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On the orders of vanishing elements of finite groups. (English) Zbl 07341190
Summary: Let $$G$$ be a finite group and $$p$$ be a prime. Let $$\operatorname{Vo}(G)$$ denote the set of the orders of vanishing elements, $$\operatorname{Vo}_p(G)$$ be the subset of $$\operatorname{Vo}(G)$$ consisting of those orders of vanishing elements divisible by $$p$$ and $$\operatorname{Vo}_{p^\prime}(G)$$ be the subset of $$\operatorname{Vo}(G)$$ consisting of those orders of vanishing elements not divisible by $$p$$. Dolfi, Pacifi, Sanus and Spiga proved that if $$a$$ is not a $$p$$-power for all $$a \in \operatorname{Vo}(G)$$, then $$G$$ has a normal Sylow $$p$$-subgroup. In another article, the same authors also show that if $$\operatorname{Vo}_{p^\prime}(G) = \varnothing$$, then $$G$$ has a normal nilpotent $$p$$-complement. These results are variations of the well known Ito-Michler and Thompson theorems. In this article we study solvable groups such that $$| \operatorname{Vo}_p(G) | = 1$$ and show that $$P^\prime$$ is subnormal. This is analogous to the work of Isaacs, Moréto, Navarro and Tiep where they considered groups with just one character degree divisible by $$p$$. We also study certain finite groups $$G$$ such that $$| \operatorname{Vo}_{p^\prime}(G) | = 1$$ and we prove that $$G$$ has a normal subgroup $$L$$ such that $$G / L$$ a normal $$p$$-complement and $$L$$ has a normal $$p$$-complement. This is analogous to the recent work of Giannelli, Rizo and Schaeffer Fry on character degrees with a few $$p^\prime$$-character degrees. Bubboloni, Dolfi and Spiga studied finite groups such that every vanishing element is of order $$p^m$$ for some integer $$m \geqslant 1$$. As a generalization, we investigate groups such that $$\gcd(a, b) = p^m$$ for some integer $$m \geqslant 0$$, for all $$a, b \in \operatorname{Vo}(G)$$. We also study finite solvable groups whose irreducible characters vanish only on elements of prime power order.
##### MSC:
 20C15 Ordinary representations and characters
GAP
Full Text:
##### References:
 [1] Bechtell, H., The Theory of Groups, New Hampshire (1971), Addison-Wesley [2] 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 [3] Brough, J., On vanishing criteria that control finite group structure, J. Algebra, 458, 207-215 (2016) · Zbl 1353.20014 [4] Brown, R., Frobenius groups and classical maximal orders, Mem. Amer. Math. Soc., 151, 717 (2001), viii+110 · Zbl 0976.20002 [5] 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 [6] Bugeaud, Y.; Cao, Z.; Mignotte, M., On simple $$K_4$$-groups, J. Algebra, 241, 658-668 (2001) · Zbl 0989.20017 [7] 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 [8] 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 [9] 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 [10] 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 [11] Dornhoff, L., Group Representation Theory, Part A: Ordinary Representation Theory (1971), Marcel Dekker, Inc.: Marcel Dekker, Inc. New York · Zbl 0227.20002 [12] 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 [13] 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 [14] Granville, A.; Ono, K., Defect zero p-blocks for finite simple groups, Trans. Amer. Math. Soc., 348, 331-347 (1996) · Zbl 0855.20007 [15] Herzog, M., On finite simple groups of order divisible by three primes only, J. Algebra, 10, 383-388 (1968) · Zbl 0167.29101 [16] Higman, G., Finite groups in which every element has prime power order, J. London Math. Soc., 32, 335-342 (1957) · Zbl 0079.03204 [17] Isaacs, I. M., Character Theory of Finite Groups (2006), Amer. Math. Soc.: Amer. Math. Soc. Rhode Island · Zbl 1119.20005 [18] 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 [19] Madanha, S. Y.; Rodrigues, B. G., Finite groups with some restriction on the vanishing set, Comm. Algebra, 47, 5474-5481 (2020) · Zbl 07253635 [20] Magaard, K.; Tong-Viet, H. P., Character degree sums in finite non-solvable groups, J. Group Theory, 14, 53-57 (2011) · Zbl 1242.20012 [21] Malle, G.; Navarro, G.; Olsson, J. B., Zeros of characters of finite groups, J. Group Theory, 3, 353-368 (2002) · Zbl 0965.20003 [22] 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 [23] Robinson, D. J.S., A Course in the Theory of Finite Groups (1995), Springer Verlag: Springer Verlag New York-Berlin [24] Suzuki, M., Finite groups with nilpotent centralizers, Trans. Amer. Math. Soc., 99, 425-470 (1961) · Zbl 0101.01604 [25] The, G. A.P., GAP - groups, algorithms and programming, version 4.8.7 (2017) [26] 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
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