×

On a minimal counterexample to the Alperin-McKay conjecture. (English) Zbl 1237.20011

From the text: We prove the following Theorem. For a minimal counterexample \((G,B)\) to the Alperin-McKay conjecture, the following holds. (i) \(O_p(G)\) and \(O_{p'}(G)\) are both central in \(G\). In particular, the Fitting subgroup of \(G\) is a central subgroup of \(G\). (ii) \(G\) has a unique \(G\)-conjugacy class of components.

MSC:

20C20 Modular representations and characters
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] J. L. Alperin, The main problem of block theory, in Proceedings of the Conference on Finite Groups (Univ. Utah, Park City, Utah, 1975) , 341-356, Academic Press, New York, 1976. · Zbl 0366.20012
[2] E. C. Dade, Counting characters in blocks, II, J. Reine Angew. Math. 448 (1994), 97-190. · Zbl 0790.20020 · doi:10.1515/crll.1994.448.97
[3] C. W. Eaton and G. R. Robinson, On a minimal counterexample to Dade’s projective conjecture, J. Algebra 249 (2002), no. 2, 453-462. · Zbl 1005.20013 · doi:10.1006/jabr.2001.8989
[4] M. E. Harris and R. Knörr, Brauer correspondence for covering blocks of finite groups, Comm. Algebra 13 (1985), no. 5, 1213-1218. · Zbl 0561.20002 · doi:10.1080/00927878508823213
[5] I. M. Isaacs, G. Malle and G. Navarro, A reduction theorem for the McKay conjecture, Invent. Math. 170 (2007), no. 1, 33-101. · Zbl 1138.20010 · doi:10.1007/s00222-007-0057-y
[6] B. Külshammer and L. Puig, Extensions of nilpotent blocks, Invent. Math. 102 (1990), no. 1, 17-71. · Zbl 0739.20003 · doi:10.1007/BF01233419
[7] M. Murai, A remark on the Alperin-McKay conjecture, J. Math. Kyoto Univ. 44 (2004), no. 2, 245-254. · Zbl 1086.20007
[8] H. Nagao and Y. Tsushima, Representations of finite groups , Academic Press, New York, 1989. · Zbl 0673.20002
[9] G. R. Robinson, Dade’s projective conjecture for \(p\)-solvable groups, J. Algebra 229 (2000), no. 1, 234-248. · Zbl 0955.20006 · doi:10.1006/jabr.2000.8307
[10] B. Späth, A reduction theorem for the Alperin-McKay conjecture, to appear in J. Reine Angew. Math.
[11] J. Thévenaz, \(G\)-algebras and modular representation theory , Oxford Mathematical Monographs, Oxford Univ. Press, New York, 1995. · Zbl 0837.20015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.