Archiv der Mathematik

, Volume 98, Issue 2, pp 105–114 | Cite as

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

  • Alice C. Niemeyer
  • Sabina B. Pannek
  • Cheryl E. Praeger


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 


General linear group Proportions of elements Large irreducible subspaces 


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Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  • Alice C. Niemeyer
    • 1
  • Sabina B. Pannek
    • 2
  • Cheryl E. Praeger
    • 1
  1. 1.School of Mathematics and Statistics M019The University of Western AustraliaNedlandsAustralia
  2. 2.Lehrstuhl D fur MathematikRWTH AachenAachenGermany

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