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ASYMPTOTIC BOUNDS FOR THE SIZE OF Hom(A, GL n (q))

  • MICHAEL BATE (a1) and ALEC GULLON (a2)

Abstract

Fix an arbitrary finite group A of order a, and let X(n, q) denote the set of homomorphisms from A to the finite general linear group GL n (q). The size of X(n, q) is a polynomial in q. In this note, it is shown that generically this polynomial has degree n 2(1 – a −1) − ε r and leading coefficient mr , where ε r and mr are constants depending only on r := n mod a. We also present an algorithm for explicitly determining these constants.

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References

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1. Bate, M. E., The number of homomorphisms from finite groups to classical groups, J. Algebra 308 (2) (2007), 612628.
2. Chigira, N., Takegahara, Y. and Yoshida, T., On the number of homomorphisms from a finite group to a general linear group, J. Algebra 232 (1) (2000), 236254.
3. Curtis, C. W. and Reiner, I., Methods of representation theory – with applications to finite groups and orders, vol. I (Wiley, New York, 1981).
4. Lubotzky, A. and Magid, A. R., Varieties of representations of finitely generated groups, Mem. Amer. Math. Soc. 58 (336), (1985).
5. Liebeck, M. W. and Shalev, A., The number of homomorphisms from a finite group to a general linear group, Commun. Algebra 32 (2) (2004), 657661.
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Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
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