On the Coincidence of Gruenberg–Kegel Graphs of an Almost Simple Group and a Nonsolvable Frobenius Group

N. V. Maslova   ; K. A. Il'enko

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 28 (2022) no. 2, pp. 168-175

Résumé

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 $q$.

Détail

Keywords: finite group, Gruenberg–Kegel graph (prime graph), almost simple group.
Mots-clés : nonsolvable Frobenius group

Bibliographie

[1] Buturlakin A.A., “Spektry konechnykh lineinykh i unitarnykh grupp”, Algebra i logika, 47:2 (2008), 157–173 | MR | Zbl

[2] Gorshkov I.B., Maslova N.V., “Konechnye pochti prostye gruppy s grafami Gryunberga–Kegelya kak u razreshimykh grupp”, Algebra i logika, 57:2 (2018), 175–196 | DOI | MR | Zbl

[3] Zinoveva M.R., Kondratev A.S., “Konechnye pochti prostye gruppy s grafami prostykh chisel, vse svyaznye komponenty kotorykh yavlyayutsya klikami”, Tr. In-ta matematiki i mekhaniki UrO RAN, 21:3 (2015), 132–141

[4] Zinoveva M.R., Mazurov V.D., “O konechnykh gruppakh s nesvyaznym grafom prostykh chisel”, Tr. In-ta matematiki i mekhaniki UrO RAN, 18:3 (2012), 99–105

[5] Kondratev A.S., Khramtsov I.V., “Pismo v redaktsiyu”, Tr. In-ta matematiki i mekhaniki UrO RAN, 28:1 (2022), 276–277 | DOI | MR

[6] Kondratev A.S., Khramtsov I.V., “O konechnykh chetyreprimarnykh gruppakh”, Tr. In-ta matematiki i mekhaniki UrO RAN, 17:4 (2011), 142–159

[7] Maslova N.V., “Klassifikatsiya maksimalnykh podgrupp nechetnogo indeksa v konechnykh prostykh klassicheskikh gruppakh”, Tr. In-ta matematiki i mekhaniki UrO RAN, 14:4 (2008), 100–118

[8] Maslova N.V., “O sovpadenii grafov Gryunberga–Kegelya konechnoi prostoi gruppy i ee sobstvennoi podgruppy”, Tr. In-ta matematiki i mekhaniki UrO RAN, 20:1 (2014), 156–168

[9] Bray J.N., Holt D.F., Roney-Dougal C.M., The maximal subgroups of the low-dimensional finite classical groups, London Math. Soc. Lect. Note Ser., 407, Cambridge Univ. Press, Cambridge, 2013, 438 pp. | DOI | MR | Zbl

[10] J.H. Conway [et. al.], Atlas of finite groups, Clarendon Press, Oxford, 1985, 252 pp. | MR | Zbl

[11] Gérono G.C., “Note sur la résolution en nombres entiers et positifs de l' équation $x^m = y^n+1$”, Nouv. Ann. Math (2), 9 (1870), 469–471

[12] Gorenstein D., Finite groups, Chelsea, NY, 1968, 520 pp. | MR

[13] Gorenstein D., Lyons R., Solomon R., The classification of the finite simple groups. Number 3, Mathematical Surveys and Monographs, 40.3, American Math. Soc., Providence, RI, 1998 | MR

[14] Gruenberg K.W., Roggenkamp K.W., “Decomposition of the augmentation ideal and of the relation modules of a finite group”, Proc. London Math. Soc. (3), 31:2 (1975), 149–166 | DOI | MR | Zbl

[15] C. Jansen [et. al.], An atlas of Brauer characters, Clarendon Press, Oxford, 1995, 327 pp. | Zbl

[16] Kleidman P., Liebeck M., The subgroup structure of the finite classical groups, Cambridge Univ. Press, Cambridge, 1990, 304 pp. | MR | Zbl

[17] Mahmoudifar A., On some Frobenius groups with the same prime graph as the almost simple group $PGL(2,49)$, [e-resource], 2016, 7 pp., arXiv: 1601.00146 | MR

[18] Williams J. S., “Prime graph components of finite groups”, J. Algebra, 69:2 (1981), 487–513 | DOI | MR | Zbl

[19] Zavarnitsine A. V., “Recognition of the simple groups $L_3(q)$ by element orders”, J. Group Theory, 7:1 (2004), 81–97 | MR | Zbl

[20] Zsigmondy K., “Zur Theorie der Potenzreste”, J. Monatshefte Math. Phys., 3 (1892), 265–284 | DOI | MR | Zbl