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Integer group determinants for small groups

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Abstract

For every group of order at most 14 we determine the values taken by its group determinant when its variables are integers.

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References

  1. Apostol, T.M.: Resultants of cyclotomic polynomials. Proc. Amer. Math. Soc. 24, 457–462 (1970)

    Article  MathSciNet  Google Scholar 

  2. Bhargava, M., Hanke, J.: Universal quadratic forms and the 290-theorem, preprint

  3. Boerkoel, T., Pinner, C.: Minimal group determinants and the Lind-Lehmer problem for dihedral groups, to appear Acta Arith. arXiv:1802.07336

  4. Clem, S., Pinner, C.: The Lind Lehmer constant for 3-groups. Integers 18, A40 (2018)

    MathSciNet  MATH  Google Scholar 

  5. Conrad, K.: The origin of representation theory. Enseign. Math. 44(3–4), 361–392 (1998)

    MathSciNet  MATH  Google Scholar 

  6. Cox, D.A.: Primes of the form \(x^2+ny^2\), Fermat, Class Field Theory and Complex Multiplication. Wiley, New York (1989)

    MATH  Google Scholar 

  7. Dasbach, O., Lalín, M.: Mahler measure under variations of the base group. Forum Math. 21, 621–637 (2009)

    Article  MathSciNet  Google Scholar 

  8. Dickson, L.E.: Quatenary quadratic forms representing all integers. Amer. J. Math. 49, 39–56 (1927)

    Article  MathSciNet  Google Scholar 

  9. De Silva, D., Pinner, C.: The Lind-Lehmer constant for \({\mathbb{Z}}_p^n\). Proc. Amer. Math. Soc. 142(6), 1935–1941 (2014)

    Article  MathSciNet  Google Scholar 

  10. De Silva, D., Mossinghoff, M., Pigno, V., Pinner, C.: The Lind-Lehmer constant for certain \(p\)-groups. Math. Comp. https://doi.org/10.1090/mcom/3350

    Article  MathSciNet  Google Scholar 

  11. Formanek, E., Sibley, D.: The group determinant determines the group. Proc. Amer. Math. Soc. 112, 649–656 (1991)

    Article  MathSciNet  Google Scholar 

  12. Kaiblinger, N.: On the Lehmer constant of finite cyclic groups. Acta Arith. 142(1), 79–84 (2010)

    Article  MathSciNet  Google Scholar 

  13. Kaiblinger, N.: Progress on Olga Taussky-Todd’s circulant problem. Ramanujan J. 28(1), 45–60 (2012)

    Article  MathSciNet  Google Scholar 

  14. Lang, S.: Cyclotomic Fields I and II, Graduate Texts in Mathematics 121. Springer-Verlag, New York (1990)

    Google Scholar 

  15. Laquer, H.: Values of circulants with integer entries. In: Hoggatt, V.E. (ed.) A Collection of Manuscripts Related to the Fibonacci Sequence, pp. 212–217. Fibonacci Assoc., Santa Clara (1980)

    Google Scholar 

  16. Lehmer, D.H.: Factorization of certain cyclotomic functions. Ann. Math. 34(3), 461–479 (1933)

    Article  MathSciNet  Google Scholar 

  17. Lehmer, E.T.: A numerical function applied to cyclotomy. Bull. Amer. Math. Soc. 36, 291–298 (1930)

    Article  MathSciNet  Google Scholar 

  18. Lind, D.: Lehmer’s problem for compact abelian groups. Proc. Amer. Math. Soc. 133(5), 1411–1416 (2005)

    Article  MathSciNet  Google Scholar 

  19. Mossinghoff, M., Pigno, V., Pinner, C.: The Lind-Lehmer constant for \({\mathbb{Z}}_2^r \times {\mathbb{Z}}_4^s\), to appear Mosc. J. Comb. Number Theory. arXiv:1805.05450

  20. Newman, M.: On a problem suggested by Olga Taussky-Todd, Ill. J. Math. 24, 156–158 (1980)

    MathSciNet  MATH  Google Scholar 

  21. Newman, M.: Determinants of circulants of prime power order. Linear Multilinear Algebra 9, 187–191 (1980)

    Article  MathSciNet  Google Scholar 

  22. Pierce, T.A.: The numerical factors of the arithmetic forms \(\prod _{i=1}^n(1\pm \alpha _i^m)\). Ann. Math. 18(2), 53–64 (1916)

    Article  MathSciNet  Google Scholar 

  23. Pigno, V., Pinner, C.: The Lind-Lehmer constant for cyclic groups of order less than \(892,371,480\). Ramanujan J. 33(2), 295–300 (2014)

    Article  MathSciNet  Google Scholar 

  24. Pinner, C., Vipismakul, W.: The Lind-Lehmer constant for \({\mathbb{Z}}_{m} \times {\mathbb{Z}}^{n}_{p}\). Integers 16, A46 (2016)

    MATH  Google Scholar 

  25. Serre, J.-P.: Linear Representations of Finite Groups, Graduate Texts in Mathematics 42. Springer-Verlag, New York (1977)

    Book  Google Scholar 

  26. Vipismakul, W.: The stabilizer of the group determinant and bounds for Lehmer’s conjecture on finite abelian groups, Ph. D. Thesis, University of Texas at Austin, (2013)

  27. Washington, L.: Introduction to Cyclotomic Fields, GTM 83. Springer-Verlag, New York (1982)

    Book  Google Scholar 

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Acknowledgements

We are very grateful to the reviewer for reading the manuscript so carefully, and for suggesting numerous minor improvements. The second author also thanks the University of Edinburgh for the invitation to visit and the Edinburgh Mathematical Society for its financial support.

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Authors and Affiliations

  1. Department of Mathematics, Kansas State University, Manhattan, KS, 66506, USA

    Christopher Pinner

  2. School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Edinburgh, Scotland, EH9 3FD, UK

    Christopher Smyth

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  1. Christopher Pinner
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  2. Christopher Smyth
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Correspondence to Christopher Pinner.

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Pinner, C., Smyth, C. Integer group determinants for small groups. Ramanujan J 51, 421–453 (2020). https://doi.org/10.1007/s11139-018-0092-x

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  • DOI: https://doi.org/10.1007/s11139-018-0092-x

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