# zbMATH — the first resource for mathematics

On the orders of vanishing elements of finite groups. (English) Zbl 07341190
An element $$x$$ of a finite group $$G$$ is called a vanishing element if $$\chi(x) = 0$$ for some irreducible character $$\chi$$ of $$G$$. The paper under review is concerned with $$\mathrm{Vo}(G)$$, the set of orders of the vanishing elements in $$G$$. Let $$p$$ be a fixed prime number.
In Theorem A, the author deals with the case where $$\mathrm{Vo}(G)$$ contains precisely one number not divisible by $$p$$. He proves that $$G$$ is solvable and that – under certain additional hypotheses – the $$p$$-length of $$G$$ is at most 2.
In Theorem B, the author considers the situation where $$G$$ is solvable and $$\mathrm{Vo}(G)$$ contains exactly one number divisible by $$p$$. He proves that the commutator subgroup of a Sylow $$p$$-subgroup $$P$$ of $$G$$ is subnormal in $$G$$ and that $$P/O_p(G)$$ is cyclic.
In Theorem C, the author supposes that $$p>7$$ and that $$\mathrm{gcd}(a,b)$$ is a power of $$p$$ whenever $$a,b \in \mathrm{Vo}(G)$$ are distinct. He shows that $$G$$ is solvable and that the $$p$$-length of $$G$$ is at most 2.
Finally, in Theorem D, the author describes the finite solvable groups $$G$$ with the property that the numbers in $$\mathrm{Vo}(G)$$ are prime powers.
##### MSC:
 20C15 Ordinary representations and characters
##### Keywords:
irreducible character; vanishing element
GAP
Full Text:
##### References:
  Bechtell, H., The Theory of Groups, New Hampshire (1971), Addison-Wesley  Bianchi, M.; Chillag, D.; Lewis, M.; Pacifici, E., Character degree graphs that are complete graphs, Proc. Amer. Math. Soc., 135, 671-676 (2007) · Zbl 1112.20006  Brough, J., On vanishing criteria that control finite group structure, J. Algebra, 458, 207-215 (2016) · Zbl 1353.20014  Brown, R., Frobenius groups and classical maximal orders, Mem. Amer. Math. Soc., 151, 717 (2001), viii+110 · Zbl 0976.20002  Bubboloni, D.; Dolfi, S.; Spiga, P., Finite groups whose irreducible characters vanish only on p-elements, J. Pure Appl. Algebra, 213, 370-376 (2009) · Zbl 1162.20004  Bugeaud, Y.; Cao, Z.; Mignotte, M., On simple $$K_4$$-groups, J. Algebra, 241, 658-668 (2001) · Zbl 0989.20017  Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A., Atlas of Finite Groups (1985), Clarendon Press: Clarendon Press Oxford · Zbl 0568.20001  Dolfi, S.; Pacifici, E.; Sanus, L.; Spiga, P., On the orders of zeros of irreducible characters, J. Algebra, 321, 345-352 (2009) · Zbl 1162.20005  Dolfi, S.; Pacifici, E.; Sanus, L.; Spiga, P., On the vanishing prime graph of finite groups, J. London Math. Soc. (2), 82, 167-183 (2010) · Zbl 1203.20024  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  Dornhoff, L., Group Representation Theory, Part A: Ordinary Representation Theory (1971), Marcel Dekker, Inc.: Marcel Dekker, Inc. New York · Zbl 0227.20002  Franciosi, S.; de Giovanni, F.; Heineken, H.; Newell, M. L., On the Fitting length of a soluble product of nilpotent groups, Arch. Math., 57, 313-318 (1991) · Zbl 0774.20021  Gianelli, E.; Rizo, N.; Schaeffer Fry, A. A., Groups with few $$p^\prime$$-character degrees, J. Pure Appl. Algebra, 224, Article 106338 pp. (2020) · Zbl 07187749  Granville, A.; Ono, K., Defect zero p-blocks for finite simple groups, Trans. Amer. Math. Soc., 348, 331-347 (1996) · Zbl 0855.20007  Herzog, M., On finite simple groups of order divisible by three primes only, J. Algebra, 10, 383-388 (1968) · Zbl 0167.29101  Higman, G., Finite groups in which every element has prime power order, J. London Math. Soc., 32, 335-342 (1957) · Zbl 0079.03204  Isaacs, I. M., Character Theory of Finite Groups (2006), Amer. Math. Soc.: Amer. Math. Soc. Rhode Island · Zbl 1119.20005  Isaacs, I. M.; Moréto, A.; Navarro, G.; Tiep, P. H., Groups with just one character degree divisible by a given prime, Trans. Amer. Math. Soc., 361, 6521-6547 (2009) · Zbl 1203.20005  Madanha, S. Y.; Rodrigues, B. G., Finite groups with some restriction on the vanishing set, Comm. Algebra, 47, 5474-5481 (2020) · Zbl 07253635  Magaard, K.; Tong-Viet, H. P., Character degree sums in finite non-solvable groups, J. Group Theory, 14, 53-57 (2011) · Zbl 1242.20012  Malle, G.; Navarro, G.; Olsson, J. B., Zeros of characters of finite groups, J. Group Theory, 3, 353-368 (2002) · Zbl 0965.20003  Qian, G., Bounding the Fitting height of a finite solvable group by the number of zeros in a character table, Proc. Amer. Math. Soc., 130, 3171-3176 (2002) · Zbl 1007.20008  Robinson, D. J.S., A Course in the Theory of Finite Groups (1995), Springer Verlag: Springer Verlag New York-Berlin  Suzuki, M., Finite groups with nilpotent centralizers, Trans. Amer. Math. Soc., 99, 425-470 (1961) · Zbl 0101.01604  The, G. A.P., GAP - groups, algorithms and programming, version 4.8.7 (2017)  Zhang, J.; Li, Z.; Shao, C., Finite groups whose irreducible characters vanish only on elements of prime power order, International Electronic J. Algebra, 9, 114-123 (2011) · Zbl 1259.20010
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.