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- W2009640192 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." @default.
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- W2009640192 date "2012-02-01" @default.
- W2009640192 modified "2023-10-07" @default.
- W2009640192 title "Irreducible linear subgroups generated by pairs of matrices with large irreducible submodules" @default.
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- W2009640192 doi "https://doi.org/10.1007/s00013-012-0359-1" @default.
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