# The 2-Characters of a Group and the Weak Cayley Table

## Abstract

Group character theory has many applications, but it is not easy to characterize the information contained in the character table of a group, or give concise information which in addition to that in the character table determines a group. The first part of this chapter examines the extra information which is obtained if the irreducible 2-characters of a group are known. It is shown that this is equivalent to the knowledge of the “weak Cayley table” (WCT) of the group *G* . A list of properties which are determined by the WCT and not by the character table is obtained, but non-isomorphic groups may have the same WCT. This work gives insight into why the problems in R. Brauer, Representations of finite groups, in “Lectures in Modern Mathematics, Vol. I,” T.L. Saaty (ed.), Wiley, New York, NY, 1963, 133–175, have often been hard to answer.

There is a discussion of the work of Humphries on \(\mathcal {W}(G)\) the group of weak Cayley table isomorphisms (WCTI’s). This group contains the a subgroup \(\mathcal {W}_{0}(G)\) generated by *Aut*(*G*) and the anti-automorphism *g* → *g*^{−1}. \(\mathcal {W}(G)\) is said to be trivial if \(\mathcal {W}(G)=\mathcal {W}_{0}(G)\). For some groups, such as the symmetric groups and dihedral groups \(\mathcal {W}(G)\), is trivial, but there are many examples where \(\mathcal {W}(G)\) is much larger than \(\mathcal {W}_{0}(G)\). The discussion is not limited to the finite case.

The category of “weak morphisms” between groups is described. These are weaker than ordinary homomorphisms but preserve more structure than maps which preserve the character table. A crossed product condition on these morphisms is needed to make this category associative. This category has some interesting properties.

## References

- 30.R. Brauer, in
*Representations of Finite Groups*, ed. by T.L. Saaty. Lectures in Modern Mathematics, vol. I (Wiley, New York, 1963), pp. 133–175Google Scholar - 48.A.R. Camina, Some conditions which almost characterize Frobenius groups. Israel J. Math.
**31**, 153–160 (1978)MathSciNetCrossRefGoogle Scholar - 56.G.Y. Chen, A new characterization of finite simple groups. Chin. Sci. Bull.
**40**, 446–450 (1995)MathSciNetzbMATHGoogle Scholar - 60.D. Chillag, A. Mann, C.M. Scoppola, Generalized Frobenius groups II. Israel J. Math.
**62**, 269–282 (1988)MathSciNetCrossRefGoogle Scholar - 67.R.H. Crowell, The derived group of a permutation representation. Adv. Math.
**53**, 99–124 (1984)MathSciNetCrossRefGoogle Scholar - 70.E.C. Dade, E, Answer to a question of R. Brauer. J. Algebra
**1**, 1–4 (1964)CrossRefGoogle Scholar - 71.E.C. Dade, M.K. Yadav, Finite groups with many product conjugacy classes. Israel J.Math
**154**, 29–49 (2006)MathSciNetCrossRefGoogle Scholar - 78.J. Dénes, A.D. Keedwell,
*Latin Squares and Their Applications*(Academic, New York, 1974)zbMATHGoogle Scholar - 151.S.P. Humphries, Weak Cayley table groups. J. Algebra
**216**, 135–158 (1999)MathSciNetCrossRefGoogle Scholar - 152.S.P. Humphries, K.W. Johnson, Fusions of character tables and Schur rings of abelian groups. Commun. Algebra
**36**, 1437–1460 (2008)MathSciNetCrossRefGoogle Scholar - 153.S.P. Humphries, K.W. Johnson, Fusions of character tables. II. p-groups. Commun. Algebra
**37**, 4296–4315 (2009)MathSciNetCrossRefGoogle Scholar - 158.B. Huppert,
*Endliche Gruppen. I*. Die Grundlehren der Mathematischen Wissenschaften, vol. 134 (Springer, Berlin, 1967)Google Scholar - 163.N. Ito,
*Lectures on Frobenius and Zassenhaus Groups*(University of Illinois at Chicago, 1969)Google Scholar - 169.K.W. Johnson, On the group determinant. Math. Proc. Camb. Philos. Soc.
**109**, 299–311 (1991)MathSciNetCrossRefGoogle Scholar - 173.K.W. Johnson, S.K. Sehgal, The 2-character table is not sufficient to determine a group. Proc. Am. Math. Soc.
**119**, 1021–1027 (1993)zbMATHGoogle Scholar - 174.K.W. Johnson, S.K. Sehgal, The 2-characters of a group and the group determinant. Eur. J. Comb.
**16**, 623–631 (1995)MathSciNetCrossRefGoogle Scholar - 180.K.W. Johnson, J.D.H. Smith, On the category of weak Cayley table morphisms between groups. Sel. Math. (N.S.)
**13**, 57–67 (2007)MathSciNetCrossRefGoogle Scholar - 184.K.W. Johnson, S. Mattarei, S.K. Sehgal, Weak Cayley tables. J. Lond. Math. Soc.
**61**, 395–411 (2000)MathSciNetCrossRefGoogle Scholar - 193.W. Kimmerle, On the characterization of finite groups by characters, in
*The Atlas of Finite Groups: Ten Years on (Birmingham, 1995)*. London Mathematical Society Lecture Note Series, vol. 249 (Cambridge University Press, Cambridge, 1998), pp. 119–138Google Scholar - 194.W. Kimmerle, K.W. Roggenkamp, Non-isomorphic groups with isomorphic spectral tables and Burnside matrices. Chin. Ann. Math. Ser. B
**15**, 273–282 (1994)MathSciNetzbMATHGoogle Scholar - 195.W. Kimmerle, R. Sandling, Group-theoretic and group ring-theoretic determination of certain Sylow and Hall subgroups and the resolution of a question of R. Brauer. J. Algebra
**171**, 329–346 (1995)MathSciNetCrossRefGoogle Scholar - 211.A. Mann, C.M. Scoppola, On p-groups of Frobenius type. Arch. Math. (Basel)
**56**, 320–332 (1991)MathSciNetCrossRefGoogle Scholar - 213.J. McKay, D. Sibley, Brauer pairs with the same 2-characters. PreprintGoogle Scholar
- 216.S. Mattarei, Character tables and metabelian groups. J. Lond. Math. Soc. 46(2), 92–100 (1992)MathSciNetCrossRefGoogle Scholar
- 217.S. Mattarei, Retrieving information about a group from its character table, Ph.D Thesis, University of Warwick, 1992Google Scholar
- 218.S. Mattarei, An example of p-groups with identical character tables and different derived lengths. Arch. Math.
**62**, 12–20 (1994)MathSciNetCrossRefGoogle Scholar - 219.S. Mattarei, Retrieving information about a group from its character degrees or from its class sizes. Proc. Am. Math. Soc. 134, 2189–2195 (2006)MathSciNetCrossRefGoogle Scholar
- 264.J.D.H. Smith, A.B. Romanowska,
*Post-Modern Algebra*(Wiley, New York, 1999)CrossRefGoogle Scholar