Abstract
Let \(G\) be a finite group. Its spectrum \(\omega(G)\) is the set of all element orders of \(G\). The prime spectrum \(\pi(G)\) is the set of all prime divisors of the order of \(G\). The Gruenberg–Kegel graph (or the prime graph) \(\Gamma(G)\) is the simple graph with vertex set \(\pi(G)\) in which any two vertices \(p\) and \(q\) are adjacent if and only if \(pq\in\omega(G)\). The structural Gruenberg–Kegel theorem implies that the class of finite groups with disconnected Gruenberg–Kegel graphs widely generalizes the class of finite Frobenius groups, whose role in finite group theory is absolutely exceptional. The question of coincidence of Gruenberg–Kegel graphs of a finite Frobenius group and of an almost simple group naturally arises. The answer to the question is known in the cases when the Frobenius group is solvable and when the almost simple group coincides with its socle. In this short note we answer the question in the case when the Frobenius group is nonsolvable and the socle of the almost simple group is isomorphic to \(PSL_{2}(q)\) for some \(q\).
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REFERENCES
A. A. Buturlakin, “Spectra of finite linear and unitary groups,” Algebra Logic 47 (2), 91–99 (2008).
I. B. Gorshkov and N. V. Maslova, “Finite almost simple groups whose Gruenberg–Kegel graphs coincide with Gruenberg–Kegel graphs of solvable groups,” Algebra Logic 57 (2), 115–129 (2018). https://doi.org/10.1007/s10469-018-9484-7
M. R. Zinov’eva and A. S. Kondrat’ev, “Finite almost simple groups with prime graphs all of whose connected components are cliques,” Proc. Steklov Inst. Math. 295 (Suppl. 1), S178–S188 (2016). https://doi.org/10.1134/S0081543816090194
M. R. Zinov’eva and V. D. Mazurov, “On finite groups with disconnected prime graph,” Tr. Inst. Mat. Mekh. 18 (3), 99–105 (2012). https://doi.org/10.1134/S0081543813090149
A. S. Kondrat’ev and I. V. Khramtsov, “Letter to the Editor,” Trudy Inst. Mat. Mekh. UrO RAN 28 (1), 276–277 (2022). https://doi.org/10.21538/0134-4889-2022-28-1-276-277
A. S. Kondrat’ev and I. V. Khramtsov, “On finite tetraprimary groups,” Proc. Steklov Inst. Math. 279 (Suppl. 1), S43–S61 (2012). https://doi.org/10.1134/S0081543812090040
N. V. Maslova, “Classification of maximal subgroups of odd index in finite simple classical groups,” Proc. Steklov Inst. Math. 267 (Suppl. 1), S164–S188 (2009). https://doi.org/10.1134/S0081543809070153
N. V. Maslova, “On the coincidence of Grünberg–Kegel graphs of a finite simple group and its proper subgroup,” Proc. Steklov Inst. Math. 288 (Suppl. 1), S129–S141 (2015). https://doi.org/10.1134/S0081543815020133
J. N. Bray, D. F. Holt, and C. M. Roney-Dougal, The Maximal Subgroups of the Low-Dimensional Finite Classical Groups (Cambridge Univ. Press, Cambridge, 2013), London Math. Soc. Lecture Note Ser., Vol. 407.
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of Finite Groups (Oxford Univ. Press, Oxford, 1985).
G. C. Gerono, “Note sur la résolution en nombres entiers et positifs de l’équation \(x^{m}=y^{n}+1\),” Nouv. Ann. Math. (2) 9, 469–471 (1870).
D. Gorenstein, Finite Groups (Chelsea, New York, 1968).
D. Gorenstein, R. Lyons, and R. Solomon, The Classification of the Finite Simple Groups: Number 3 (Amer. Math. Soc., Providence, RI, 1998), Ser. Math. Surveys and Monographs, Vol. 40, issue 3.
K. W. Gruenberg and K. W. Roggenkamp, “Decomposition of the augmentation ideal and of the relation modules of a finite group,” Proc. London Math. Soc. (3) 31 (2), 149–166. (1975). https://doi.org/10.1112/plms/s3-31.2.149
C. Jansen, K. Lux, R. Parker, and R. Wilson, An Atlas of Brauer Characters (Clarendon, Oxford, 1995).
P. Kleidman and M. Liebeck, The Subgroup Structure of the Finite Classical Groups (Cambridge Univ. Press, Cambridge, 1990).
A. Mahmoudifar, “On some Frobenius groups with the same prime graph as the almost simple group \(PGL(2,49)\),” (2016), https://arxiv.org/pdf/1601.00146.pdf https://arxiv.org/pdf/1601.00146.pdf
J. S. Williams, “Prime graph components of finite groups,” J. Algebra 69 (2), 487–513 (1981). https://doi.org/10.1016/0021-8693(81)90218-0
A. V. Zavarnitsine, “Recognition of the simple groups \(L_{3}(q)\) by element orders,” J. Group Theory 7, 81–97 (2004). https://doi.org/10.1515/jgth.2003.044
K. Zsigmondy, “Zur Theorie der Potenzreste,” Monatsh. Math. Phys. 3 (1), 265–284 (1892). https://doi.org/10.1007/BF01692444
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This work was supported by the Russian Science Foundation (project no. 19-71-10067).
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Translated from Trudy Instituta Matematiki i Mekhaniki UrO RAN, Vol. 28, No. 2, pp. 168 - 175, 2022 https://doi.org/10.21538/0134-4889-2022-28-2-168-175.
Translated by E. Vasil’eva
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Maslova, N.V., Ilenko, K.A. On the Coincidence of Gruenberg–Kegel Graphs of an Almost Simple Group and a Nonsolvable Frobenius Group. Proc. Steklov Inst. Math. 317 (Suppl 1), S130–S135 (2022). https://doi.org/10.1134/S0081543822030117
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DOI: https://doi.org/10.1134/S0081543822030117