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  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 109, Issue 2
  • March 1991, pp. 299-311

On the group determinant

  • K. W. Johnson (a1)
  • DOI:
  • Published online: 24 October 2008

The original motivation for the introduction by Frobenius of group characters for non-abelian groups was the problem of the factorization of the group determinant corresponding to a finite group G. The original papers are [5] and [6] and a good historical survey of the work is given in [7] and [8]. If G is of order n, the group matrix XG is defined to be the n×n matrix {xg, h} where xg, h = xghG. Here the xg, gG, represent variables. The group determinant ΘG is defined to be det(XG), and is thus a polynomial of degree n in the xg. This determinant is the same, up to sign, as that of the matrix obtained from the unbordered multiplication table of G by replacing each element g by xg. If there is no ambiguity ΘG will be written as Θ.

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[4]E. Formanek . The invariants of n×n matrices. In Invariant Theory (editor S. S. Koh ), Lecture Notes in Math. vol. 1278 (Springer-Verlag, 1987), pp. 1843.

[7]T. Hawkins . The origins of the theory of group characters. Arch. Hist. Exact Sci. 7 (1971), 142170.

[8]T. Hawkins . New light on Frobenius' creation of the theory of group characters. Arch. Hist. Exact Sci. 12 (1974), 217243.

[9]H.-J. Hoehnke . Über Beziehungen zwischen Probleme von H. Brandt aus der Theorie der Algebren und den Automorphismen der Normenform. Math. Nachr. 34 (1967), 229255.

[11]R. Sandling . The isomorphism problem for group rings: a survey. In Orders and Their Applications, Lecture Notes in Math. vol. 1142 (Springer-Verlag, 1985), pp. 256288.

[12]R. P. Stanley . Invariants of finite groups and their applications to combinatorics. Bull. Amer. Math. Soc. (N.S.) 1 (1979), 475511.

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