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The group determinant determines the group

Authors: Edward Formanek and David Sibley
Journal: Proc. Amer. Math. Soc. 112 (1991), 649-656
MSC: Primary 20C15; Secondary 15A15, 20C20
MathSciNet review: 1062831
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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\vert {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 [Enhancements On Off] (What's this?)

  • [1] J. Dieudonne, Sur une generalisation du groupe orthogonal a quatre variables, Arch. Math. 1 (1949), 282-287. MR 0029360 (10:586l)
  • [2] W. Feit, The representation theory of finite groups, North-Holland, Amsterdam, 1982. MR 661045 (83g:20001)
  • [3] 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.
  • [4] K. W. Johnson, Latin square determinants, Algebraic, Extremal, and Metric Combinatorics (M.-M. Deza, P. Frankl, and I. G. Rosenberg, eds.), pp. 146-154; London Math. Soc. Lecture Notes, No. 131, Cambridge Univ. Press, Cambridge, 1988. MR 1052664 (91h:05026)
  • [5] -, On the group determinant, preprint.

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Keywords: Group determinant
Article copyright: © Copyright 1991 American Mathematical Society

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