Invariants of finite solvable groups

V. S. Monakhov ; A. A. Trofimuk

Problemy fiziki, matematiki i tehniki, no. 1 (2010), pp. 63-81.

Résumé

The review of the results connected with invariants of finite solvable groups is given in this article. These results adjoin to the researches of authors. Section 1 contains the list of used designations and definitions. The data about invariants of solvable groups with given structure of Sylow subgroups are collected in Section 2. The estimations of the invariants depending on indexes of the maximal subgroups are listed in Section 3. The information about nilpotent $\pi$-length of $\pi$-solvable groups are contained in Section 4. Open questions are formulated.

Détail

Mots-clés : solvable group
Keywords: derived length, nilpotent length, $p$-length, $\pi$-length, nilpotent $\pi$-length, chief rank, $p$-rank.

Bibliographie

[1] S. A. Chunikhin, Podgruppy konechnykh grupp, Nauka i tekhnika, Minsk, 1964, 158 pp. | MR | Zbl

[2] L. A. Shemetkov, Formatsii konechnykh grupp, Nauka, M., 1978, 272 pp. | MR | Zbl

[3] V. S. Monakhov, Vvedenie v teoriyu konechnykh grupp i ikh klassov, Vysheishaya shkola, Minsk, 2006, 207 pp.

[4] B. Huppert, Endliche Gruppen, v. I, Springer, Berlin–Heidelberg–New York, 1967 | MR | Zbl

[5] P. Hall, G. Higman, “The $p$-length of a $p$-soluble groups and reduction theorems for Burnside's problem”, Proc. London Math. Soc., 3:7 (1956), 1–42 | DOI | MR | Zbl

[6] B. Huppert, N. Blackburn, Endliche Gruppen, v. II, Springer, Berlin–Heidelberg–New York, 1982

[7] A. G. Anischenko, V. S. Monakhov, “Tsentralnye peresecheniya i $p$-dlina $p$-razreshimykh grupp”, Doklady AN BSSR, 21:11 (1977), 968–971 | MR

[8] V. D. Mazurov, “O $p$-dline razreshimykh grupp”, VI Vsesoyuznyi simpozium po teorii grupp, sb. nauch. st. (Kiev, 1980), 50–60 | Zbl

[9] L. A. Shemetkov, “O chastichno razreshimykh konechnykh gruppakh”, Matematicheskii sbornik, 72:1 (1967), 97–107 | MR

[10] S. A. Chunikhin, L. A. Shemetkov, “Konechnye gruppy”, Algebra. Topologiya. Geometriya, Sb. nauch. st., VINITI AN SSSR, M., 1971, 7–70

[11] L. A. Shemetkov, “O $p$-dline proizvolnykh konechnykh grupp”, Doklady AN BSSR, 13:5 (1969), 394–395 | Zbl

[12] Kourovskaya tetrad (nereshennye voprosy teorii grupp), Izdanie 16, Institut matematiki im. S. L. Soboleva, Novosibirsk, 2006

[13] V. D. Mazurov, “O konechnykh gruppakh s metatsiklicheskimi silovskimi 2-podgruppami”, Sib. matem. zh., 8:5 (1967), 966–982 | MR | Zbl

[14] A. R. Camina, T. M. Gagen, “Groups with metacyclic Sylow 2-subgroups”, Canad. J. Math., 21 (1969), 1234–1237 | DOI | MR | Zbl

[15] D. Chillag, J. Sonn, “Sylow-metacyclic groups and $\mathcal{Q}$-admissibility”, Isr. J. Math., 40:3–4 (1981), 307–323 | DOI | MR | Zbl

[16] B. Huppert, “Uber das Produkt von paarweise vertauschbaren zyklischen Gruppen”, Math. Z., 58 (1953), 243–264 | DOI | MR | Zbl

[17] N. Ito, “Uber das Produkt von zwei zyklischen 2-gruppen”, Pub. Math., 4 (1956), 517–520 | MR | Zbl

[18] N. Ito, A. Oxara, “Sur les Groupes Factorisables par Deux 2 -Groupes Cycliques. I; II”, Pros. Japan Acad. Tokyo, 32 (1956), 736–740 ; 741–743 | DOI | MR | Zbl

[19] N. Blackburn, “Uber das Produkt von zwei zyklischen 2-gruppen”, Math. Zeitschr., 68 (1958), 422–427 | DOI | MR | Zbl

[20] J. Douglas, “On the supersolvability of bicyclic groups”, Proc. Nat. Acad. Sci., 49 (1961), 1493–1495 | DOI | MR

[21] Konovalov A. B., Sistema kompyuternoi algebry GAP 4.4.12, , 2009 http://www.gap-system.org/ukrgap/gapbook/manual.pdf

[22] V. S. Monakhov, E. E. Gribovskaya, “O maksimalnykh i silovskikh podgruppakh konechnykh razreshimykh grupp”, Matematicheskie zametki, 70:4 (2001), 603–612 | DOI | MR | Zbl

[23] V. S. Monakhov, A. A. Trofimuk, “Konechnye razreshimye gruppy, silovskie $p$-podgruppy kotorykh libo bitsiklicheskie, libo imeyut $p^3$”, Fundamentalnaya i prikladnaya matematika, 15:2 (2009), 121–131 | MR

[24] B. Huppert, N. Blackburn, Endliche Gruppen, v. III, Springer, Berlin–Heidelberg–New York, 1982

[25] Ya. G. Berkovich, “O razreshimykh gruppakh konechnogo poryadka”, Matem. sb., 74 (116):1 (1967), 75–92 | MR

[26] R. Maier, “Faktorisierte $p$-auflösbare Gruppen”, Arch. Math., 27 (1975), 576–583 | DOI | MR

[27] V. D. Mazurov, Zamechaniya o konechnykh gruppakh, Redkollegiya Sib. mat. zh., Novosibirsk, 1974, 7 pp.

[28] V. D. Mazurov, XII Vsesoyuzn. algebr. koll., Tez. soobsch. Tetrad 1 (Sverdlovsk, 1973)

[29] V. S. Monakhov, “O chastichnoi sverkhrazreshimosti konechnoi faktorizuemoi gruppy”, Doklady NAN Belarusi, 45:3 (2001), 32–36 | MR | Zbl

[30] A. R. Camina, T. M. Gagen, “Some non-solvable factorizable groups”, J. Math., 14:1 (1970), 91–98 | MR | Zbl

[31] V. S. Monakhov, “O proizvedenii dvukh grupp, odna iz kotorykh soderzhit tsiklicheskuyu podgruppu indeksa 2”, Matematicheskie zametki, 16:2 (1974), 285–295

[32] V. S. Monakhov, “O razreshimykh konechnykh gruppakh s silovskimi podgruppami malogo ranga”, Doklady NAN Belarusi, 46:2 (2002), 25–28 | MR | Zbl

[33] V. S. Monakhov, “Ob indeksakh maksimalnykh podgrupp konechnykh razreshimykh grupp”, Algebra i logika, 43:4 (2004), 411–424 | MR | Zbl

[34] V. S. Monakhov, A. A. Trofimuk, “Konechnye gruppy, silovskie podgruppy kotorykh libo tsiklicheskie, libo poryadka $p^2$”, Izvestiya Gomelskogo gosuniversiteta im. F. Skoriny, 47:2 (2008), 139–145 | MR

[35] A. A. Trofimuk, “O proizvodnoi dline konechnykh grupp s zadannymi svoistvami silovskikh podgrupp”, Maltsevskie chteniya, Materialy mezhdunarodnoi algebraicheskoi konferentsii (Novosibirsk, 24–28 avgusta 2009 g.), Institut matematiki im. S. L. Soboleva, Novosibirsk, 2009, 92

[36] H. Zassenhaus, “Beweis eines Zatzes uber diskrete Gruppen”, Abh. Math. Sem. Univ. Hamburg, 12 (1938), 289–312 | DOI

[37] J. D. Dixon, “The solvable length of a solvable linear groups”, Math. Z., 12 (1968), 151–158 | DOI | MR

[38] Y. Berkovich, “Solvable permutation groups of maximal derived length”, Algebra Colloquium, 4:2 (1997), 175–186 | MR | Zbl

[39] M. F. Newman, “The solvable length of a solvable linear groups”, Math. Z., 126 (1972), 59–70 | DOI | MR | Zbl

[40] P. P. Palfy, “Bounds for linear groups of odd order”, Rend. Circ. mat. Palermo. Ser. 2, 39:23 (1990), 253–263 | MR

[41] V. S. Monakhov, A. A. Trofimuk, “On a finite group having a normal series with restriction on its factors”, Ukrainian Mathematical Congress, Abstracts of talks of 7th Int. Algebraic Conference (Ukraine, Kharkov, 18–23 August 2009), ed. G. N. Zholtkevich, Institute of Mathematics of National Academy of Sciences of Ukraine, Kiev; Taras Shevchenko Kiev National University; V. N. Karazin Kharkov National University, 2009, 98–99

[42] B. Huppert, “Normalteiler und maximale Untergruppen endlicher Gruppen”, Math. Zeitschr., 60 (1954), 409–434 | DOI | MR | Zbl

[43] M. Kholl, Teoriya grupp, IL, M., 1962, 468 pp.

[44] L. Ya. Polyakov, “O vliyanii svoistv maksimalnykh podgrupp na razreshimost konechnoi gruppy”, Konechnye gruppy, Minsk, 1966, 89–97 | Zbl

[45] L. A. Shemetkov, “O konechnykh razreshimykh gruppakh”, Izvestiya AN SSSR. Cer. matem., 32:3 (1968), 533–559 | MR

[46] V. S. Monakhov, M. V. Selkin, “O razreshimosti normalnykh podgrupp konechnykh grupp”, Matem. zametki, 51:3 (1992), 85–90 | MR | Zbl

[47] V. S. Monakhov, M. V. Selkin, “O stroenii normalnykh podgrupp konechnykh grupp”, Voprosy algebry, 1993, no. 6, 96–100 | MR | Zbl

[48] S. F. Kamornikov, M. V. Selkin, “O razreshimykh podgruppakh konechnykh grupp”, Vestsi Akad. navuk Belarusi. Ser. fiz-mat. navuk, 1994, no. 2, 53–57 | MR

[49] S. F. Kamornikov, M. V. Selkin, “O vliyanii maksimalnykh podgrupp primarnogo indeksa na stroenie konechnoi gruppy”, Izv. vyssh. ucheb. zaved. Matematika, 1995, no. 6, 24–28 | MR

[50] M. V. Selkin, Maksimalnye podgruppy v teorii klassov konechnykh grupp, Belaruskaya navuka, Minsk, 1997, 144 pp. | MR

[51] D. A. Khodanovich, “O razreshimosti konechnoi gruppy s ogranichennymi indeksami nenilpotentnykh maksimalnykh podgrupp”, Vestnik PGU. Ceriya C. Fundamentalnye nauki, 2005, no. 4, 18–22

[52] S. F. Kamornikov, “K teoreme F. Kholla”, Voprosy algebry, 1990, no. 5, 45–52 | MR | Zbl

[53] V. S. Monakhov, M. V. Selkin, E. E. Gribovskaya, “O razreshimykh normalnykh podgruppakh konechnykh grupp”, Ukrainskii matematicheskii zhurnal, 54:7 (2002), 940–950 | MR

[54] J. Köhler, “Finite groups with all maximal subgroups of prime or prime square index”, Canad. J. Math., 16:3 (1964), 435–442 | DOI | MR | Zbl

[55] R. Guralnick, “Subgroups of prime power index in a simple group”, J. Algebra, 81:2 (1983), 304–311 | DOI | MR | Zbl

[56] L. S. Kazarin, “O gruppakh, predstavimykh v vide proizvedeniya dvukh razreshimykh podgrupp”, Commun. Algebra, 14:6 (1986), 1001–1066 | DOI | MR | Zbl

[57] E. E. Gribovskaya, “Konechnye razreshimye gruppy s indeksami maksimalnykh podgrupp, ravnymi $p$, $p^2$ ili 8”, Vestsi NAN Belarusi. Ser. fiz.-mat. navuk, 2001, no. 4, 11–14 | MR

[58] E. E. Gribovskaya, “O razreshimykh normalnykh podgruppakh konechnykh grupp s indeksami maksimalnykh podgrupp $p$, $p^2$ ili 8”, Vesnik Vitsebskaga dzyarzhaŭnaga universiteta, 21:3 (2001), 98–103 | MR

[59] A. A. Trofimuk, “O $p$-dline konechnykh razreshimykh grupp s ogranicheniyami na indeksy maksimalnykh podgrupp”, Lomonosovskie chteniya, Materialy mezhdunarodnoi nauchnoi konferentsii studentov, aspirantov i molodykh uchenykh, eds. V. A. Trifonova i dr., Filial MGU v g. Sevastopole, Sevastopol, 2009, 54–55

[60] E. E. Gribovskaya, Indeksy maksimalnykh podgrupp v teorii klassov sostavnykh konechnykh grupp, Dis. ... na soiskanie uchenoi stepeni kand. fiz.-mat. nauk: 01.01.06, Gomel, 2002, 65 pp.

[61] E. E. Gribovskaya, V. S. Monakhov, Konechnye gruppy s indeksami maksimalnykh podgrupp, ne delyaschimisya na $p^4$, Preprint No 91, GGU im. F. Skoriny, Gomel, 2000, 13 pp. | MR

[62] V. S. Monakhov, D. A. Khodanovich, “O konechnykh gruppakh s ogranichennymi indeksami maksimalnykh podgrupp”, Doklady NAN Belarusi, 50:3 (2006), 20–24 | MR | Zbl

[63] D. A. Khodanovich, Konechnye gruppy s ogranichennymi indeksami neformatsionnykh maksimalnykh podgrupp, Dis. ... na soiskanie uchenoi stepeni kand. fiz.-mat. nauk: 01.01.06, Gomel, 2008, 66 pp.

[64] Y. Berkovich, L. Kazarin, “Indices of elements and normal structure of finite groups”, J. Algebra, 283 (2005), 564–583 | DOI | MR | Zbl

[65] V. S. Monakhov, V. N. Tyutyanov, “O konechnykh gruppakh s nekotorymi podgruppami prostykh indeksov”, Sib. matem. zhurnal, 48:4 (2007), 833–836 | MR | Zbl

[66] V. C. Monakhov, “K teoreme Khupperta–Shemetkova”, Trudy instituta matematiki, 16:1 (2008), 64–66

[67] V. S. Monakhov, “Zamechaniya o maksimalnykh podgruppakh konechnykh grupp”, Doklady NAN Belarusi, 47:4 (2003), 31–33 | MR | Zbl

[68] W. Gashütz, “Existenz und Konjugiertsein von Untergruppen, die in endlichen auflosbaren Gruppen durch gewisse Indexschranken definiert sind”, J. Algebra, 53:2 (1978), 389–394 | DOI | MR

[69] V. S. Monakhov, D. A. Khodanovich, “O vliyanii indeksov maksimalnykh podgrupp na nilpotentnuyu dlinu konechnoi razreshimoi gruppy”, Doklady NAN Belarusi, 50:4 (2006), 23–28 | MR | Zbl

[70] M. A. Gribovskaya, “Gruppy nechetnogo poryadka s ogranichennymi indeksami maksimalnykh podgrupp”, Izvestiya Gomelskogo gosudarstvennogo universiteta im. F. Skoriny, 14:5 (2002), 80–84 | Zbl

[71] M. A. Gribovskaya, “O nilpotentnosti subnormalnoi podgruppy konechnoi gruppy”, Izvestiya Gomelskogo gosudarstvennogo universiteta im. F. Skoriny, 15:6 (2002), 158–160

[72] M. A. Gribovskaya, “Normalnye podgruppy konechnoi gruppy i indeksy ne soderzhaschikh eë maksimalnykh podgrupp”, Vesnik Vitsebskaga dzyarzhaŭnaga universiteta, 26:4 (2002), 82–86

[73] M. A. Gribovskaya, “O gruppakh nechetnogo poryadka s nebolshimi indeksami maksimalnykh podgrupp”, Algebra i lineinaya optimizatsiya, Trudy mezhdunarodnogo seminara, posvyaschennogo 90-letiyu so dnya rozhdeniya S. N. Chernikova, UrO RAN, Ekaterinburg, 2002, 92–98 | MR

[74] M. A. Gribovskaya, “Podgruppa Fittinga i sverkhrazreshimost konechnoi razreshimoi gruppy”, Tvorchestvo molodykh, Sb. nauch. rabot stud. i aspir., GGU im. F. Skoriny, Gomel, 2002, 82–84

[75] M. A. Gribovskaya, V. S. Monakhov, Naibolshii prostoi delitel poryadka gruppy i indeksy maksimalnykh podgrupp, Preprint No 38, GGU im. F. Skoriny, Gomel, 2000, 16 pp. | MR

[76] M. A. Gribovskaya, Konechnye gruppy s ogranichennymi indeksami otdelnykh maksimalnykh podgrupp, Dis. ... na soiskanie uchenoi stepeni kand. fiz.-mat. nauk: 01.01.06, Gomel, 2003

[77] W. Gaschütz, “Uber die $\Phi$-Untergruppe endlicher Gruppen”, Math. Z., 58 (1953), 160–170 | DOI | MR

[78] V. S. Monakhov, “Zamechanie o peresechenii nenormalnykh maksimalnykh podgruppakh konechnykh grupp”, Izvestiya Gomelskogo gosuniversiteta im. F. Skoriny, 27:6 (2004), 81 | MR

[79] V. S. Monakhov, “O maksimalnykh podgruppakh konechnykh grupp”, Izvestiya Gomelskogo gosuniversiteta im. F. Skoriny, 47:2 (2008), 146–148

[80] M. Isaacs, Finite group theory, American Mathematical Society, Providence, Rhode Island, 2008 | MR

[81] V. M. Levchuk, A. G. Likharev, “Konechnye prostye gruppy s dopolnyaemymi maksimalnymi podgruppami”, Sib. mat. zh-al, 47:4 (2006), 798–810 | MR | Zbl

[82] V. N. Tyutyanov, “Konechnye gruppy s dopolnyaemymi podgruppami”, Izvestiya GGU im. F. Skoriny, 36:6 (2006), 178–183

[83] V. S. Monakhov, “Konechnye $\pi$-razreshimye gruppy s khollovymi maksimalnymi”, Matematicheskie zametki, 84:3 (2008), 390–394 | DOI | MR | Zbl

[84] N. S. Chernikov, A. P. Petravchuk, “O $\pi$-dline konechnykh $\pi$-razreshimykh grupp”, Beskonechnye gruppy i primykayuschie algebraicheskie struktury, Tr. Instituta matematiki AN Ukrainy, 1993, 393–405 | MR

[85] L. S. Kazarin, “Soluble product of groups”, Infinite groups 94, 1995, 111–123 | MR

[86] R. Carter, B. Fischer, T. Hawkes, “Extreme Classes of finite soluble groups”, J. Algebra, 9:3 (1968), 285–313 | DOI | MR

[87] M. Numata, “On the $\pi$-nilpotent length of $\pi$-solvable groups”, Osaka J. Math., 8 (1971), 447–451 | MR | Zbl

[88] V. S. Monakhov, “O konechnykh razreshimykh gruppakh”, Algebra i teoriya chisel. Sovremennye problemy i prilozheniya, Tezisy dokladov VI Mezhdunarodnoi konferentsii, posvyaschennoi 100-letiyu N. G. Chudakova (Saratov, 13–17 sentyabrya 2004 g.), Saratov, 2004, 83

[89] V. S. Monakhov, O. A. Shpyrko, “O nilpotetnoi $\pi$-dline konechnoi $\pi$-razreshimoi gruppy”, Diskretnaya matematika, 13:3 (2001), 145–152 | DOI | MR | Zbl

[90] O. A. Shpyrko, “Khollovy podgruppy i nilpotentnaya $\pi$-dlina $\pi$-razreshimoi gruppy”, Vestsi NAN Belarusi. Cep. fiz.- mat. navuk, 2001, no. 2, 48–50 | MR

[91] K. Doerk, “Über die nilpotente Länge maximaler Untergruppen bei endlichen auflösbaren Gruppen”, Rend. Sem. Mat. Univ. Padova, 91 (1994), 19–21 | MR | Zbl

[92] A. Ballester-Bolinches, M. D. Perez-Ramos, “A note on the $\mathfrak{F}$-length of maximal subgroups in finite soluble groups”, Math. Nachr., 166 (1994), 67–70 | DOI | MR | Zbl

[93] K. Doerk, T. Hawkes, Finite soluble groups, Walter de Gruyter, Berlin–New York, 1992, 889 pp. | MR

[94] V. S. Monakhov, O. A. Shpyrko, “The nilpotent $\pi$-length of maximum subgroups in finite $\pi$-soluble groups”, Moscow University Mathematics Bulletin, 64:6 (2009), 229–234 | DOI | MR

[95] O. A. Shpyrko, “O konechnykh $\pi$-razreshimykh gruppakh s zadannymi silovskimi podgruppami”, Izvestiya Gomelskogo gosuniversiteta im. F. Skoriny, 1:1 (15) (1999), 127–134 | MR

[96] O. A. Shpyrko, “O peresecheniyakh khollovykh podgrupp i $\pi$-dline $\pi$-razreshimoi gruppy”, Vesnik Vitsebskaga dzyarzhaŭnaga universiteta, 18:4 (2000), 82–89

[97] V. S. Monakhov, O. A. Shpyrko, Nilpotentnaya $\pi$-dlina konechnykh $\pi$-razreshimykh grupp, Preprint No 92, GGU im. F. Skoriny, Gomel, 2000, 12 pp. | MR

[98] O. A. Shpyrko, Nilpotentnaya $\pi$-dlina konechnykh $\pi$-razreshimykh gruppa, Dis. ... na soiskanie uchenoi stepeni kand. fiz.-mat. nauk: 01.01.06, Gomel, 2001, 81 pp.

[99] O. A. Shpyrko, “O nilpotentnoi $\pi$-dline konechnykh $\pi$-razreshimykh grupp s ogranicheniyami na podgruppy”, Vestnik Moskovskogo gosuniversiteta im. M. V. Lomonosova. Matematika. Mekhanika, 63:1 (2008), 21–25 | MR | Zbl