Formations of finite $C_\pi$-groups
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E. P. Vdovin, D. O. Revin and L. A. Shemetkov
Translated by: E. P. Vdovin - St. Petersburg Math. J. 24 (2013), 29-37
- DOI: https://doi.org/10.1090/S1061-0022-2012-01230-6
- Published electronically: November 15, 2012
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Abstract:
It is proved that the class of all finite $C_\pi$-groups is closed under subdirect products. Conditions for a given formation of $C_\pi$-groups to be saturated or partially saturated are found.References
- P. Hall, Theorems like Sylow’s, Proc. London Math. Soc. (3) 6 (1956), 286–304. MR 77533, DOI 10.1112/plms/s3-6.2.286
- Danila O. Revin and Evgeny P. Vdovin, An existence criterion for Hall subgroups of finite groups, J. Group Theory 14 (2011), no. 1, 93–101. MR 2764926, DOI 10.1515/JGT.2010.037
- Danila Olegovitch Revin and Evgenii Petrovitch Vdovin, Hall subgroups of finite groups, Ischia group theory 2004, Contemp. Math., vol. 402, Amer. Math. Soc., Providence, RI, 2006, pp. 229–263. MR 2258669, DOI 10.1090/conm/402/07585
- Dieter Blessenohl, Über Formationen und Halluntergruppen endlicher, auflösbarer Gruppen, Math. Z. 142 (1975), 299–300 (German). MR 372021, DOI 10.1007/BF01183053
- L. A. Šemetkov, Formatsii konechnykh grupp, Sovremennaya Algebra. [Modern Algebra], “Nauka”, Moscow, 1978 (Russian). MR 519875
- L. A. Shemetkov and A. F. Vasil′ev, Nonlocal formations of finite groups, Dokl. Akad. Nauk Belarusi 39 (1995), no. 4, 5–8, 122 (Russian, with English and Russian summaries). MR 1384403
- E. P. Vdovin and D. O. Revin, A conjugacy criterion for Hall subgroups in finite groups, Sibirsk. Mat. Zh. 51 (2010), no. 3, 506–516 (Russian, with Russian summary); English transl., Sib. Math. J. 51 (2010), no. 3, 402–409. MR 2683093, DOI 10.1007/s11202-010-0041-4
- Michio Suzuki, Group theory. II, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 248, Springer-Verlag, New York, 1986. Translated from the Japanese. MR 815926, DOI 10.1007/978-3-642-86885-6
- S. A. Čunihin, Podgruppy konechnykh grupp, Izdat. “Nauka i Tehnika”, Minsk, 1964 (Russian). MR 0212082
- L. A. Shemetkov, On partially saturated formations and residuals of finite groups, Comm. Algebra 29 (2001), no. 9, 4125–4137. Special issue dedicated to Alexei Ivanovich Kostrikin. MR 1857031, DOI 10.1081/AGB-100105992
- A. N. Skiba and L. A. Shemetkov, Multiply $\omega$-local formations and Fitting classes of finite groups, Mat. Tr. 2 (1999), no. 2, 114–147 (Russian, with Russian summary). MR 1767827
- A. N. Skiba and L. A. Shemetkov, Multiply $\mathfrak {L}$-compositional formations of finite groups, Ukraïn. Mat. Zh. 52 (2000), no. 6, 783–797 (Russian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 52 (2000), no. 6, 898–913 (2001). MR 1819684, DOI 10.1007/BF02591784
- L. A. Shemetkov, Frattini extensions of finite groups and formations, Comm. Algebra 25 (1997), no. 3, 955–964. MR 1433445, DOI 10.1080/00927879708825900
- Klaus Doerk and Trevor Hawkes, Finite soluble groups, De Gruyter Expositions in Mathematics, vol. 4, Walter de Gruyter & Co., Berlin, 1992. MR 1169099, DOI 10.1515/9783110870138
- V. D. Mazurov and D. O. Revin, On the Hall $D_\pi$-property for finite groups, Sibirsk. Mat. Zh. 38 (1997), no. 1, 125–134, iii (Russian, with Russian summary); English transl., Siberian Math. J. 38 (1997), no. 1, 106–113. MR 1446679, DOI 10.1007/BF02674906
- D. O. Revin, Hall $\pi$-subgroups of finite Chevalley groups whose characteristic belongs to $\pi$, Mat. Tr. 2 (1999), no. 1, 160–208 (Russian, with Russian summary). MR 1762623
- E. P. Vdovin and D. O. Revin, Hall subgroups of odd order in finite groups, Algebra Logika 41 (2002), no. 1, 15–56, 118 (Russian, with Russian summary); English transl., Algebra Logic 41 (2002), no. 1, 8–29. MR 1924596, DOI 10.1023/A:1014653900781
- D. O. Revin, The $D_\pi$-property in a class of finite groups, Algebra Logika 41 (2002), no. 3, 335–370, 387–388 (Russian, with Russian summary); English transl., Algebra Logic 41 (2002), no. 3, 187–206. MR 1934540, DOI 10.1023/A:1016029025936
- V. A. Vedernikov, Subdirect products of finite groups with Hall $\pi$-subgroups, Mat. Zametki 59 (1996), no. 2, 311–314 (Russian); English transl., Math. Notes 59 (1996), no. 1-2, 219–221. MR 1391846, DOI 10.1007/BF02310964
- Peter Schmid, Über die Automorphismengruppen endlicher Gruppen, Arch. Math. (Basel) 23 (1972), 236–242 (German). MR 308266, DOI 10.1007/BF01304876
Bibliographic Information
- E. P. Vdovin
- Affiliation: Sobolev Institute of mathematics, pr. Academica Koptyuga 4, Novosibirsk 630090, Russia
- Email: vdovin@math.nsc.ru
- D. O. Revin
- Affiliation: Sobolev Institute of mathematics, pr. Academica Koptyuga 4, Novosibirsk 630090, Russia
- Email: revin@math.nsc.ru
- L. A. Shemetkov
- Affiliation: Francisk Skorina Gomel State University, ul. Sovetskaya 104, Gomel 246019, Belarus
- Email: shemetkov@gsu.by
- Received by editor(s): October 6, 2010
- Published electronically: November 15, 2012
- Additional Notes: The first two authors were partially supported by RFBR (projects nos. 10-01-00391 and 10-01-90007) and by the Federal Target Program “Scientific and educational personnel of innovation Russia” for 2009–2013 (government contract no. 14.740.11.0346). The first author was also supported by the Deligne 2004 Balzan Prize in Mathematics. The third author was supported by BRFFR (grant no. $\Phi$10P-231) and by the Ministry of Education of Belarus (grant no. 2008470).
- © Copyright 2012 American Mathematical Society
- Journal: St. Petersburg Math. J. 24 (2013), 29-37
- MSC (2010): Primary 20F17
- DOI: https://doi.org/10.1090/S1061-0022-2012-01230-6
- MathSciNet review: 3013293