Local definitions of formations of finite groups

L. A. Shemetkov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 16 (2010) no. 8, pp. 229-244.

Résumé

A problem of constructing local definitions for formations of finite groups is discussed in the article. The author analyzes relations between local definitions of various types. A new proof of the existence of an $\omega$-composition satellite of an $\omega$-solubly saturated formation is obtained. It is proved that if a nonempty formation of finite groups is $\mathfrak X$-local by Förster, then it has an $\mathfrak X$-composition satellite.

Bibliographie

[1] Ballester-Bolinshe A., Kalvo K., Shemetkov L. A., “O chastichno nasyschennykh formatsiyakh konechnykh grupp”, Mat. sb., 198:6 (2007), 3–24 | DOI | MR | Zbl

[2] Skiba A. N., Shemetkov L. A., “O chastichno lokalnykh formatsiyakh”, Dokl. AN Belarusi, 39:3 (1995), 9–11 | MR

[3] Skiba A. N., Shemetkov L. A., “Kratno $\omega$-lokalnye formatsii i klassy Fittinga konechnykh grupp”, Mat. tr., 2:2 (1999), 114–147 | MR | Zbl

[4] Skiba A. N., Shemetkov L. A., “Kratno $\mathfrak L$-kompozitsionnye formatsii konechnykh grupp”, Ukr. mat. zhurn., 52:6 (2000), 783–797 | MR | Zbl

[5] Syskin S. A., “Sverkhrazreshimye gruppy avtomorfizmov konechnykh razreshimykh grupp”, Algebra i logika, 7:3 (1968), 105–107 | MR | Zbl

[6] Shemetkov L. A., “Stupenchatye formatsii grupp”, Mat. sb., 94(136):4 (1974), 628–648 | MR | Zbl

[7] Shemetkov L. A., “Dva napravleniya v razvitii neprostykh konechnykh grupp”, Doklad, prochitannyi na XII Vsesoyuznom algebraicheskom kollokviume v Sverdlovske v sentyabre 1973 g., Uspekhi mat. nauk, 30:2 (1975), 179–198 | MR | Zbl

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

[9] Shemetkov L. A., “O proizvedenii formatsii”, Dokl. AN BSSR, 28:2 (1984), 101–103 | MR | Zbl

[10] Ballester A., “Remarks on formations”, Israel J. Math., 73:1 (1991), 97–106 | DOI | MR | Zbl

[11] Ballester-Bolinches A., Calvo C., Esteban-Romero R., “Products of formations of finite groups”, J. Algebra, 299:2 (2000), 602–615 | DOI | MR

[12] Ballester-Bolinches A., Calvo C., Esteban-Romero R., “$\mathfrak X$-saturated formations of finite groups”, Commun. Algebra, 33:4 (2005), 1053–1064 | DOI | MR | Zbl

[13] Ballester-Bolinches A., Ezquerro L. M., Classes of Finite Groups, Springer, Dordrecht, 2006 | MR | Zbl

[14] Ballester-Bolinches A., Shemetkov L. A., “On lattices of $p$-local formations of finite groups”, Math. Nachr., 186:1 (1997), 57–65 | DOI | MR | Zbl

[15] Doerk K., Hawkes T., Finite Soluble Groups, Walter de Gruyter, Berlin, 1992 | MR

[16] Förster P., “Projektive Klassen endlicher Gruppen IIa. Gesättigte Formationen: ein allgemeiner Satz von Gaschütz–Lubeseder–Baer Typ”, Publ. Sec. Mat. Univ. Autònoma Barcelona, 29:2–3 (1985), 39–76 | MR | Zbl

[17] Gaschütz W., “Zur Theorie der endlichen auflösbaren Gruppen”, Math. Z., 80:4 (1963), 300–305 | MR | Zbl

[18] Griess R. L., Schmid P., “Frattini module”, Arch. Math., 30:3 (1978), 256–266 | DOI | MR | Zbl

[19] Huppert B., Blackburn N., Finite Groups, v. II, Grundlehren Math. Wiss., 242, Springer, Berlin, 1982 | MR | Zbl

[20] Schmid P., “Über die Automorphismengruppen endlicher Gruppen”, Arch. Math., 23:3 (1972), 236–242 | DOI | MR | Zbl

[21] Schmid P., “Lokale Formationen endlicher Gruppen”, Math. Z., 137:1 (1974), 31–48 | DOI | MR | Zbl

[22] Shemetkov L. A., “Frattini extensions of finite groups and formations”, Commun. Algebra, 25:3 (1997), 955–964 | DOI | MR | Zbl

[23] Shemetkov L. A., “On partially saturated formations and residuals of finite groups”, Commun. Algebra, 29:9 (2001), 4125–4137 | DOI | MR | Zbl

[24] Shemetkov L. A., “A note on $\mathfrak X$-local formations”, Problemy fiziki, matematiki i tekhniki, 2010, no. 4(5), 64–65