Abstract
Let p be an odd prime number. In this paper, we characterize the nonabelian composition factors of a finite group with odd p-Sylow automizers, and then prove that the McKay conjecture, the Alperin weight conjecture, and the Alperin–McKay conjecture hold for such a group.
Keywords
Finite groups Composition factors Irreducible ordinary charactersMathematics Subject Classification
20CPreview
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Notes
Acknowledgements
When preparing the paper, the second author is supported by NSFC (Nos. 11471131 and 11625104). The authors thank the referees for their carefully reading the paper and their helpful comments to improve the paper.
References
- 1.Cabanes, M., Späth, B.: On the inductive Alperin–McKay condition for simple groups of type A. J. Algebra 442, 104–123 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
- 2.Gorenstein, D.: Finite Groups. Harper-Row, New York (1968)zbMATHGoogle Scholar
- 3.Gorenstein, D., Lyons, R.: The local structure of finite groups of characteristic 2 type. Mem. Am. Math. Soc. 42, vii+731 (1983)Google Scholar
- 4.Gorenstein, D., Lyons, R., Solomon, R.: The Classification of the Finite Simple Groups. Number 3. Part I. Chapter A: Almost simple K-groups, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI (1998)Google Scholar
- 5.Guralnick, R.M., Malle, G., Navarro, G.: Self-normalizing Sylow subgroups. Proc. Am. Math. Soc. 132, 973–979 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
- 6.Guralnick, R.M., Navarro, G., Tiep, P.H.: Finite groups with odd Sylow normalizers. Proc. Am. Math. Soc. 144, 5129–5139 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
- 7.Isaacs, I.M., Malle, G., Navarro, G.: A reduction theorem for the McKay conjecture. Invent. Math. 170, 33–101 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
- 8.Koshitani, S., Späth, B.: The inductive Alperin–McKay and blockwise Alperin weight conditions for blocks with cyclic defect groups and odd primes. J. Group Theory 19, 777–813 (2016)MathSciNetzbMATHGoogle Scholar
- 9.Liebeck, M.W., Saxl, J., Seitz, G.M.: Subgroups of maximal rank in finite exceptional groups of Lie type. Proc. Lond. Math. Soc. (3) 65, 297–325 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
- 10.Malle, G., Späth, B.: Characters of odd degree. Ann. Math. (2) 184, 869–908 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
- 11.Malle, G., Testerman, D.: Linear Algebraic Groups and Finite Groups of Lie Type. Cambridge Studies in Advanced Mathematics, vol. 133. Cambridge University Press, Cambridge (2011)CrossRefzbMATHGoogle Scholar
- 12.Navarro, G.: The McKay conjecture and Galois automorphisms. Ann. Math. (2) 160, 1129–1140 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
- 13.Navarro, G., Tiep, P.H., Vallejo, C.: McKay natural correspondences on characters. Algebra Number Theory 8, 1839–1856 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
- 14.Sawabe, M.: A note on finite simple groups with abelian Sylow \(p\)-subgroups. Tokyo J. Math. 30, 293–304 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
- 15.Sawabe, M., Watanabe, A.: On the principal blocks of finite groups with abelian Sylow \(p\)-subgroups. J. Algebra 237, 719–734 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
- 16.Späth, B.: A reduction theorem for the blockwise Alperin weight conjecture. J. Group Theory 16, 159–220 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
- 17.Späth, B.: A reduction theorem for the Alperin–Mckay conjecture. J. Reine Angew. Math. 680, 153–189 (2013)MathSciNetzbMATHGoogle Scholar
- 18.Robinson, D.J.S.: A Course in the Theory of Groups. Springer, Berlin (1995)zbMATHGoogle Scholar
- 19.Tiep, P.H., Zalesski, A.E.: Real conjugacy classes in algebraic groups and finite groups of Lie type. J. Group Theory 8, 291–315 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
- 20.Wilson, R.A.: The Mckay conjecture is true for the sporadic simple groups. J. Algebra 207, 294–305 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
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