The 2-character table does not determine a group

Johnson, Kenneth W.; Sehgal, Surinder K.

doi:10.1090/S0002-9939-1993-1166358-X

The $2$-character table does not determine a group
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by Kenneth W. Johnson and Surinder K. Sehgal PDF

Proc. Amer. Math. Soc. 119 (1993), 1021-1027 Request permission

Abstract:

Frobenius had defined the group determinant of a group $G$ which is a polynomial in $n = |G|$ variables. Formanek and Sibley have shown that the group determinant determines the group. Hoehnke and Johnson show that the $3$-characters (a part of the group determinant) determine the group. In this paper it is shown that the $2$-characters do not determine the group. If we start with a group $G$ of a certain type then a group $H$ with the same $2$-character table must form a Brauer pair with $G$. A complete description of such an $H$ is available in Comm. Algebra 9 (1981), 627-640.

References

Richard Brauer, Representations of finite groups, Lectures on Modern Mathematics, Vol. I, Wiley, New York, 1963, pp. 133–175. MR 0178056
Gerald H. Cliff and Surinder K. Sehgal, On groups having the same character tables, Comm. Algebra 9 (1981), no. 6, 627–640. MR 608509, DOI 10.1080/00927878108822606

Über die Primfaktoren der Gruppendeterminante

Edward Formanek and David Sibley, The group determinant determines the group, Proc. Amer. Math. Soc. 112 (1991), no. 3, 649–656. MR 1062831, DOI 10.1090/S0002-9939-1991-1062831-1
H.-J. Hoehnke and K. W. Johnson, $3$-characters are sufficient for the group determinant, Second International Conference on Algebra (Barnaul, 1991) Contemp. Math., vol. 184, Amer. Math. Soc., Providence, RI, 1995, pp. 193–206. MR 1332286, DOI 10.1090/conm/184/02115
Bertram Huppert, Zweifach transitive, auflösbare Permutationsgruppen, Math. Z. 68 (1957), 126–150 (German). MR 94386, DOI 10.1007/BF01160336
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
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Richard Mansfield, A group determinant determines its group, Proc. Amer. Math. Soc. 116 (1992), no. 4, 939–941. MR 1123661, DOI 10.1090/S0002-9939-1992-1123661-6