Geodesic
Finite groups with three classes of maximal subgroups
Sbornik. Mathematics, Tome 59 (1988) no. 1, pp. 223-236
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Using the classification of finite simple groups, a description is obtained of finite groups having exactly three classes of conjugate maximal subgroups. If such a group is not solvable, then its factor group modulo its Frattini subgroup is isomorphic to $\mathrm{PSL}(2,7)$ or $\mathrm{PSL}(2,2^p)$, where $p$ is a prime. To prove this result, it was necessary to describe finite groups having at most two classes of conjugate nonnormal maximal subgroups. Bibliography: 28 titles. 
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V. A. Belonogov. Finite groups with three classes of maximal subgroups. Sbornik. Mathematics, Tome 59 (1988) no. 1, pp. 223-236. http://geodesic.mathdoc.fr/item/SM_1988_59_1_a11/

[1] Belonogov V. A., “Konechnye gruppy s edinstvennym klassom nenilpotentnykh maksimalnykh podgrupp”, Sib. matem. zhurn., V:5 (1964), 987–995 | MR

[2] Belonogov V. A., “Konechnye gruppy s edinstvennym klassom neinvariantnykh maksimalnykh podgrupp”, Izv. AN BSSR. Ser. fiz.-matem. nauk, no. 3, 1969, 114–117 | MR | Zbl

[3] Kazarin L. S., “O probleme S. A. Chunikhina”, Issledovaniya po teorii grupp, UNTs AN SSSR, Sverdlovsk, 1984, 81–99 | MR

[4] Levchuk V. M., Nuzhin Ya. N., “O stroenii grupp Ri”, Algebra i logika, 24:1 (1985), 26–41 | MR | Zbl

[5] Mazurov V. D., Mazurova N. P., “Shirokie podgruppy sporadicheskikh grupp, I”, Strukturnye voprosy teorii grupp, UNTs AN SSSR, Sverdlovsk, 1986, 71–84 | MR

[6] Putilov S. V., “Abnormalnye maksimalnye podgruppy i razreshimost konechnykh grupp”, Matem. zametki, 34:3 (1983), 347–353 | MR | Zbl

[7] Steinberg R., Lektsii o gruppakh Shevalle, Mir, M., 1975 | MR | Zbl

[8] Syskin S. A., “Abstraktnye svoistva prostykh sporadicheskikh grupp”, UMN, 35:5 (1980), 181–212 | MR | Zbl

[9] Ball R. W., “Maximal subgroups of symmetric groups”, Trans. Amer. Math. Soc., 121:2 (1966), 393–407 | DOI | MR | Zbl

[10] Bloom D. M., “The subgroups of $\text{PSL}(3,q)$ for odd $q$”, Trans. Amer. Math. Soc., 127:1 (1967), 150–178 | DOI | MR | Zbl

[11] Carter R. W., Simple groups of Lie type, John Wiley and Sons, London, 1972 | MR | Zbl

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

[13] Davis S. L., Solomon R., “Some sporadic characterizations”, Commun. Algebra, 9:17 (1981), 1725–1742 | DOI | MR | Zbl

[14] Dickson L. E., Linear groups with an exposition of the Galois field theory, Dover, New York, 1958 | MR | Zbl

[15] Fischer J., McKay J., “The nonabelian simple groups $G$, $|G|10^6$ – maximal subgroups”, Math. Comput., 32:144 (1978), 1293–1302 | DOI | MR | Zbl

[16] Gorenstein D., Finite groups, Harper and Row, New York, 1968 | MR | Zbl

[17] Gorenstein D., The classification of finite simple groups, Vol. 1: Groups of noncharacteristic 2 type, Plenum Press, New York, 1983 | MR | Zbl

[18] Gorenstein D., Lyons R., “The local structure of finite groups of characteristic 2 type”, Memoirs of Amer. Math. Soc., 42:276 (1983), 1–731 | MR

[19] Hartley R. W., “Determination of the ternary collineation groups whose coefficients lie in the $\text{GF}(2^n)$”, Ann. Math., 27:2 (1925), 140–158 | DOI | MR | Zbl

[20] Huppert B., Endlichen gruppen, Vol. 1, Springer, Berlin, 1967 | MR | Zbl

[21] Kantor W. M., “Subgroups of classical groups generated by long root elements”, Trans. Amer. Math. Soc., 248:2 (1979), 347–379 | DOI | MR | Zbl

[22] Kantor W. M., Liebler R. A., “The rank 3 permutation representations of the finite classical groups”, Trans. Amer. Math. Soc., 271:1 (1982), 1–71 | DOI | MR | Zbl

[23] Pazderski G., “Über maximal Untergruppen endlicher gruppen”, Math. Nachr., 26:6 (1964), 307–319 | MR | Zbl

[24] Seitz G. M., “Flag-transitive subgroups of Chevalley groups”, Ann. Math., 97:1 (1973), 27–56 | DOI | MR | Zbl

[25] Seitz G. M., “Subgroups of finite groups of Lie tvpe”, J. Algebra, 61 (1979), 16–27 | DOI | MR | Zbl

[26] Smith S. D., “Large extraspecial subgroups of widths 4 and 6”, J. Algebra, 58 (1979), 251–281 | DOI | MR | Zbl

[27] Suzuki M., “On class of doubly transitive groups”, Ann. Math., 75:1 (1962), 105–145 | DOI | MR | Zbl

[28] Tchakerian K. B., “The maximal subgroups of the Tits simple group”, Dokl. Bolg. AN, 34:12 (1981), 1637 | MR | Zbl