The 1-, 2-, and 3-characters determine a group

Hoehnke, H.-J.; Johnson, K. W.

doi:10.1090/S0273-0979-1992-00302-6

The 1-, 2-, and 3-characters determine a group
HTML articles powered by AMS MathViewer

by H.-J. Hoehnke and K. W. Johnson PDF

Bull. Amer. Math. Soc. 27 (1992), 243-245 Request permission

Abstract:

A set of invariants for a finite group is described. These arise naturally from Frobenius’ early work on the group determinant and provide an answer to a question of Brauer. Whereas it is well known that the ordinary character table of a group does not determine the group uniquely, it is a consequence of the results presented here that a group is determined uniquely by its "3-character" table.

References

H. Brandt, Idealtheorie in Quaternionenalgebren, Math. Ann. 99 (1928), no. 1, 1–29 (German). MR 1512436, DOI 10.1007/BF01459083

Algèbre

Richard Brauer, Representations of finite groups, Lectures on Modern Mathematics, Vol. I, Wiley, New York, 1963, pp. 133–175. MR 0178056
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

Über die Primfaktoren der Gruppendeterminante

Über die Darstellung der endlichen Gruppen durch lineare Substitutionen

Thomas Hawkins, The origins of the theory of group characters, Arch. History Exact Sci. 7 (1971), no. 2, 142–170. MR 1554141, DOI 10.1007/BF00411808
Thomas Hawkins, New light on Frobenius’ creation of the theory of group characters, Arch. History Exact Sci. 12 (1974), 217–243. MR 0446837, DOI 10.1007/BF00357245
Hans-Jürgen Hoehnke, Über komponierbare Formen und konkordante hyperkomplexe Grössen, Math. Z. 70 (1958), 1–12 (German). MR 99353, DOI 10.1007/BF01558573
Hans-Jürgen Hoehnke, Über Beziehungen zwischen Problemen von H. Brandt aus der Theorie der Algebren und den Automorphismen der Normenform, Math. Nachr. 34 (1967), 229–255 (German). MR 218392, DOI 10.1002/mana.19670340115
K. W. Johnson, On the group determinant, Math. Proc. Cambridge Philos. Soc. 109 (1991), no. 2, 299–311. MR 1085397, DOI 10.1017/S0305004100069760
Kenneth W. Johnson and Surinder K. Sehgal, The $2$-character table does not determine a group, Proc. Amer. Math. Soc. 119 (1993), no. 4, 1021–1027. MR 1166358, DOI 10.1090/S0002-9939-1993-1166358-X
Moshe Roitman, A complete set of invariants for finite groups and other results, Adv. in Math. 41 (1981), no. 3, 301–311. MR 630923, DOI 10.1016/0001-8708(81)90023-2