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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The group determinant determines the group
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by Edward Formanek and David Sibley PDF
Proc. Amer. Math. Soc. 112 (1991), 649-656 Request permission

Abstract:

Let $G = \left \{ {{g_1}, \ldots ,{g_n}} \right \}$ be a finite group of order $n$, let $K$ be a field whose characteristic is prime to $n$, and let $\left \{ {{x_g}\left | {g \in G} \right .} \right \}$ be independent commuting variables over $K$. The group determinant of $G$ is the determinant of the $n \times n$ matrix $\left ( {{x_{{g_i}g_j^{ - 1}}}} \right )$. We show that two groups with the same group determinant are isomorphic.
References
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  • Walter Feit, The representation theory of finite groups, North-Holland Mathematical Library, vol. 25, North-Holland Publishing Co., Amsterdam-New York, 1982. MR 661045
  • F. G. Frobenius, Über die Darstellung der endlichen Gruppen durch lineare Substitutionen, Sitz. Kön. Preuss. Akad. Wiss. Berlin (1897), 944-1015; Band III of F. G. Frobenius-Gesammelte Abhandlungen, Springer-Verlag, Berlin, 1968, pp. 82-118.
  • K. W. Johnson, Latin square determinants, Algebraic, extremal and metric combinatorics, 1986 (Montreal, PQ, 1986) London Math. Soc. Lecture Note Ser., vol. 131, Cambridge Univ. Press, Cambridge, 1988, pp. 146–154. MR 1052664
  • —, On the group determinant, preprint.
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Additional Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 112 (1991), 649-656
  • MSC: Primary 20C15; Secondary 15A15, 20C20
  • DOI: https://doi.org/10.1090/S0002-9939-1991-1062831-1
  • MathSciNet review: 1062831