Characters and generation of Sylow 2-subgroups

Navarro, Gabriel; Rizo, Noelia; Schaeffer Fry, A.; Vallejo, Carolina

doi:10.1090/ert/555

Characters and generation of Sylow 2-subgroups
HTML articles powered by AMS MathViewer

by Gabriel Navarro, Noelia Rizo, A. A. Schaeffer Fry and Carolina Vallejo

Represent. Theory 25 (2021), 142-165

DOI: https://doi.org/10.1090/ert/555

Published electronically: February 25, 2021

PDF | Request permission

Abstract:

We show that the character table of a finite group $G$ determines whether a Sylow 2-subgroup of $G$ is generated by 2 elements, in terms of the Galois action on characters. Our proof of this result requires the use of the Classification of Finite Simple Groups and provides new evidence for the so-far elusive Alperin–McKay–Navarro conjecture.

References

J. L. Alperin, Isomorphic blocks, J. Algebra 43 (1976), no. 2, 694–698. MR 422396, DOI 10.1016/0021-8693(76)90135-6
Richard Brauer, Representations of finite groups, Lectures on Modern Mathematics, Vol. I, Wiley, New York, 1963, pp. 133–175. MR 0178056
W. Burnside, Theory of groups of finite order, Dover Publications, Inc., New York, 1955. 2d ed. MR 0069818
Roger Carter and Paul Fong, The Sylow $2$-subgroups of the finite classical groups, J. Algebra 1 (1964), 139–151. MR 166271, DOI 10.1016/0021-8693(64)90030-4
Marc Cabanes and Michel Enguehard, Representation theory of finite reductive groups, New Mathematical Monographs, vol. 1, Cambridge University Press, Cambridge, 2004. MR 2057756, DOI 10.1017/CBO9780511542763
Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR 794307
David A. Craven, Representation theory of finite groups: a guidebook, Universitext, Springer, Cham, 2019. MR 3970262, DOI 10.1007/978-3-030-21792-1
E. C. Dade, Remarks on isomorphic blocks, J. Algebra 45 (1977), no. 1, 254–258. MR 576548, DOI 10.1016/0021-8693(77)90371-4
E. C. Dade, Blocks with cyclic defect groups, Ann. of Math. (2) 84 (1966), 20–48. MR 200355, DOI 10.2307/1970529
F. Digne, G. I. Lehrer, and J. Michel, On Gel′fand-Graev characters of reductive groups with disconnected centre, J. Reine Angew. Math. 491 (1997), 131–147. MR 1476090
The GAP Group, GAP Groups, Algorithms, and Programming, Version 4.4; 2004, http://www.gap-system.org.
Meinolf Geck, Gerhard Hiss, Frank Lübeck, Gunter Malle, and Götz Pfeiffer, CHEVIE—a system for computing and processing generic character tables, Appl. Algebra Engrg. Comm. Comput. 7 (1996), no. 3, 175–210. Computational methods in Lie theory (Essen, 1994). MR 1486215, DOI 10.1007/BF01190329
Meinolf Geck and Götz Pfeiffer, Characters of finite Coxeter groups and Iwahori-Hecke algebras, London Mathematical Society Monographs. New Series, vol. 21, The Clarendon Press, Oxford University Press, New York, 2000. MR 1778802
Eugenio Giannelli, Noelia Rizo, and A. A. Schaeffer Fry, Groups with few $p’$-character degrees, J. Pure Appl. Algebra 224 (2020), no. 8, 106338, 15. MR 4074578, DOI 10.1016/j.jpaa.2020.106338
Daniel Gorenstein and Koichiro Harada, Finite simple groups of low $2$-rank and the families $G_{2}(q),\,D_{4}^{2}(q),\,q$ odd, Bull. Amer. Math. Soc. 77 (1971), 829–862. MR 306301, DOI 10.1090/S0002-9904-1971-12794-5
Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups. Number 5. Part III. Chapters 1–6. The generic case, stages 1–3a, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 2002. MR 1923000, DOI 10.1090/surv/040.5
I. Martin Isaacs, Character theory of finite groups, AMS Chelsea Publishing, Providence, RI, 2006. Corrected reprint of the 1976 original [Academic Press, New York; MR0460423]. MR 2270898, DOI 10.1090/chel/359
I. Martin Isaacs, Finite group theory, Graduate Studies in Mathematics, vol. 92, American Mathematical Society, Providence, RI, 2008. MR 2426855, DOI 10.1090/gsm/092
Andrei Jaikin-Zapirain, The number of finite $p$-groups with bounded number of generators, Finite groups 2003, Walter de Gruyter, Berlin, 2004, pp. 209–217. MR 2125074
F. Lübeck, Character degrees and their multiplicities for some groups of Lie type of rank $<9$ (webpage, 2007). http://www.math.rwth-aachen.de/~Frank.Luebeck/chev/DegMult/index.html.
G. Lusztig, On the representations of reductive groups with disconnected centre, Astérisque 168 (1988), 10, 157–166. Orbites unipotentes et représentations, I. MR 1021495
G. Lusztig, Rationality properties of unipotent representations, J. Algebra 258 (2002), no. 1, 1–22. Special issue in celebration of Claudio Procesi’s 60th birthday. MR 1958895, DOI 10.1016/S0021-8693(02)00514-8
I. G. Macdonald, On the degrees of the irreducible representations of symmetric groups, Bull. London Math. Soc. 3 (1971), 189–192. MR 289677, DOI 10.1112/blms/3.2.189
Gunter Malle, Height 0 characters of finite groups of Lie type, Represent. Theory 11 (2007), 192–220. MR 2365640, DOI 10.1090/S1088-4165-07-00312-3
Gunter Malle, Extensions of unipotent characters and the inductive McKay condition, J. Algebra 320 (2008), no. 7, 2963–2980. MR 2442005, DOI 10.1016/j.jalgebra.2008.06.033
Gunter Malle and Britta Späth, Characters of odd degree, Ann. of Math. (2) 184 (2016), no. 3, 869–908. MR 3549625, DOI 10.4007/annals.2016.184.3.6
G. Navarro, Characters and blocks of finite groups, London Mathematical Society Lecture Note Series, vol. 250, Cambridge University Press, Cambridge, 1998. MR 1632299, DOI 10.1017/CBO9780511526015
Gabriel Navarro, Linear characters of Sylow subgroups, J. Algebra 269 (2003), no. 2, 589–598. MR 2015855, DOI 10.1016/S0021-8693(03)00391-0
Gabriel Navarro, The McKay conjecture and Galois automorphisms, Ann. of Math. (2) 160 (2004), no. 3, 1129–1140. MR 2144975, DOI 10.4007/annals.2004.160.1129
Gabriel Navarro, Character theory and the McKay conjecture, Cambridge Studies in Advanced Mathematics, vol. 175, Cambridge University Press, Cambridge, 2018. MR 3753712, DOI 10.1017/9781108552790
Gabriel Navarro, Benjamin Sambale, and Pham Huu Tiep, Characters and Sylow 2-subgroups of maximal class revisited, J. Pure Appl. Algebra 222 (2018), no. 11, 3721–3732. MR 3806747, DOI 10.1016/j.jpaa.2018.02.002
Gabriel Navarro and Pham Huu Tiep, Sylow subgroups, exponents, and character values, Trans. Amer. Math. Soc. 372 (2019), no. 6, 4263–4291. MR 4009430, DOI 10.1090/tran/7816
Gabriel Navarro and Pham Huu Tiep, Brauer’s height zero conjecture for the 2-blocks of maximal defect, J. Reine Angew. Math. 669 (2012), 225–247. MR 2980457, DOI 10.1515/crelle.2011.147
Gabriel Navarro and Pham Huu Tiep, Characters of relative $p’$-degree over normal subgroups, Ann. of Math. (2) 178 (2013), no. 3, 1135–1171. MR 3092477, DOI 10.4007/annals.2013.178.3.7
Gabriel Navarro, Pham Huu Tiep, and Alexandre Turull, $p$-rational characters and self-normalizing Sylow $p$-subgroups, Represent. Theory 11 (2007), 84–94. MR 2306612, DOI 10.1090/S1088-4165-07-00263-4
Gabriel Navarro, Pham Huu Tiep, and Alexandre Turull, Brauer characters with cyclotomic field of values, J. Pure Appl. Algebra 212 (2008), no. 3, 628–635. MR 2365337, DOI 10.1016/j.jpaa.2007.06.019
Gabriel Navarro, Pham Huu Tiep, and Carolina Vallejo, Brauer correspondent blocks with one simple module, Trans. Amer. Math. Soc. 371 (2019), no. 2, 903–922. MR 3885165, DOI 10.1090/tran/7458
Jørn B. Olsson, On the $p$-blocks of symmetric and alternating groups and their covering groups, J. Algebra 128 (1990), no. 1, 188–213. MR 1031917, DOI 10.1016/0021-8693(90)90049-T
Noelia Rizo, A. A. Schaeffer Fry, and Carolina Vallejo, Galois action on the principal block and cyclic Sylow subgroups, Algebra Number Theory 14 (2020), no. 7, 1953–1979. MR 4150255, DOI 10.2140/ant.2020.14.1953
A. A. Schaeffer Fry, Galois automorphisms on Harish-Chandra series and Navarro’s self-normalizing Sylow 2-subgroup conjecture, Trans. Amer. Math. Soc. 372 (2019), no. 1, 457–483. MR 3968776, DOI 10.1090/tran/7590
Amanda A. Schaeffer Fry and Jay Taylor, On self-normalising Sylow 2-subgroups in type A, J. Lie Theory 28 (2018), no. 1, 139–168. MR 3694099
Peter Schmid, Extending the Steinberg representation, J. Algebra 150 (1992), no. 1, 254–256. MR 1174899, DOI 10.1016/S0021-8693(05)80060-2
Ken-ichi Shinoda, The conjugacy classes of Chevalley groups of type $(F_{4})$ over finite fields of characteristic $2$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 21 (1974), 133–159. MR 0349863
Harold N. Ward, On Ree’s series of simple groups, Bull. Amer. Math. Soc. 69 (1963), 113–114. MR 141707, DOI 10.1090/S0002-9904-1963-10885-X