Characters and generation of Sylow 2-subgroups
Authors:
Gabriel Navarro, Noelia Rizo, A. A. Schaeffer Fry and Carolina Vallejo
Journal:
Represent. Theory 25 (2021), 142-165
MSC (2020):
Primary 20C20, 20C15
DOI:
https://doi.org/10.1090/ert/555
Published electronically:
February 25, 2021
MathSciNet review:
4220651
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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.
- J. L. Alperin, Isomorphic blocks, J. Algebra 43 (1976), no. 2, 694–698. MR 422396, DOI https://doi.org/10.1016/0021-8693%2876%2990135-6
- 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 https://doi.org/10.1016/0021-8693%2864%2990030-4
- Marc Cabanes and Michel Enguehard, Representation theory of finite reductive groups, New Mathematical Monographs, vol. 1, Cambridge University Press, Cambridge, 2004. MR 2057756
- 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
- E. C. Dade, Remarks on isomorphic blocks, J. Algebra 45 (1977), no. 1, 254–258. MR 576548, DOI https://doi.org/10.1016/0021-8693%2877%2990371-4
- E. C. Dade, Blocks with cyclic defect groups, Ann. of Math. (2) 84 (1966), 20–48. MR 200355, DOI https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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
- 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
- I. Martin Isaacs, Finite group theory, Graduate Studies in Mathematics, vol. 92, American Mathematical Society, Providence, RI, 2008. MR 2426855
- 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 https://doi.org/10.1016/S0021-8693%2802%2900514-8
- I. G. Macdonald, On the degrees of the irreducible representations of symmetric groups, Bull. London Math. Soc. 3 (1971), 189–192. MR 289677, DOI https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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
- Gabriel Navarro, Linear characters of Sylow subgroups, J. Algebra 269 (2003), no. 2, 589–598. MR 2015855, DOI https://doi.org/10.1016/S0021-8693%2803%2900391-0
- Gabriel Navarro, The McKay conjecture and Galois automorphisms, Ann. of Math. (2) 160 (2004), no. 3, 1129–1140. MR 2144975, DOI https://doi.org/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
- 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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/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 https://doi.org/10.1016/0021-8693%2890%2990049-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 https://doi.org/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 https://doi.org/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 https://doi.org/10.1016/S0021-8693%2805%2980060-2
- 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. I A 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 https://doi.org/10.1090/S0002-9904-1963-10885-X
Retrieve articles in Representation Theory of the American Mathematical Society with MSC (2020): 20C20, 20C15
Retrieve articles in all journals with MSC (2020): 20C20, 20C15
Additional Information
Gabriel Navarro
Affiliation:
Departament de Matemàtiques, Universitat de València, 46100 Burjassot, València, Spain
MR Author ID:
129760
Email:
gabriel@uv.es
Noelia Rizo
Affiliation:
Departament de Matemàtiques, Universitat de València, 46100 Burjassot, València, Spain
MR Author ID:
1070925
Email:
noelia.rizo@uv.es
A. A. Schaeffer Fry
Affiliation:
Department of Mathematics and Statistics, MSU Denver, Denver, Colorado 80217
MR Author ID:
899206
Email:
aschaef6@msudenver.edu
Carolina Vallejo
Affiliation:
Departamento de Matemáticas, Edificio Sabatini, Universidad Carlos III de Madrid, Av. Universidad 30, 28911, Leganés. Madrid, Spain
MR Author ID:
1001337
ORCID:
0000-0003-3363-3376
Email:
carolina.vallejo@uc3m.es
Keywords:
Sylow 2-subgroups,
character tables,
principal blocks,
Alperin–Galois–McKay conjecture
Received by editor(s):
March 18, 2020
Received by editor(s) in revised form:
April 1, 2020
Published electronically:
February 25, 2021
Additional Notes:
The first, second and fourth authors were partially supported by the Spanish Ministerio de Ciencia e Innovación PID2019-103854GB-I00 and FEDER funds. The third author was partially supported by the National Science Foundation under Grant No. DMS-1801156. The fourth author also acknowledges support by Spanish Ministerio de Ciencia e Innovación MTM2017-82690-P and the ICMAT Severo Ochoa project SEV-2015-0554. Part of this work was supported by the National Security Agency under Grant No. H98230-19-1-0119, The Lyda Hill Foundation, The McGovern Foundation, and Microsoft Research, while the second, third, and fourth authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the summer of 2019.
Article copyright:
© Copyright 2021
American Mathematical Society