Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Journal of Group Theory

Editor-in-Chief: Parker, Christopher W.

Managing Editor: Wilson, John S. / Khukhro, Evgenii I. / Kramer, Linus

6 Issues per year


IMPACT FACTOR 2017: 0.581

CiteScore 2017: 0.53

SCImago Journal Rank (SJR) 2017: 0.778
Source Normalized Impact per Paper (SNIP) 2017: 0.893

Mathematical Citation Quotient (MCQ) 2017: 0.45

Online
ISSN
1435-4446
See all formats and pricing
More options …
Volume 20, Issue 6

Issues

Blocks of small defect in alternating groups and squares of Brauer character degrees

Xiaoyou Chen / James P. Cossey / Mark L. Lewis / Hung P. Tong-Viet
Published Online: 2017-07-18 | DOI: https://doi.org/10.1515/jgth-2017-0025

Abstract

Let p be a prime. We show that other than a few exceptions, alternating groups will have p-blocks with small defect for p equal to 2 or 3. Using this result, we prove that a finite group G has a normal Sylow p-subgroup P and G/P is nilpotent if and only if φ(1)2 divides |G:ker(φ)| for every irreducible Brauer character φ of G.

References

  • [1]

    D. Benson, Spin modules for symmetric groups, J. Lond. Math. Soc. (2) 38 (1988), no. 2, 250–262. Google Scholar

  • [2]

    C. Bessenrodt and H. Weber, On p-blocks of symmetric and alternating groups with all irreducible Brauer characters of prime power degree, J. Algebra 320 (2008), no. 6, 2405–2421. Web of ScienceCrossrefGoogle Scholar

  • [3]

    R. Brauer, On a conjecture by Nakayama, Trans. Roy. Soc. Canada. Sect. III. (3) 41 (1947), 11–19. Google Scholar

  • [4]

    J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Oxford University Press, Oxford, 1985. Google Scholar

  • [5]

    S. Dolfi, Large orbits in coprime actions of solvable groups, Trans. Amer. Math. Soc. 360 (2008), no. 1, 135–152. CrossrefGoogle Scholar

  • [6]

    A. Espuelas and G. Navarro, Blocks of small defect, Proc. Amer. Math. Soc. 114 (1992), no. 4, 881–885. CrossrefGoogle Scholar

  • [7]

    S. M. Gagola, Jr., A character theoretic condition for F(G)>1, Comm. Algebra 33 (2005), no. 5, 1369–1382. Google Scholar

  • [8]

    S. M. Gagola, Jr. and M. L. Lewis, A character-theoretic condition characterizing nilpotent groups, Comm. Algebra 27 (1999), no. 3, 1053–1056. CrossrefGoogle Scholar

  • [9]

    A. Granville and K. Ono, Defect zero p-blocks for finite simple groups, Trans. Amer. Math. Soc. 348 (1996), no. 1, 331–347. CrossrefGoogle Scholar

  • [10]

    Z. Halasi and K. Podoski, Every coprime linear group admits a base of size two, Trans. Amer. Math. Soc. 368 (2016), no. 8, 5857–5887. Google Scholar

  • [11]

    I. M. Isaacs, Characters of π-separable groups, J. Algebra 86 (1984), no. 1, 98–128. CrossrefGoogle Scholar

  • [12]

    I. M. Isaacs, Characters of solvable groups, The Arcata Conference on Representations of Finite Groups (Arcata 1986), Proc. Sympos. Pure Math. 47, American Mathematical Society, Providence (1987), 103–109. Google Scholar

  • [13]

    I. M. Isaacs, The π-character theory of solvable groups, J. Aust. Math. Soc. Ser. A 57 (1994), no. 1, 81–102. CrossrefGoogle Scholar

  • [14]

    I. M. Isaacs, Large orbits in actions of nilpotent groups, Proc. Amer. Math. Soc. 127 (1999), no. 1, 45–50. CrossrefGoogle Scholar

  • [15]

    G. James and A. Kerber, The Representation Theory of the Symmetric Group, Encyclopedia Math. Appl. 16, Addison-Wesley, Reading, 1981. Google Scholar

  • [16]

    M. L. Lewis, Bπ-characters and quotients, preprint (2016), https://arxiv.org/abs/1609.02029.

  • [17]

    A. Moretó and T. R. Wolf, Orbit sizes, character degrees and Sylow subgroups, Adv. Math. 184 (2004), no. 1, 18–36. CrossrefGoogle Scholar

  • [18]

    G. Navarro, Characters and Blocks of Finite Groups, London Math. Soc. Lecture Note Ser. 250, Cambridge University Press, Cambridge, 1998. Google Scholar

  • [19]

    J. B. Olsson, Lower defect groups in symmetric groups, J. Algebra 104 (1986), no. 1, 37–56. CrossrefGoogle Scholar

  • [20]

    J. B. Olsson, On the p-blocks of symmetric and alternating groups and their covering groups, J. Algebra 128 (1990), no. 1, 188–213. CrossrefGoogle Scholar

  • [21]

    G. D. B. Robinson, On a conjecture by Nakayama, Trans. Roy. Soc. Canada. Sect. III. (3) 41 (1947), 20–25. Google Scholar

  • [22]

    E. P. Vdovin, Regular orbits of solvable linear p-groups, Sib. Èlektron. Mat. Izv. 4 (2007), 345–360. Google Scholar

  • [23]

    T. R. Wolf, Large orbits of supersolvable linear groups, J. Algebra 215 (1999), no. 1, 235–247. CrossrefGoogle Scholar

  • [24]

    The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.8.5, 2016, http://www.gap-system.org. Google Scholar

About the article


Received: 2017-02-10

Revised: 2016-06-23

Published Online: 2017-07-18

Published in Print: 2017-11-01


Funding Source: Henan University of Technology

Award identifier / Grant number: 2014JCYJ14

Award identifier / Grant number: 2016JJSB074

Award identifier / Grant number: 26510009

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11571129

The first author thanks the support of China Scholarship Council, Department of Mathematical Sciences of Kent State University for its hospitality, Funds of Henan University of Technology (2014JCYJ14, 2016JJSB074, 26510009), the Project of Department of Education of Henan Province (17A110004), the Projects of Zhengzhou Municipal Bureau of Science and Technology (20150249, 20140970), and the NSFC (11571129).


Citation Information: Journal of Group Theory, Volume 20, Issue 6, Pages 1155–1173, ISSN (Online) 1435-4446, ISSN (Print) 1433-5883, DOI: https://doi.org/10.1515/jgth-2017-0025.

Export Citation

© 2017 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
J. Siemons and A. Zalesski
Journal of Algebraic Combinatorics, 2018
[2]
Hung P. Tong-Viet
Journal of Algebra, 2018, Volume 503, Page 265

Comments (0)

Please log in or register to comment.
Log in