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Levchuk's Problem on Diagonal Matrices in SL_n(q) (Kourovka Notebook 7.27): An Affirmative Answer as a Corollary of Theorems of Fulman and Guralnick

Year: 2026
Type: preprint
Abstract: V. M. Levchuk asked in 1980 (Kourovka Notebook, Problem 7.27) whether, for all sufficiently large q, the group SL_n(q) contains a diagonal matrix lying in no proper irreducible subgroup except block-monomial ones. We show that the answer is yes for every n, as a short corollary of density theorems of Fulman and Guralnick; it is not presented as a new theorem. For fixed n and q → ∞, almost all regular diagonal matrices d of SL_n(q) have the property that every proper subgroup containing d that is irreducible over F_q permutes the summands of a decomposition of F_q^n into coordinate subspaces. The deduction combines the density theorems with an elementary count of the classes of regular diagonal matrices, a description of the maximal subgroups of maximal rank, and an eigenvalue argument excluding extension field subgroups; the bound for q is not explicit. For n = 2, Dickson's list of subgroups gives the complete answer: a diagonal matrix with this property exists if and only if q ∈ {4, 8} or q ≥ 13 (if irreducibility is meant absolutely, also for q = 2), and diag(w, w⁻¹) works for every generator w of F_q^* when q ≥ 13. No printed statement of the general answer was found, and no priority is claimed. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: KOU-7.27 (Kourovka Notebook, Problem 7.27, V. M. Levchuk, 7th issue, 1980).
DOI: 10.5281/zenodo.23445979
Language: English
Open Access status: green
OpenAlex ID: W7222877288
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