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On Brauer’s height 0 conjecture

Published online by Cambridge University Press:  22 January 2016

T.R. Berger  and
R. Knörr
T.R. Berger
Affiliation:
Department of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA
R. Knörr
Affiliation:
Department of Mathematics, University of Essen, 4300 Essen, West, Germany
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R. Brauer not only laid the foundations of modular representation theory of finite groups, he also raised a number of questions and made conjectures (see [1], [2] for instance) which since then have attracted the interest of many people working in the field and continue to guide the research efforts to a good extent. One of these is known as the “Height zero conjecture”. It may be stated as follows:

CONJECTURE. Let B be a p-block of the finite group G. All irreducible ordinary characters of G belonging to B are of height 0 if and only if a defect group of B is abelian.

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 109 , March 1988 , pp. 109 - 116
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1988

References

[1] Brauer, R., Number theoretical investigations on groups of finite order, Proc. Internat. Symp. on Algebraical Number Theory, Tokyo and Nikko 1955, 55–62.Google Scholar
[1] Brauer, R., On some conjectures concerning finite simple groups, Studies in Math. Analysis and Related Topics, Stanford Univ. Press 1962, 56–61.Google Scholar
[1] Brauer, R., Representations of finite groups, Lectures on Modern Mathematics, Vol. 1, Wiley, New York 1963, 133–175.Google Scholar
[1] Cline, E., Stable Clifford Theory, J. Algebra, 22 (1972), 350–364.Google Scholar
[1] Feit, W., The representation theory of finite groups, North-Holland, Amsterdam 1982.Google Scholar
[1] Fong, P., Some properties of characters of finite solvable groups, Bull. Amer. Math. Soc, 66 (1960), 116–117.Google Scholar
[1] Fong, P. and Srinivasan, B., The blocks of finite general linear and unitary groups, Invent, math., 69 (1982), 109–153.Google Scholar
[1] Gluck, D. and Wolf, T. R., Defect groups and character heights in blocks of solvable groups, II. Preprint 1983.CrossRefGoogle Scholar
[1] Gluck, D. and Wolf, T. R., Brauer’s height conjecture for p-solvable groups, preprint 1983.Google Scholar
[10] Isaacs, I. M., Character theory of finite groups, Academic Press, New York 1976.Google Scholar
[11] Knörr, R., Blocks, vertices and normal subgroups, Math. Z., 148 (1976), 53–60.CrossRefGoogle Scholar
[12] Knörr, R., On the vertices of irreducible modules, Ann. of Math., (2) 110 (1979), 487–499.Google Scholar
[13] Michler, G. and Olsson, J. B., Character correspondences in finite general linear, unitary and symmetric groups, preprint 1983.Google Scholar
[14] Reynolds, W. F., Blocks and normal subgroups of finite groups, Nagoya Math. J., 22 (1963), 15–32.Google Scholar
[15] Wolf, T. R., Defect groups and character heights in blocks of solvable groups, J. Algebra, 72 (1981), 183–209.Google Scholar