Abstract
We consider the problem of classifying all finite basis-transitive matroids and reduce it to the classification of the finite basis-transitive and point-primitive simple matroids (or geometric lattices, or dimensional linear spaces). Our main result shows how a basis- and point-transitive simple matroid is decomposed into a so-called supersum. In particular each block of imprimitivity bears the structure of two closely related simple matroids, and the set of blocks of imprimitivity bears the structure of a point- and basis-transitive matroid.
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Delandtsheer, A., Li, H. Basis-Transitive Matroids. Journal of Algebraic Combinatorics 3, 285–290 (1994). https://doi.org/10.1023/A:1022411901450
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DOI: https://doi.org/10.1023/A:1022411901450