×

zbMATH — the first resource for mathematics

Finite groups with some restriction on the vanishing set. (English) Zbl 07253635
Summary: Let \(x\) be an element of a finite group \(G\) and denote the order of \(x\) by \(\text{ord}(x).\) We consider a finite group \(G\) such that \(\text{gcd(ord}(x), \text{ord}(y)) \leqslant 2\) for any two vanishing elements \(x\) and \(y\) contained in distinct conjugacy classes. We show that such a group \(G\) is solvable. When \(G\) with the property above is supersolvable, we show that \(G\) has a normal metabelian 2-complement.

MSC:
20C15 Ordinary representations and characters
Software:
GAP
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bechtell, H., The Theory of Groups (1971), New Hampshire: Addison-Wesley, New Hampshire · Zbl 0516.20009
[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, 451-461 (2000) · Zbl 0986.20006
[3] Bianchi, M.; Chillag, D.; Lewis, M.; Pacifici, E., Character degree graphs that are complete graphs, Proc. Am. Math. Soc, 135, 3, 671-676 (2007) · Zbl 1112.20006
[4] Brough, J., On vanishing criteria that control finite group structure, J. Algebra, 458, 207-215 (2016) · Zbl 1353.20014
[5] Brown, R., Frobenius groups and classical maximal orders, Mem. Am. Math. Soc, 151, 717, viii+ (2001) · Zbl 0976.20002
[6] Bubboloni, D.; Dolfi, S.; Spiga, P., Finite groups whose irreducible characters vanish only on p-elements, J. Pure Appl. Algebra, 213, 3, 370-376 (2009) · Zbl 1162.20004
[7] Chillag, D., On zeros of characters of finite groups, Proc. Am. Math. Soc, 127, 4, 977-983 (1999) · Zbl 0917.20007
[8] Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A., Atlas of Finite Groups (1985), Oxford: Clarendon Press, Oxford · Zbl 0568.20001
[9] Dolfi, S.; Pacifici, E.; Sanus, L.; Spiga, P., On the orders of zeros of irreducible characters, J. Algebra, 321, 1, 345-352 (2009) · Zbl 1162.20005
[10] Dolfi, S.; Pacifici, E.; Sanus, L.; Spiga, P., On the vanishing prime graph of finite groups, J. London Math. Soc, 82, 1, 167-183 (2010) · Zbl 1203.20024
[11] Dolfi, S.; Pacifici, E.; Sanus, L.; Spiga, P., On the vanishing prime graph of solvable groups, J. Group Theory, 13, 189-206 (2010) · Zbl 1196.20029
[12] GAP Group., GAP-Groups, Algorithms and Programming, Version 4.8.4 (2016)
[13] Granville, A.; Ono, K., Defect zero p-blocks for finite simple groups, Trans. Am. Math. Soc, 348, 1, 331-347 (1996) · Zbl 0855.20007
[14] Isaacs, I. M., Character Theory of Finite Groups (2006), Rhode Island: American Mathematical Society, Rhode Island · Zbl 1119.20005
[15] Isaacs, I. M.; Navarro, G.; Wolf, T. R., Finite group elements where no irreducible character vanishes, J. Algebra, 222, 2, 413-423 (1999) · Zbl 0959.20009
[16] Li, Z.; Shao, C.; Zhang, J.; Li, Z., Finite groups whose irreducible characters vanish only on elements of prime power order, Int. Electron. J. Algebra, 9, 114-123 (2001) · Zbl 1259.20010
[17] Madanha, S. Y., Zeros of primitive characters of finite groups, J. Group Theory, 23, 2, 193-216 (2020) · Zbl 07177053
[18] Magaard, K.; Tong-Viet, H. P., Character degree sums in finite non-solvable groups, J. Group Theory, 14, 53-57 (2011) · Zbl 1242.20012
[19] Malle, G.; Navarro, G.; Olsson, J. B., Zeros of characters of finite groups, J. Group Theory, 3, 353-368 (2000) · Zbl 0965.20003
[20] Qian, G., Bounding the fitting height of a finite solvable group by the number of zeros in a character table, Proc. Am. Math. Soc, 130, 11, 3171-3176 (2002) · Zbl 1007.20008
[21] Robati, S. M., Groups whose set of vanishing elements is the union of at most three conjugacy classes, Bull. Belg. Math. Soc. Simon Stevin, 26, 1, 85-89 (2019) · Zbl 07060317
[22] Robinson, D. J. S., A course in the theory of finite groups (1995), New York, Berlin: Springer Verlag, New York, Berlin
[23] Suzuki, M., Finite groups with nilpotent centralizers, Trans. Am. Math. Soc, 99, 3, 425-470 (1961) · Zbl 0101.01604
[24] Suzuki, M., On a class of double transitive groups, Ann. Math, 75, 1, 105-145 (1962) · Zbl 0106.24702
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.