof 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 fatelements.

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