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Algebra Logika, 2008, Volume 47, Number 3, Pages 364–394 (Mi al363)  

This article is cited in 18 scientific papers (total in 18 papers)

The $D_\pi$-property in finite simple groups

D. O. Revin

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: Let $\pi$ be some set of primes. A finite group is said to possess the $D_\pi$-property if all of its maximal $\pi$-subgroups are conjugate. It is not hard to show that this property is equivalent to satisfaction of the complete analog of Sylow's theorem for Hall $\pi$-subgroups of a group. In the paper, we bring to a close an arithmetic description of finite simple groups with the $D_\pi$-property, for any set $\pi$ of primes. Previously, it was proved that a finite group possesses the $D_\pi$-property iff each composition factor of the group has this property. Therefore, the results obtained mean in fact that the question of whether a given group enjoys the $D_\pi$-property becomes purely arithmetic.

Keywords: finite group, $D_\pi$-property, Sylow theorem.

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English version:
Algebra and Logic, 2008, 47:3, 210–227

Bibliographic databases:

UDC: 512.542
Received: 27.08.2007
Revised: 09.01.2008

Citation: D. O. Revin, “The $D_\pi$-property in finite simple groups”, Algebra Logika, 47:3 (2008), 364–394; Algebra and Logic, 47:3 (2008), 210–227

Citation in format AMSBIB
\by D.~O.~Revin
\paper The $D_\pi$-property in finite simple groups
\jour Algebra Logika
\yr 2008
\vol 47
\issue 3
\pages 364--394
\jour Algebra and Logic
\yr 2008
\vol 47
\issue 3
\pages 210--227

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. D. O. Revin, “Vokrug gipotezy F. Kholla”, Sib. elektron. matem. izv., 6 (2009), 366–380  mathnet  mathscinet
    2. Revin D.O., Vdovin E.P., “On the number of classes of conjugate Hall subgroups in finite simple groups”, J. Algebra, 324:12 (2010), 3614–3652  crossref  mathscinet  zmath  isi  elib  scopus
    3. D. O. Revin, “On Baer–Suzuki $\pi$-theorems”, Siberian Math. J., 52:2 (2011), 340–347  mathnet  crossref  mathscinet  isi
    4. Revin D.O., Vdovin E.P., “An existence criterion for Hall subgroups of finite groups”, J Group Theory, 14:1 (2011), 93–101  crossref  mathscinet  zmath  isi  elib  scopus
    5. E. P. Vdovin, D. O. Revin, “Theorems of Sylow type”, Russian Math. Surveys, 66:5 (2011), 829–870  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. D. O. Revin, “On a relation between the Sylow and Baer–Suzuki theorems”, Siberian Math. J., 52:5 (2011), 904–913  mathnet  crossref  mathscinet  isi
    7. E. P. Vdovin, N. Ch. Manzaeva, D. O. Revin, “On the heritability of the property $D_\pi$ by subgroups”, Proc. Steklov Inst. Math. (Suppl.), 279, suppl. 1 (2012), 130–138  mathnet  crossref  isi  elib
    8. N. Ch. Manzaeva, “Reshenie problemy Vilanda dlya sporadicheskikh grupp”, Sib. elektron. matem. izv., 9 (2012), 294–305  mathnet
    9. E. P. Vdovin, D. O. Revin, “On the pronormality of Hall subgroups”, Siberian Math. J., 54:1 (2013), 22–28  mathnet  crossref  mathscinet  isi
    10. A. A. Galt, W. Guo, E. M. Averkin, D. O. Revin, “On the local case in the Aschbacher theorem for linear and unitary groups”, Siberian Math. J., 55:2 (2014), 239–245  mathnet  crossref  mathscinet  isi
    11. N. Ch. Manzaeva, “Heritability of the property $\mathcal D_\pi$ by overgroups of $\pi$-Hall subgroups in the case where $2\in\pi$”, Algebra and Logic, 53:1 (2014), 17–28  mathnet  crossref  mathscinet  isi
    12. W. Guo, D. O. Revin, “On the class of groups with pronormal Hall $\pi$-subgroups”, Siberian Math. J., 55:3 (2014), 415–427  mathnet  crossref  mathscinet  isi  elib  elib
    13. Guo W., Revin D.O., Vdovin E.P., “Confirmation For Wielandt'S Conjecture”, J. Algebra, 434 (2015), 193–206  crossref  mathscinet  zmath  isi  elib  scopus
    14. I. B. Gorshkov, “On Thompson's conjecture for alternating and symmetric groups of degree greater than 1361”, Proc. Steklov Inst. Math. (Suppl.), 293, suppl. 1 (2016), 58–65  mathnet  crossref  mathscinet  isi  elib
    15. A. A. Galt, D. O. Revin, “Lokalnyi sluchai v teoreme Ashbakhera dlya lineinykh i unitarnykh grupp”, Sib. elektron. matem. izv., 13 (2016), 1207–1218  mathnet  crossref
    16. Guo W. Revin D.O., “Classification and Properties of the -Submaximal Subgroups in Minimal Nonsolvable Groups”, Bull. Math. Sci., 8:2 (2018), 325–351  crossref  mathscinet  zmath  isi  scopus
    17. W. Guo, D. O. Revin, “Maximal and submaximal $\mathfrak X$-subgroups”, Algebra and Logic, 57:1 (2018), 9–28  mathnet  crossref  crossref  isi
    18. Guo W., Revin D.O., “Pronormality and Submaximal (Sic)-Subgroups on Finite Groups”, Commun. Math. Stat., 6:3, SI (2018), 289–317  crossref  mathscinet  zmath  isi  scopus
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