On zeros of characters of finite groups

Jinshan Zhang; Zhencai Shen; Dandan Liu

Czechoslovak Mathematical Journal (2010)

  • Volume: 60, Issue: 3, page 801-816
  • ISSN: 0011-4642

Abstract

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For a finite group G and a non-linear irreducible complex character χ of G write υ ( χ ) = { g ∈ G ∣ χ ( g ) = 0 } . In this paper, we study the finite non-solvable groups G such that υ ( χ ) consists of at most two conjugacy classes for all but one of the non-linear irreducible characters χ of G . In particular, we characterize a class of finite solvable groups which are closely related to the above-mentioned question and are called solvable ϕ -groups. As a corollary, we answer Research Problem 2 in [Y. Berkovich and L. Kazarin: Finite groups in which the zeros of every non-linear irreducible character are conjugate modulo its kernel. Houston J. Math. 24 (1998), 619–630.] posed by Y. Berkovich and L. Kazarin.

How to cite

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Zhang, Jinshan, Shen, Zhencai, and Liu, Dandan. "On zeros of characters of finite groups." Czechoslovak Mathematical Journal 60.3 (2010): 801-816. <http://eudml.org/doc/38042>.

@article{Zhang2010,
abstract = {For a finite group $G$ and a non-linear irreducible complex character $\chi $ of $G$ write $\upsilon (\chi )=\lbrace g\in G\mid \chi (g)=0\rbrace $. In this paper, we study the finite non-solvable groups $G$ such that $\upsilon (\chi )$ consists of at most two conjugacy classes for all but one of the non-linear irreducible characters $\chi $ of $G$. In particular, we characterize a class of finite solvable groups which are closely related to the above-mentioned question and are called solvable $\varphi $-groups. As a corollary, we answer Research Problem $2$ in [Y. Berkovich and L. Kazarin: Finite groups in which the zeros of every non-linear irreducible character are conjugate modulo its kernel. Houston J. Math. 24 (1998), 619–630.] posed by Y. Berkovich and L. Kazarin.},
author = {Zhang, Jinshan, Shen, Zhencai, Liu, Dandan},
journal = {Czechoslovak Mathematical Journal},
keywords = {finite groups; characters; zeros; irreducible characters; zeros of characters; finite soluble groups; conjugacy classes},
language = {eng},
number = {3},
pages = {801-816},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On zeros of characters of finite groups},
url = {http://eudml.org/doc/38042},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Zhang, Jinshan
AU - Shen, Zhencai
AU - Liu, Dandan
TI - On zeros of characters of finite groups
JO - Czechoslovak Mathematical Journal
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 60
IS - 3
SP - 801
EP - 816
AB - For a finite group $G$ and a non-linear irreducible complex character $\chi $ of $G$ write $\upsilon (\chi )=\lbrace g\in G\mid \chi (g)=0\rbrace $. In this paper, we study the finite non-solvable groups $G$ such that $\upsilon (\chi )$ consists of at most two conjugacy classes for all but one of the non-linear irreducible characters $\chi $ of $G$. In particular, we characterize a class of finite solvable groups which are closely related to the above-mentioned question and are called solvable $\varphi $-groups. As a corollary, we answer Research Problem $2$ in [Y. Berkovich and L. Kazarin: Finite groups in which the zeros of every non-linear irreducible character are conjugate modulo its kernel. Houston J. Math. 24 (1998), 619–630.] posed by Y. Berkovich and L. Kazarin.
LA - eng
KW - finite groups; characters; zeros; irreducible characters; zeros of characters; finite soluble groups; conjugacy classes
UR - http://eudml.org/doc/38042
ER -

References

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  1. Berkovich, Y., Kazarin, L., Finite groups in which the zeros of every nonlinear irreducible character are conjugate modulo its kernel, Houston J. Math. 24 (1998), 619-630. (1998) Zbl0969.20004MR1686628
  2. Bianchi, M., Chillag, D., Gillio, A., Finite groups in which every irreducible character vanishes on at most two conjugacy classes, Houston J. Math. 26 (2000), 451-461. (2000) Zbl0986.20006MR1811932
  3. Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., Wilson, R. A., Atlas of Finite Groups, Oxford Univ. Press, Oxford and New York (1985). (1985) Zbl0568.20001MR0827219
  4. Gagola, S. M., 10.2140/pjm.1983.109.363, Pacific J. Math. 109 (1983), 363-385. (1983) Zbl0536.20005MR0721927DOI10.2140/pjm.1983.109.363
  5. Gallagher, P. X., 10.1016/0021-8693(66)90048-2, J. Algebra 4 (1965), 42-45. (1965) MR0200356DOI10.1016/0021-8693(66)90048-2
  6. Gorenstein, D., Finite Groups, Harper-Row (1968). (1968) Zbl0185.05701MR0231903
  7. Huppert, B., Endliche Gruppen I, Springer-Verlag, Berlin-Heidelberg-New York (1967). (1967) Zbl0217.07201MR0224703
  8. Huppert, B., Blackburn, N., Finite groups III, Springer-Verlag, Berlin, New York (1982). (1982) Zbl0514.20002MR0662826
  9. Isaacs, I. M., Character Theory of Finite Groups, Academic Press, New York (1976). (1976) Zbl0337.20005MR0460423
  10. Isaacs, I. M., 10.4153/CJM-1989-003-2, Canada J. Math. 41 (1989), 68-82. (1989) Zbl0686.20002MR0996718DOI10.4153/CJM-1989-003-2
  11. Kuisch, E. B., Waall, R. W. Van Der, 10.1016/0021-8693(92)90027-J, J. Algebra 149 (1992), 454-471. (1992) MR1172440DOI10.1016/0021-8693(92)90027-J
  12. Macdonald, I. D., 10.1007/BF02761376, Israel J. Math. 40 (1981), 350-364. (1981) Zbl0486.20016MR0654591DOI10.1007/BF02761376
  13. Manz, O., 10.1016/0021-8693(85)90210-8, J. Algebra 94 (1985), 211-255. (1985) MR0789547DOI10.1016/0021-8693(85)90210-8
  14. Manz, O., Staszewski, R., 10.1007/BF01164012, Math. Z. 192 (1986), 383-389. (1986) Zbl0606.20011MR0845210DOI10.1007/BF01164012
  15. Manz, O., Wolf, T. R., Representations of solvable groups, Cambridge University Press, Cambridge (1993). (1993) Zbl0928.20008MR1261638
  16. Mazurov, V. D., 10.1023/A:1022676707499, Algebra and Logic 42 (2003), 29-36. (2003) Zbl1035.20025MR1988023DOI10.1023/A:1022676707499
  17. Noritzsch, T., 10.1006/jabr.1995.1213, J. Algebra 175 (1995), 767-798. (1995) MR1341745DOI10.1006/jabr.1995.1213
  18. Qian, G. H., Shi, W. J., A characterization of L 2 ( 2 f ) in terms of the number of character zeros, Contributions to Algebra and Geometry 1 (2009), 1-9. (2009) MR2499777
  19. Qian, G. H., Shi, W. J., You, X. Z., 10.1081/AGB-200039286, Comm. Algebra 32 (2004), 4809-4820. (2004) Zbl1094.20013MR2111598DOI10.1081/AGB-200039286
  20. Ren, Y. C., Zhang, J. S., On zeros of characters of finite groups and solvable ϕ -groups, Adv. Math. (China) 37 (2008), 426-436. (2008) MR2463235
  21. Seitz, G., 10.1090/S0002-9939-1968-0222160-X, Proc. Amer. Soc. 19 (1968), 459-461. (1968) Zbl0244.20010MR0222160DOI10.1090/S0002-9939-1968-0222160-X
  22. Suzuki, M., 10.1090/S0002-9947-1961-0131459-5, Soc. Trans. Amer. Math. Soc. 99 (1961), 425-470. (1961) Zbl0101.01604MR0131459DOI10.1090/S0002-9947-1961-0131459-5
  23. Veralopez, A., Veralopez, J., 10.1007/BF02764723, Israel J. Math. 51 (1985), 305-338. (1985) MR0804489DOI10.1007/BF02764723
  24. Willems, W., 10.1016/0021-8693(88)90176-7, J. Algebra 113 (1988), 511-522. (1988) Zbl0653.20014MR0929777DOI10.1016/0021-8693(88)90176-7
  25. Wong, W. J., 10.1017/S1446788700004511, J. Austral. Math. Soc. 7 (1967), 570-576. (1967) Zbl0203.02902MR0220821DOI10.1017/S1446788700004511
  26. Zhang, J. S., Shi, J. T., Shen, Z. C., Finite groups whose irreducible characters vanish on at most three conjugacy classes, (to appear) in J. Group Theory. MR2736158
  27. Zhang, J. S., Shi, W. J., 10.1515/JGT.2008.045, J. Group Theory. 11 (2008), 697-708. (2008) Zbl1159.20010MR2446151DOI10.1515/JGT.2008.045

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