Irreducible linear subgroups generated by pairs of matrices with large irreducible submodules
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We call an element of a finite general linear group GL(d, q) 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 SL(d, q) ≤ G ≤ 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 × G, is less than q −d+1. We also prove that the conditional probability to obtain a pair (g 1, g 2) in G × 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.
Mathematics Subject Classification (2010)Primary 20G40 Secondary 20P05
KeywordsGeneral linear group Proportions of elements Large irreducible subspaces
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- 1.Peter J. Cameron: Combinatorics: topics, techniques, algorithms. Cambridge University Press, Cambridge (1994)Google Scholar
- 3.Charles R. Leedham-Green, The computational matrix group project, In: Groups and computation, III (Columbus, OH, 1999), volume 8 of Ohio State Univ. Math. Res. Inst. Publ., pp. 229–247, de Gruyter, Berlin, 2001.Google Scholar
- 6.R. A. Parker, The computer calculation of modular characters (the meat-axe), Computational group theory (Durham, 1982), pp. 267–274. Academic Press, London, 1984.Google Scholar
- 7.Donald E. Taylor: The geometry of the classical groups, volume 9 of Sigma Series in Pure Mathematics. Heldermann Verlag, Berlin (1992)Google Scholar