The -character table does not determine a group
Authors:
Kenneth W. Johnson and Surinder K. Sehgal
Journal:
Proc. Amer. Math. Soc. 119 (1993), 1021-1027
MSC:
Primary 20C15
DOI:
https://doi.org/10.1090/S0002-9939-1993-1166358-X
MathSciNet review:
1166358
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Abstract | References | Similar Articles | Additional Information
Abstract: Frobenius had defined the group determinant of a group which is a polynomial in
variables. Formanek and Sibley have shown that the group determinant determines the group. Hoehnke and Johnson show that the
-characters (a part of the group determinant) determine the group. In this paper it is shown that the
-characters do not determine the group. If we start with a group
of a certain type then a group
with the same
-character table must form a Brauer pair with
. A complete description of such an
is available in Comm. Algebra 9 (1981), 627-640.
- [1] R. Brauer, Representations of finite groups, Lectures in Modern Mathematics (T. L. Saaty, ed.), vol. 1, Wiley, New York, 1963, pp. 133-175. MR 0178056 (31:2314)
- [2] G. Cliff and S. Sehgal, On groups having the same character tables, Comm. Algebra 9 (1981), 627-640. MR 608509 (82f:20023)
- [3] G. Frobenius, Über die Primfaktoren der Gruppendeterminante, Sber. Akad. Wiss. Berlin (1896), 1343-1382.
- [4] E. Formanek and D. Sibley, The group determinant determines the group, Proc. Amer. Math. Soc. 112 (1991), 649-656. MR 1062831 (91j:20021)
- [5]
H. J. Hoehnke and K. W. Johnson, The
-characters are sufficient for the group determinant, Proc. Ring Theory Conference (Barnaul, Siberia), 1991 (to appear). MR 1332286 (96c:20018)
- [6] B. Huppert, Zweifach transitive, auflosbäre Permutationgruppen, Math. Z. 68 (1957), 126-150. MR 0094386 (20:904)
- [7] K. W. Johnson, Latin square determinants, Algebraic, Extremal and Metric Combinatorics 1986, London Math. Soc. Lecture Notes Ser., vol. 131, Cambridge Univ. Press, London and New York, 1988, pp. 146-154. MR 1052664 (91h:05026)
- [8] -, On the group determinant, Math. Proc. Cambridge Philos. Soc. 109 (1991), 299-311. MR 1085397 (91j:20022)
- [9] R. Mansfield, A group determinant determines its group, preprint. MR 1123661 (93b:20018)
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Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1993-1166358-X
Keywords:
Group determinant,
character table,
Brauer pairs
Article copyright:
© Copyright 1993
American Mathematical Society