Problems on characters solvable groups
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Gabriel Navarro
Universitat de València. Departament de Matemàtiques
I review some problems on characters of finite solvable groups, while introducing new ones.
Paraules clau
characters, solvable groups
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Com citar
Navarro, Gabriel. «Problems on characters: solvable groups». Publicacions Matemàtiques, 2023, vol.VOL 67, núm. 1, p. 173–198, https://raco.cat/index.php/PublicacionsMatematiques/article/view/412624.
Referències
R. Brauer, Representations of finite groups, in: “Lectures on Modern Mathematics”, Vol. I, Wiley, New York, 1963, pp. 133–175.
M. Broue and Ll. Puig, A Frobenius theorem for blocks, Invent. Math. 56(2) (1980), 117–128. DOI: 10.1007/BF01392547
O. Brunat and G. Malle, Characters of positive height in blocks of finite quasi-simple groups, Int. Math. Res. Not. IMRN 2015(17) (2015), 7763–7786. DOI: 10.1093/imrn/rnu171
C. Casolo and S. Dolfi, Products of primes in conjugacy class sizes and irreducible character degrees, Israel J. Math. 174 (2009), 403–418. DOI: 10.1007/s11856-009-0120-z
E. J. Cassell, Conjugacy classes in finite groups, commuting graphs and character degrees, Thesis (Ph.D.)-University of Birmingham (2013). Available on https://etheses.bham.ac.uk/id/eprint/4628/1/Cassell13PhD.pdf.
X. Chen and G. Navarro, Brauer characters, degrees and subgroups, Bull. Lond. Math. Soc. 54(3) (2022), 891–893. DOI: 10.1112/blms.12592
J. P. Cossey, Bounds on the number of lifts of a Brauer character in a p-solvable group, J. Algebra 312(2) (2007), 699–708. DOI: 10.1016/j.jalgebra.2007.03.023
S. Dolfi, Large orbits in coprime actions of solvable groups, Trans. Amer. Math. Soc. 360(1) (2008), 135–152. DOI: 10.1090/S0002-9947-07-04155-4
S. Dolfi and G. Navarro, Large orbits of elements centralized by a Sylow subgroup, Arch. Math. (Basel) 93(4) (2009), 299–304.
DOI: 10.1007/s00013-009-0016-5
S. Dolfi, G. Navarro, E. Pacifici, L. Sanus, and P. H. Tiep, Non-vanishing elements of finite groups, J. Algebra 323(2) (2010), 540–545. DOI: 10.1016/j.jalgebra.2009.08.014
C. W. Eaton and A. Moreto´, Extending Brauer’s height zero conjecture to blocks with nonabelian defect groups, Int. Math. Res. Not. IMRN 2014(20) (2014), 5581–5601. DOI: 10.1093/imrn/rnt131
E. Farias e Soares, Big primes and character values for solvable groups, J. Algebra 100(2) (1986), 305–324. DOI: 10.1016/0021-8693(86)90079-7
B. Fein and B. Gordon, Fields generated by characters of finite groups, J. London Math. Soc. (2) 4 (1972), 735–740. DOI: 10.1112/jlms/s2-4.4.735
Z. Feng, Y. Liu, and J. Zhang, On heights of characters of finite groups, J. Algebra 556 (2020), 106–135. DOI: 10.1016/j.jalgebra.2020.02.035
M. Geck, Green functions and Glauberman degree-divisibility, Ann. of Math. (2) 192(1) (2020), 229–249. DOI: 10.4007/annals.2020.192.1.4
E. Giannelli, A. Kleshchev, G. Navarro, and P. H. Tiep, Restriction of odd degree characters and natural correspondences, Int. Math. Res. Not. IMRN 2017(20) (2017), 6089–6118. DOI: 10.1093/imrn/rnw174
E. Giannelli, S. Law, J. Long, and C. Vallejo, Sylow branching coefficients and a conjecture of Malle and Navarro, Bull. Lond. Math. Soc. 54(2) (2022), 552–567. DOI: 10.1112/blms.12584
E. Giannelli and G. Navarro, Restricting irreducible characters to Sylow p-subgroups, Proc. Amer. Math. Soc. 146(5) (2018), 1963–1976. DOI: 10.1090/proc/13970
D. Gluck, Large p 0 -orbits for p-nilpotent linear groups, Math. Z. 284(3-4) (2016), 1035–1052. DOI: 10.1007/s00209-016-1686-x
Z. Halasi and K. Podoski, Every coprime linear group admits a base of size two, Trans. Amer. Math. Soc. 368(8) (2016), 5857–5887. DOI: 10.1090/tran/6544
K. Harada, Revisiting character theory of finite groups, Bull. Inst. Math. Acad. Sin. (N.S.) 13(4) (2018), 383–395. DOI: 10.21915/BIMAS.2018402
M. Hertweck, A counterexample to the isomorphism problem for integral group rings, Ann. of Math. (2) 154(1) (2001), 115–138. DOI: 10.2307/3062112
L. Hethelyi and B. K ´ ulshammer ¨ , On the number of conjugacy classes of a finite solvable group, Bull. London Math. Soc. 32(6) (2000), 668–672. DOI: 10.1112/S0024609300007499
I. M. Isaacs, Characters of solvable and symplectic groups, Amer. J. Math. 95(3) (1973), 594–635. DOI: 10.2307/2373731
I. M. Isaacs, “Character Theory of Finite Groups”, Corrected reprint of the 1976 original, AMS Chelsea Publishing, Providence, RI, 2006. DOI: 10.1090/chel/359
I. M. Isaacs, “Finite Group Theory”, Graduate Studies in Mathematics 92, American Mathematical Society, Providence, RI, 2008. DOI: 10.1090/gsm/092
I. M. Isaacs, Carter subgroups, characters and composition series, Trans. Amer. Math. Soc. Ser. B 9 (2022), 470–498. DOI: 10.1090/btran/93
I. M. Isaacs, T. M. Keller, U. Meierfrankenfeld, and A. Moreto´, Fixed point spaces, primitive character degrees and conjugacy class sizes, Proc. Amer.
Math. Soc. 134(11) (2006), 3123–3130. DOI: 10.1090/S0002-9939-06-08383-3
I. M. Isaacs, G. Malle, and G. Navarro, A reduction theorem for the McKay conjecture, Invent. Math. 170(1) (2007), 33–101.
DOI: 10.1007/s00222-007-0057-y
I. M. Isaacs and G. Navarro, Sylow 2-subgroups of rational solvable groups, Math. Z. 272(3-4) (2012), 937–945. DOI: 10.1007/s00209-011-0965-9
I. M. Isaacs and G. Navarro, Primes and conductors of characters, Preprint (2021).
I. M. Isaacs, G. Navarro, and T. R. Wolf, Finite group elements where no irreducible character vanishes, J. Algebra 222(2) (1999), 413–423. DOI: 10.1006/jabr.1999.8007
P. Jin and Y. Yang, Primitive character degrees of solvable groups, J. Algebra 573 (2021), 532–538. DOI: 10.1016/j.jalgebra.2020.11.025
R. Kessar, M. Linckelmann, J. Lynd, and J. Semeraro, Weight conjectures for fusion systems, Adv. Math. 357 (2019), 106825, 40 pp. DOI: 10.1016/j.aim.2019.106825
R. Kessar and G. Malle, Quasi-isolated blocks and Brauer’s height zero conjecture, Ann. of Math. (2) 178(1) (2013), 321–384. DOI: 10.4007/annals.2013.178.1.6
G. Malle, On the number of characters in blocks of quasi-simple groups, Algebr. Represent. Theory 23(3) (2020), 513–539.
DOI: 10.1007/s10468-019-09860-0
G. Malle and G. Navarro, Inequalities for some blocks of finite groups, Arch. Math. (Basel) 87(5) (2006), 390–399. DOI: 10.1007/s00013-006-1769-8
G. Malle and G. Navarro, Blocks with equal height zero degrees, Trans. Amer. Math. Soc. 363(12) (2011), 6647–6669. DOI: 10.1090/S0002-9947-2011-05333-X
G. Malle, G. Navarro, and B. Sambale, On defects of characters and decomposition numbers, Algebra Number Theory 11(6) (2017), 1357–1384. DOI: 10.2140/ant.2017.11.1357
G. Malle, G. Navarro, and P. H. Tiep, Zeros of characters and element orders, in preparation.
G. Malle and B. Spath ¨ , Characters of odd degree, Ann. of Math. (2) 184(3) (2016), 869–908. DOI: 10.4007/annals.2016.184.3.6
C. Marchi, Primitive characters of odd order groups, J. Algebra 547 (2020), 345–357. DOI: 10.1016/j.jalgebra.2019.11.024
A. Maroti, A lower bound for the number of conjugacy classes of a finite group,
Adv. Math. 290 (2016), 1062–1078. DOI: 10.1016/j.aim.2015.12.020
A. R. Miller, On roots of unity and character values, Preprint (2020). arXiv:2003.13238
M. Murai, A note on the number of irreducible characters in a p-block of a finite
group, Osaka J. Math. 21(2) (1984), 387–398.
G. Navarro, “Characters and Blocks of Finite Groups”, London Mathematical Society Lecture Note Series 250, Cambridge University Press, Cambridge, 1998. DOI: 10.1017/CBO9780511526015
G. Navarro, Irreducible restriction and zeros of characters, Proc. Amer. Math. Soc. 129(6) (2001), 1643–1645. DOI: 10.1090/S0002-9939-00-05747-6
G. Navarro, Problems in character theory, in: “Character Theory of Finite Groups”, Contemp. Math. 524, Amer. Math. Soc., Providence, RI, 2010,
pp. 97–125. DOI: 10.1090/conm/524/10350
G. Navarro, The set of character degrees of a finite group does not determine its solvability, Proc. Amer. Math. Soc. 143(3) (2015), 989–990.
DOI: 10.1090/S0002-9939-2014-12321-5
G. Navarro, “Character Theory and the McKay Conjecture”, Cambridge Studies in Advanced Mathematics 175, Cambridge University Press, Cambridge, 2018. DOI: 10.1017/9781108552790
G, Navarro and P. H. Tiep, Characters of relative p 0-degree over normal subgroups, Ann. of Math. (2) 178(3) (2013), 1135–1171. DOI: 10.4007/annals.2013.178.3.7
G, Navarro and P. H. Tiep, The fields of values of characters of degree not divisible by p, Forum Math. Pi 9 (2021), Paper no. e2, 28 pp. DOI: 10.1017/fmp.2021.1
G, Navarro, P. H. Tiep, and C. Vallejo, McKay natural correspondences on characters, Algebra Number Theory 8(8) (2014), 1839–1856. DOI: 10.2140/ant.2014.8.1839
T. Okuyama and Y. Tsushima, Local properties of p-block algebras of finite groups, Osaka Math. J. 20(1) (1983), 33–41. DOI: 10.18910/8689
G. Qian, Element orders and character codegrees, Bull. Lond. Math. Soc. 53(3) (2021), 820–824. DOI: 10.1112/blms.12462
D. Rossi and B. Sambale, Restrictions of characters in p-solvable groups, J. Algebra 587 (2021), 130–141. DOI: 10.1016/j.jalgebra.2021.07.034
B. Sambale, The Alperin–McKay conjecture for metacyclic, minimal nonabelian defect groups, Proc. Amer. Math. Soc. 143(10) (2015), 4291–4304.
DOI: 10.1090/S0002-9939-2015-12637-8
M. Broue and Ll. Puig, A Frobenius theorem for blocks, Invent. Math. 56(2) (1980), 117–128. DOI: 10.1007/BF01392547
O. Brunat and G. Malle, Characters of positive height in blocks of finite quasi-simple groups, Int. Math. Res. Not. IMRN 2015(17) (2015), 7763–7786. DOI: 10.1093/imrn/rnu171
C. Casolo and S. Dolfi, Products of primes in conjugacy class sizes and irreducible character degrees, Israel J. Math. 174 (2009), 403–418. DOI: 10.1007/s11856-009-0120-z
E. J. Cassell, Conjugacy classes in finite groups, commuting graphs and character degrees, Thesis (Ph.D.)-University of Birmingham (2013). Available on https://etheses.bham.ac.uk/id/eprint/4628/1/Cassell13PhD.pdf.
X. Chen and G. Navarro, Brauer characters, degrees and subgroups, Bull. Lond. Math. Soc. 54(3) (2022), 891–893. DOI: 10.1112/blms.12592
J. P. Cossey, Bounds on the number of lifts of a Brauer character in a p-solvable group, J. Algebra 312(2) (2007), 699–708. DOI: 10.1016/j.jalgebra.2007.03.023
S. Dolfi, Large orbits in coprime actions of solvable groups, Trans. Amer. Math. Soc. 360(1) (2008), 135–152. DOI: 10.1090/S0002-9947-07-04155-4
S. Dolfi and G. Navarro, Large orbits of elements centralized by a Sylow subgroup, Arch. Math. (Basel) 93(4) (2009), 299–304.
DOI: 10.1007/s00013-009-0016-5
S. Dolfi, G. Navarro, E. Pacifici, L. Sanus, and P. H. Tiep, Non-vanishing elements of finite groups, J. Algebra 323(2) (2010), 540–545. DOI: 10.1016/j.jalgebra.2009.08.014
C. W. Eaton and A. Moreto´, Extending Brauer’s height zero conjecture to blocks with nonabelian defect groups, Int. Math. Res. Not. IMRN 2014(20) (2014), 5581–5601. DOI: 10.1093/imrn/rnt131
E. Farias e Soares, Big primes and character values for solvable groups, J. Algebra 100(2) (1986), 305–324. DOI: 10.1016/0021-8693(86)90079-7
B. Fein and B. Gordon, Fields generated by characters of finite groups, J. London Math. Soc. (2) 4 (1972), 735–740. DOI: 10.1112/jlms/s2-4.4.735
Z. Feng, Y. Liu, and J. Zhang, On heights of characters of finite groups, J. Algebra 556 (2020), 106–135. DOI: 10.1016/j.jalgebra.2020.02.035
M. Geck, Green functions and Glauberman degree-divisibility, Ann. of Math. (2) 192(1) (2020), 229–249. DOI: 10.4007/annals.2020.192.1.4
E. Giannelli, A. Kleshchev, G. Navarro, and P. H. Tiep, Restriction of odd degree characters and natural correspondences, Int. Math. Res. Not. IMRN 2017(20) (2017), 6089–6118. DOI: 10.1093/imrn/rnw174
E. Giannelli, S. Law, J. Long, and C. Vallejo, Sylow branching coefficients and a conjecture of Malle and Navarro, Bull. Lond. Math. Soc. 54(2) (2022), 552–567. DOI: 10.1112/blms.12584
E. Giannelli and G. Navarro, Restricting irreducible characters to Sylow p-subgroups, Proc. Amer. Math. Soc. 146(5) (2018), 1963–1976. DOI: 10.1090/proc/13970
D. Gluck, Large p 0 -orbits for p-nilpotent linear groups, Math. Z. 284(3-4) (2016), 1035–1052. DOI: 10.1007/s00209-016-1686-x
Z. Halasi and K. Podoski, Every coprime linear group admits a base of size two, Trans. Amer. Math. Soc. 368(8) (2016), 5857–5887. DOI: 10.1090/tran/6544
K. Harada, Revisiting character theory of finite groups, Bull. Inst. Math. Acad. Sin. (N.S.) 13(4) (2018), 383–395. DOI: 10.21915/BIMAS.2018402
M. Hertweck, A counterexample to the isomorphism problem for integral group rings, Ann. of Math. (2) 154(1) (2001), 115–138. DOI: 10.2307/3062112
L. Hethelyi and B. K ´ ulshammer ¨ , On the number of conjugacy classes of a finite solvable group, Bull. London Math. Soc. 32(6) (2000), 668–672. DOI: 10.1112/S0024609300007499
I. M. Isaacs, Characters of solvable and symplectic groups, Amer. J. Math. 95(3) (1973), 594–635. DOI: 10.2307/2373731
I. M. Isaacs, “Character Theory of Finite Groups”, Corrected reprint of the 1976 original, AMS Chelsea Publishing, Providence, RI, 2006. DOI: 10.1090/chel/359
I. M. Isaacs, “Finite Group Theory”, Graduate Studies in Mathematics 92, American Mathematical Society, Providence, RI, 2008. DOI: 10.1090/gsm/092
I. M. Isaacs, Carter subgroups, characters and composition series, Trans. Amer. Math. Soc. Ser. B 9 (2022), 470–498. DOI: 10.1090/btran/93
I. M. Isaacs, T. M. Keller, U. Meierfrankenfeld, and A. Moreto´, Fixed point spaces, primitive character degrees and conjugacy class sizes, Proc. Amer.
Math. Soc. 134(11) (2006), 3123–3130. DOI: 10.1090/S0002-9939-06-08383-3
I. M. Isaacs, G. Malle, and G. Navarro, A reduction theorem for the McKay conjecture, Invent. Math. 170(1) (2007), 33–101.
DOI: 10.1007/s00222-007-0057-y
I. M. Isaacs and G. Navarro, Sylow 2-subgroups of rational solvable groups, Math. Z. 272(3-4) (2012), 937–945. DOI: 10.1007/s00209-011-0965-9
I. M. Isaacs and G. Navarro, Primes and conductors of characters, Preprint (2021).
I. M. Isaacs, G. Navarro, and T. R. Wolf, Finite group elements where no irreducible character vanishes, J. Algebra 222(2) (1999), 413–423. DOI: 10.1006/jabr.1999.8007
P. Jin and Y. Yang, Primitive character degrees of solvable groups, J. Algebra 573 (2021), 532–538. DOI: 10.1016/j.jalgebra.2020.11.025
R. Kessar, M. Linckelmann, J. Lynd, and J. Semeraro, Weight conjectures for fusion systems, Adv. Math. 357 (2019), 106825, 40 pp. DOI: 10.1016/j.aim.2019.106825
R. Kessar and G. Malle, Quasi-isolated blocks and Brauer’s height zero conjecture, Ann. of Math. (2) 178(1) (2013), 321–384. DOI: 10.4007/annals.2013.178.1.6
G. Malle, On the number of characters in blocks of quasi-simple groups, Algebr. Represent. Theory 23(3) (2020), 513–539.
DOI: 10.1007/s10468-019-09860-0
G. Malle and G. Navarro, Inequalities for some blocks of finite groups, Arch. Math. (Basel) 87(5) (2006), 390–399. DOI: 10.1007/s00013-006-1769-8
G. Malle and G. Navarro, Blocks with equal height zero degrees, Trans. Amer. Math. Soc. 363(12) (2011), 6647–6669. DOI: 10.1090/S0002-9947-2011-05333-X
G. Malle, G. Navarro, and B. Sambale, On defects of characters and decomposition numbers, Algebra Number Theory 11(6) (2017), 1357–1384. DOI: 10.2140/ant.2017.11.1357
G. Malle, G. Navarro, and P. H. Tiep, Zeros of characters and element orders, in preparation.
G. Malle and B. Spath ¨ , Characters of odd degree, Ann. of Math. (2) 184(3) (2016), 869–908. DOI: 10.4007/annals.2016.184.3.6
C. Marchi, Primitive characters of odd order groups, J. Algebra 547 (2020), 345–357. DOI: 10.1016/j.jalgebra.2019.11.024
A. Maroti, A lower bound for the number of conjugacy classes of a finite group,
Adv. Math. 290 (2016), 1062–1078. DOI: 10.1016/j.aim.2015.12.020
A. R. Miller, On roots of unity and character values, Preprint (2020). arXiv:2003.13238
M. Murai, A note on the number of irreducible characters in a p-block of a finite
group, Osaka J. Math. 21(2) (1984), 387–398.
G. Navarro, “Characters and Blocks of Finite Groups”, London Mathematical Society Lecture Note Series 250, Cambridge University Press, Cambridge, 1998. DOI: 10.1017/CBO9780511526015
G. Navarro, Irreducible restriction and zeros of characters, Proc. Amer. Math. Soc. 129(6) (2001), 1643–1645. DOI: 10.1090/S0002-9939-00-05747-6
G. Navarro, Problems in character theory, in: “Character Theory of Finite Groups”, Contemp. Math. 524, Amer. Math. Soc., Providence, RI, 2010,
pp. 97–125. DOI: 10.1090/conm/524/10350
G. Navarro, The set of character degrees of a finite group does not determine its solvability, Proc. Amer. Math. Soc. 143(3) (2015), 989–990.
DOI: 10.1090/S0002-9939-2014-12321-5
G. Navarro, “Character Theory and the McKay Conjecture”, Cambridge Studies in Advanced Mathematics 175, Cambridge University Press, Cambridge, 2018. DOI: 10.1017/9781108552790
G, Navarro and P. H. Tiep, Characters of relative p 0-degree over normal subgroups, Ann. of Math. (2) 178(3) (2013), 1135–1171. DOI: 10.4007/annals.2013.178.3.7
G, Navarro and P. H. Tiep, The fields of values of characters of degree not divisible by p, Forum Math. Pi 9 (2021), Paper no. e2, 28 pp. DOI: 10.1017/fmp.2021.1
G, Navarro, P. H. Tiep, and C. Vallejo, McKay natural correspondences on characters, Algebra Number Theory 8(8) (2014), 1839–1856. DOI: 10.2140/ant.2014.8.1839
T. Okuyama and Y. Tsushima, Local properties of p-block algebras of finite groups, Osaka Math. J. 20(1) (1983), 33–41. DOI: 10.18910/8689
G. Qian, Element orders and character codegrees, Bull. Lond. Math. Soc. 53(3) (2021), 820–824. DOI: 10.1112/blms.12462
D. Rossi and B. Sambale, Restrictions of characters in p-solvable groups, J. Algebra 587 (2021), 130–141. DOI: 10.1016/j.jalgebra.2021.07.034
B. Sambale, The Alperin–McKay conjecture for metacyclic, minimal nonabelian defect groups, Proc. Amer. Math. Soc. 143(10) (2015), 4291–4304.
DOI: 10.1090/S0002-9939-2015-12637-8