Finite 4-primary groups with disconnected Gruenberg--Kegel graph containig a triangle

A. S. Kondrat'ev

Algebra i logika, Tome 62 (2023) no. 1, pp. 76-92.

Résumé

We give a description of finite 4-primary groups with disconnected Gruenberg–Kegel graph containing a triangle. As a corollary, finite groups whose Gruenberg–Kegel graph coincides with the Gruenberg–Kegel graph of $^3D_4(2)$ are exemplified, which generalizes V. D. Mazurov' description of finite groups isospectral to the group $^3D_4(2)$.

Détail

Keywords: finite group, 4-primary group, chief factor, Gruenberg–Kegel graph (prime graph), Brauer character.

Bibliographie

[1] A. S. Kondratev, “Konechnye gruppy s zadannymi svoistvami ikh grafov prostykh chisel”, Algebra i logika, 55:1 (2016), 113–120 | MR | Zbl

[2] A. S. Kondrat'ev, N. A. Minigulov, “Finite almost simple groups whose Gruenberg–Kegel graphs as abstract graphs are isomorphic to subgraphs of the Gruenberg–Kegel graph of the alternating group $A_{10}$”, Sib. elektron. matem. izv., 15 (2018), 1378–1382 | MR | Zbl

[3] A. S. Kondrat'ev, N. A. Minigulov, “On finite non-solvable groups whose Gruenberg–Kegel graphs are isomorphic to the paw”, Commun. Math. Stat., 10:4 (2022), 653–667 | DOI | MR | Zbl

[4] A. S. Kondrat'ev, N. A. Minigulov, Finite solvable groups whose Gruenberg–Kegel graphs are isomorphic to the paw, Tr. IMM UrO RAN, 28, no. 2, 2022, 269–273 | MR

[5] A. S. Kondratev, I. V. Khramtsov, “O konechnykh triprimarnykh gruppakh”, Tr. IMM UrO RAN, 16, no. 3, 2010, 150–158

[6] A. S. Kondratev, I. V. Khramtsov, “O konechnykh chetyreprimarnykh gruppakh”, Tr. IMM UrO RAN, 17, no. 4, 2011, 142—159 ; письмо в редакцию, Тр. ИММ УрО РАН, 28, No 1, 2022, 276–277

[7] A. S. Kondratev, I. D. Suprunenko, I. V. Khramtsov, “O konechnykh 4-primarnykh gruppakh s nesvyaznym grafom Gryunberga–Kegelya i kompozitsionnym faktorom, izomorfnym $L_3(17)$ ili $Sp_4(4)$”, Tr. IMM UrO RAN, 28, no. 1, 2022, 139–155

[8] V. D. Mazurov, “Neraspoznavaemost konechnoi prostoi gruppy $^3D_4(2)$ po spektru”, Algebra i logika, 52:5 (2013), 601–605 | MR | Zbl

[9] S. Dolfi, E. Dzhabara, M. S. Lyuchido, “$C55$-gruppy”, Sib. matem. zh., 45:6 (2004), 1285–1298 | MR | Zbl

[10] S. Astill, K. Parker, R. Valdeker, “O gruppakh, v kotorykh tsentralizatory elementov poryadka 5 yavlyayutsya 5-gruppami”, Sib. matem. zh., 53:5 (2012), 967–977 | MR | Zbl

[11] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of finite groups, Clarendon Press, Oxford, 1985 | MR | Zbl

[12] D. Gorienstein, Finite groups, Harper's Ser. Modern Math., Harper Row Publ, New York–Evanston–London, 1968 | MR

[13] C. Jansen, K. Lux, R. Parker, R. Wilson, An atlas of Brauer characters, Lond. Math. Soc. Monogr., New Ser., 11, Clarendon Press, Oxford, 1995 | MR | Zbl

[14] V. D. Mazurov, “O mnozhestve poryadkov elementov konechnoi gruppy”, Algebra i logika, 33:1 (1994), 81–89 | MR | Zbl

[15] R. Steinberg, Lektsii o gruppakh Shevalle, Biblioteka sb. "‘Matematika"’, Mir, M., 1975

[16] A. V. Zavarnitsine, “Fixed points of large prime-order elements in the equicharacteristic action of linear and unitary groups”, Sib. elektron. matem. izv., 8 (2011), 333–340 | MR | Zbl

[17] A. S. Kondratev, A. A. Osinovskaya, I. D. Suprunenko, “O povedenii elementov prostogo poryadka iz tsikla Zingera v predstavleniyakh spetsialnoi lineinoi gruppy”, Tr. IMM UrO RAN, 19, no. 3, 2013, 179–186

[18] J. S. Williams, “Prime graph components of finite groups”, J. Algebra, 69:4 (1981), 487–513 | DOI | MR | Zbl

[19] V. M. Busarkin, Yu. M. Gorchakov, Konechnye rasscheplyaemye gruppy, Nauka, M., 1968

[20] I. V. Khramtsov, “O konechnykh gruppakh s nesvyaznym grafom prostykh chisel, imeyuschim kompozitsionnyi faktor, izomorfnyi gruppe $L_2(81)$”, Teoriya grupp i ee prilozheniya, Tr. mezhd. shkoly-konf. po teorii grupp, posvyasch. 70-letiyu V. V. Kabanova, izd-vo KGBU, Nalchik, 2014, 56–58