# Article

Full entry | PDF   (0.2 MB)
Keywords:
finite groups; characters; zeros
Summary:
For a finite group $G$ and a non-linear irreducible complex character $\chi$ of $G$ write $\upsilon (\chi )=\{g\in G\mid \chi (g)=0\}$. 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.
References:
[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. MR 1686628 | Zbl 0969.20004
[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. MR 1811932 | Zbl 0986.20006
[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). MR 0827219 | Zbl 0568.20001
[4] Gagola, S. M.: Characters vanishing on all but two conjugacy classes. Pacific J. Math. 109 (1983), 363-385. DOI 10.2140/pjm.1983.109.363 | MR 0721927 | Zbl 0536.20005
[5] Gallagher, P. X.: Zeros of characters of finite groups. J. Algebra 4 (1965), 42-45. DOI 10.1016/0021-8693(66)90048-2 | MR 0200356
[6] Gorenstein, D.: Finite Groups. Harper-Row (1968). MR 0231903 | Zbl 0185.05701
[7] Huppert, B.: Endliche Gruppen I. Springer-Verlag, Berlin-Heidelberg-New York (1967). MR 0224703 | Zbl 0217.07201
[8] Huppert, B., Blackburn, N.: Finite groups III. Springer-Verlag, Berlin, New York (1982). MR 0662826 | Zbl 0514.20002
[9] Isaacs, I. M.: Character Theory of Finite Groups. Academic Press, New York (1976). MR 0460423 | Zbl 0337.20005
[10] Isaacs, I. M.: Coprime group actions fixing all nonlinear irreducible characters. Canada J. Math. 41 (1989), 68-82. DOI 10.4153/CJM-1989-003-2 | MR 0996718 | Zbl 0686.20002
[11] Kuisch, E. B., Waall, R. W. Van Der: Homogeneous character induction. J. Algebra 149 (1992), 454-471. DOI 10.1016/0021-8693(92)90027-J | MR 1172440
[12] Macdonald, I. D.: Some $p$-groups of Frobenius and extra-special type. Israel J. Math. 40 (1981), 350-364. DOI 10.1007/BF02761376 | MR 0654591 | Zbl 0486.20016
[13] Manz, O.: Endliche auflosbare Gruppen deren samtliche charactergrade primzahl-potenzen sind. J. Algebra 94 (1985), 211-255. DOI 10.1016/0021-8693(85)90210-8 | MR 0789547
[14] Manz, O., Staszewski, R.: Some applications of a fundamental theorem by Gluck and Wolf in the character theory of finite groups. Math. Z. 192 (1986), 383-389. DOI 10.1007/BF01164012 | MR 0845210 | Zbl 0606.20011
[15] Manz, O., Wolf, T. R.: Representations of solvable groups. Cambridge University Press, Cambridge (1993). MR 1261638 | Zbl 0928.20008
[16] Mazurov, V. D.: Groups containing a self-centralizing subgroup of order $3$. Algebra and Logic 42 (2003), 29-36. DOI 10.1023/A:1022676707499 | MR 1988023 | Zbl 1035.20025
[17] Noritzsch, T.: Groups having three irreducible character degrees. J. Algebra 175 (1995), 767-798. DOI 10.1006/jabr.1995.1213 | MR 1341745
[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. MR 2499777
[19] Qian, G. H., Shi, W. J., You, X. Z.: Conjugacy classes outside a normal subgroup. Comm. Algebra 32 (2004), 4809-4820. DOI 10.1081/AGB-200039286 | MR 2111598 | Zbl 1094.20013
[20] Ren, Y. C., Zhang, J. S.: On zeros of characters of finite groups and solvable $\varphi$-groups. Adv. Math. (China) 37 (2008), 426-436. MR 2463235
[21] Seitz, G.: Finite groups having only one irreducible representation of degree greater than one. Proc. Amer. Soc. 19 (1968), 459-461. DOI 10.1090/S0002-9939-1968-0222160-X | MR 0222160 | Zbl 0244.20010
[22] Suzuki, M.: Finite groups with nilpotent centralizers. Soc. Trans. Amer. Math. Soc. 99 (1961), 425-470. DOI 10.1090/S0002-9947-1961-0131459-5 | MR 0131459 | Zbl 0101.01604
[23] Veralopez, A., Veralopez, J.: Classification of finite groups according to the number of conjugacy classes. Israel J. Math. 51 (1985), 305-338. DOI 10.1007/BF02764723 | MR 0804489
[24] Willems, W.: Blocks of defect zero in finite simple groups. J. Algebra 113 (1988), 511-522. DOI 10.1016/0021-8693(88)90176-7 | MR 0929777 | Zbl 0653.20014
[25] Wong, W. J.: Finite groups with a self-centralizing subgroup of order $4$. J. Austral. Math. Soc. 7 (1967), 570-576. DOI 10.1017/S1446788700004511 | MR 0220821 | Zbl 0203.02902
[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. MR 2736158
[27] Zhang, J. S., Shi, W. J.: Two dual questions on zeros of characters of finite groups. J. Group Theory. 11 (2008), 697-708. DOI 10.1515/JGT.2008.045 | MR 2446151 | Zbl 1159.20010

Partner of