On Non-Congruence Subgroups of the Analogue of the Modular Group in Characteristic p

  • A. W. Mason
Part of the Developments in Mathematics book series (DEVM, volume 10)


Let k[t] be the polynomial ring over a finite field k. The group SL 2(k[t]) is often referred to as the analogue, in characteristic p, of the classical modular group SL 2(ℤ), where ℤ is the ring of rational integers. It is well-known that the smallest index of a non-congruence subgroup of SL2(ℤ) is 7. Here we compute this index for SL 2(k[t]). (In all but 6 cases it turns out to be 1 + q, where g is the order of k.)

Key words

Special linear group polynomial ring non-congruence subgroup minimal index 


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  1. 1.
    H. Bass, J. Milnor, and J.-P. Serre, “Solution of the congruence subgroup problem for SL 2(n > 3) and Sp 2(n > 2),” Inst. Hautes Etudes Sci. Publ. Math. 33 (1967), 59–137.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    E Grunewald and J. Schwermer, “On the concept of level for subgroups of SL 2 over orders of arithmetic type,” Israel J. Math. 114 (1999), 205–220.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    B. Liehl, “On the groups SL 2 over orders of arithmetic type,” J. Reine Angew. Math. 323 (1981), 153–171.MathSciNetzbMATHGoogle Scholar
  4. 4.
    A.W. Mason, “Normal subgroups of SL 2(k[t]) with or without free quotients,” J. Algebra 150 (1992), 281–295.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    A.W. Mason and S.J. Pride, “Normal subgroups of prescribed order and zero level of the modular group and related groups,” J. London Math. Soc. 42 (2) (1990), 465–474.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    A.W. Mason and Andreas Schweizer, “The minimum index of a non-congruence subgroup of SL 2 over an arithmetic domain,” Israel J. Math. 133 (2003), 29–44.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    J.-P. Serre, “Le problème des groupes de congruence pour SL 2, ” Ann. of Math. 92 (1970), 489–527.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    M. Suzuki, Group Theory I, Springer-Verlag, 1982.Google Scholar
  9. 9.
    L.N. Vaserstein, “On the group SL 2 over Dedekind rings of arithmetic type,” Math. USSR-Sb 18 (1972), 321–332.CrossRefGoogle Scholar
  10. 10.
    A. Weil, “On the analogue of the modular group in characteristic p, in functional analysis and related fields,” in Proc. Conf. for M. Stone, Univ. Chicago, 1968, Springer-Verlag, 1970, pp. 211–223.Google Scholar
  11. 11.
    K. Wohlfahrt, “An extension of F. Klein’s level concept,” Illinois J. Math. 8 (1964), 529–535.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

  • A. W. Mason
    • 1
  1. 1.Department of MathematicsUniversity of GlasgowGlasgowScotland, UK

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