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Sibirsk. Mat. Zh., 2005, Volume 46, Number 3, Pages 511–522 (Mi smj983)  

This article is cited in 68 scientific papers (total in 68 papers)

On connection between the structure of a finite group and the properties of Its prime graph

A. V. Vasil'ev

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

Abstract: It is shown that the condition of nonadjacency of 2 and at least one odd prime in the Gruenberg–Kegel graph of a finite group $G$ under some natural additional conditions suffices to describe the structure of $G$; in particular, to prove that $G$ has a unique nonabelian composition factor. Applications of this result to the problem of recognition of finite groups by spectrum are also considered.

Keywords: finite group, finite simple group, prime graph of a finite group, spectrum of a group, recognition by spectrum

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English version:
Siberian Mathematical Journal, 2005, 46:3, 396–404

Bibliographic databases:

UDC: 512.542
Received: 29.11.2004

Citation: A. V. Vasil'ev, “On connection between the structure of a finite group and the properties of Its prime graph”, Sibirsk. Mat. Zh., 46:3 (2005), 511–522; Siberian Math. J., 46:3 (2005), 396–404

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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. A. V. Vasil'ev, M. A. Grechkoseeva, “On recognition by spectrum of finite simple linear groups over fields of characteristic 2”, Siberian Math. J., 46:4 (2005), 593–600  mathnet  crossref  mathscinet  zmath  isi
    2. A. V. Vasil'ev, E. P. Vdovin, “An Adjacency Criterion for the Prime Graph of a Finite Simple Group”, Algebra and Logic, 44:6 (2005), 381–406  mathnet  crossref  mathscinet  zmath  elib  elib
    3. M. A. Grechkoseeva, M. S. Lucido, V. D. Mazurov, A. R. Moghaddamfar, A. V. Vasil'ev, “On recognition of the projective special linear groups over binary fields”, Sib. elektron. matem. izv., 2 (2005), 253–263  mathnet  mathscinet  zmath
    4. Grechkoseeva M.A., Shi Wujie, Vasilev A.V., “Recognition by spectrum of $L_{16}(2^m)$”, Algebra Colloq., 14:4 (2007), 585–591  crossref  mathscinet  zmath  isi  elib  scopus
    5. V. D. Mazurov, G. Y. Chen, “Recognizability of finite simple groups $L_4(2^m)$ and $U_4(2^m)$ by spectrum”, Algebra and Logic, 47:1 (2008), 49–55  mathnet  crossref  mathscinet  zmath  isi
    6. M. A. Grechkoseeva, “Recognition by spectrum for finite linear groups over fields of characteristic 2”, Algebra and Logic, 47:4 (2008), 229–241  mathnet  crossref  mathscinet  zmath  isi
    7. A. V. Vasil'ev, M. A. Grechkoseeva, “Recognition by spectrum for finite simple linear groups of small dimensions over fields of characteristic 2”, Algebra and Logic, 47:5 (2008), 314–320  mathnet  crossref  mathscinet  zmath  isi
    8. Khosravi B, Khosravi B., Khosravi B., “A characterization of the finite simple group $L_{16}(2)$ by its prime graph”, Manuscripta Math., 126:1 (2008), 49–58  crossref  mathscinet  zmath  isi  elib  scopus
    9. O. A. Alekseeva, A. S. Kondratev, “Raspoznavaemost po spektru grupp $ ^2D_p(3)$ dlya nechetnogo prostogo chisla $p$”, Tr. IMM UrO RAN, 14, no. 4, 2008, 3–11  mathnet  elib
    10. Grechkoseeva M.A., Shi Wujie, Vasilev A.V., “Recognition by spectrum for finite simple groups of Lie type”, Front. Math. China, 3:2 (2008), 275–285  crossref  mathscinet  zmath  isi  elib  scopus
    11. Darafsheh M.R., “On the recognition of the simple groups $L_7(3)$ and $L_8(3)$ by the spectrum”, Internat. J. Algebra Comput., 18:5 (2008), 925–933  crossref  mathscinet  zmath  isi  elib  scopus
    12. V. D. Mazurov, “O raspoznavanii konechnykh prostykh grupp po spektru”, Tr. In-ta matem., 16:1 (2008), 59–63  mathnet
    13. A. V. Vasil'ev, I. B. Gorshkov, “On recognition of finite simple groups with connected prime graph”, Siberian Math. J., 50:2 (2009), 233–238  mathnet  crossref  mathscinet  isi  elib  elib
    14. B. Khosravi, “Quasirecognition by prime graph of $L_{10}(2)$”, Siberian Math. J., 50:2 (2009), 355–359  mathnet  crossref  mathscinet  isi  elib  elib
    15. O. A. Alekseeva, A. S. Kondrat'ev, “On recognizability of some finite simple orthogonal groups by spectrum”, Proc. Steklov Inst. Math. (Suppl.), 265, suppl. 1 (2009), S10–S23  mathnet  crossref  isi  elib
    16. A. V. Vasil'ev, I. B. Gorshkov, M. A. Grechkoseeva, A. S. Kondrat'ev, A. M. Staroletov, “On recognizability by spectrum of finite simple groups of types $B_n$, $C_n$, and $ ^2D_n$ for$n=2^k$”, Proc. Steklov Inst. Math. (Suppl.), 267, suppl. 1 (2009), S218–S233  mathnet  crossref  isi  elib
    17. A. V. Vasil'ev, M. A. Grechkoseeva, V. D. Mazurov, “On finite groups isospectral to simple symplectic and orthogonal groups”, Siberian Math. J., 50:6 (2009), 965–981  mathnet  crossref  mathscinet  isi  elib  elib
    18. A. V. Vasil'ev, M. A. Grechkoseeva, V. D. Mazurov, “Characterization of the finite simple groups by spectrum and order”, Algebra and Logic, 48:6 (2009), 385–409  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    19. Khosravi B., “Some characterizations of $L_9(2)$ related to its prime graph”, Publ. Math. Debrecen, 75:3-4 (2009), 375–385  mathscinet  zmath  isi
    20. Kondrat'ev A.S., “Recognition by spectrum of the groups $^2D_{2^m+1}(3)$”, Sci. China Ser. A, 52:2 (2009), 293–300  crossref  mathscinet  zmath  isi  scopus
    21. A. S. Kondratev, “O raspoznavaemosti po spektru konechnykh prostykh ortogonalnykh grupp, II”, Vladikavk. matem. zhurn., 11:4 (2009), 32–43  mathnet  elib
    22. Vasil'ev A.V., Grechkoseeva M.A., Mazurov V.D., “Recognizability of the finite simple groups by spectrum and order”, Dokl. Math., 81:2 (2010), 216–218  crossref  mathscinet  zmath  isi  elib  scopus
    23. Monfared M.D., Darafsheh M.R., “A characterization of the groups PSU(4,4) and PSL(4,4) by non-commuting graph”, Util. Math., 81 (2010), 165–185  mathscinet  zmath  isi
    24. Wang Lingli, “Characterization of finite simple group $D_n(2)$”, Front. Math. China, 5:1 (2010), 179–190  crossref  mathscinet  zmath  isi  scopus
    25. A. M. Staroletov, “Groups isospectral to the degree 10 alternating group”, Siberian Math. J., 51:3 (2010), 507–514  mathnet  crossref  mathscinet  zmath  isi
    26. I. B. Gorshkov, “Raspoznavanie po spektru konechnykh prostykh grupp, prostye deliteli poryadkov kotorykh ne prevoskhodyat $17$”, Sib. elektron. matem. izv., 7 (2010), 14–20  mathnet  mathscinet
    27. I. A. Vakula, “On the structure of finite groups isospectral to an alternating group”, Proc. Steklov Inst. Math. (Suppl.), 272, suppl. 1 (2011), S271–S286  mathnet  crossref  isi  elib
    28. M. R. Zinoveva, “Raspoznavanie po spektru prostykh grupp $C_p(3)$”, Tr. IMM UrO RAN, 16, no. 3, 2010, 88–95  mathnet  elib
    29. Xu Mingchun, “Recognition of finite simple linear groups $L_4(2^k)$ by spectrum”, Algebra Colloq., 17:3 (2010), 469–474  crossref  mathscinet  zmath  isi  scopus
    30. Shao Ch., Jiang Q., “A new characterization of $A_{22}$ by its spectrum”, Comm. Algebra, 38:6 (2010), 2138–2141  crossref  mathscinet  zmath  isi  scopus
    31. V. D. Mazurov, “Periodic groups with prescribed element orders”, Tr. In-ta matem., 18:1 (2010), 72–78  mathnet
    32. M. A. Grechkoseeva, “Quasirecognizability of simple unitary groups over fields of even order”, Sib. elektron. matem. izv., 7 (2010), 435–444  mathnet
    33. A. V. Vasil'ev, M. A. Grechkoseeva, A. M. Staroletov, “On finite groups isospectral to simple linear and unitary groups”, Siberian Math. J., 52:1 (2011), 30–40  mathnet  crossref  mathscinet  isi
    34. Darafsheh M.R., Monfared M.D., “Characterization of A$_{16}$ by a noncommuting graph”, Ukrainian Math. J., 62:11 (2011), 1673–1679  crossref  mathscinet  isi  scopus
    35. Khosravi B., Moradi H., “Quasirecognition by prime graph of finite simple groups L (n) (2) and U (n) (2)”, Acta Math Hungar, 132:1–2 (2011), 140–153  crossref  mathscinet  zmath  isi  elib  scopus
    36. Khosravi B., Akhlaghi Z., Khatami M., “Quasirecognition by prime graph of simple group D-n(3)”, Publ Math Debrecen, 78:2 (2011), 469–484  crossref  mathscinet  zmath  isi  scopus
    37. A. Babai, B. Khosravi, “Recognition by prime graph of $^2D_{2m+1}(3)$”, Siberian Math. J., 52:5 (2011), 788–795  mathnet  crossref  mathscinet  isi
    38. A. M. Staroletov, “Sporadic composition factors of finite groups isospectral to simple groups”, Sib. elektron. matem. izv., 8 (2011), 268–272  mathnet
    39. Ghasemabadi M.F., Iranmanesh A., “Quasirecognition by the Prime Graph of the Group C-n(2), Where n not equal 3 is Odd”, Bulletin of the Malaysian Mathematical Sciences Society, 34:3 (2011), 529–540  mathscinet  zmath  isi
    40. A. M. Staroletov, “On recognition by spectrum of the simple groups $B_3(q)$, $C_3(q)$, and $D_4(q)$”, Siberian Math. J., 53:3 (2012), 532–538  mathnet  crossref  mathscinet  isi
    41. M. A. Grechkoseeva, D. V. Lytkin, “Almost recognizability by spectrum of finite simple linear groups of prime dimension”, Siberian Math. J., 53:4 (2012), 645–655  mathnet  crossref  mathscinet  isi
    42. N. Ahanjideh, “Thompson's conjecture for some finite simple groups with connected prime graph”, Algebra and Logic, 51:6 (2013), 451–478  mathnet  crossref  mathscinet  zmath  isi
    43. Ghasemabadi M.F. Iranmanesh A., “2-Quasirecognizability of the Simple Groups B-N(P) and C-N(P) by Prime Graph”, Bull. Iran Math. Soc., 38:3 (2012), 647–668  mathscinet  zmath  isi
    44. Nosratpour P. Darafsheh M.R., “Recognition of the Groups l (5)(4) and U (4)(4) by the Prime Graph”, Ukr. Math. J., 64:2 (2012), 238–246  crossref  mathscinet  zmath  isi  elib  scopus
    45. Yan Ya., Chen G., Wang L., “Od-Characterization of the Automorphism Groups of O-10(+/-)(2)”, Indian J. Pure Appl. Math., 43:3 (2012), 183–195  crossref  mathscinet  zmath  isi  scopus
    46. Khosravi B. Moradi H., “Quasirecognition by Prime Graph of Some Orthogonal Groups Over the Binary Field”, J. Algebra. Appl., 11:3 (2012), 1250056  crossref  mathscinet  zmath  isi  elib  scopus
    47. A. K. Asboei, “A characterization of the simple sporadic groups”, Sib. elektron. matem. izv., 10 (2013), 200–204  mathnet
    48. Zhang L., Shi W., Yu D., Wang J., “Recognition of Finite Simple Groups Whose First Prime Graph Components Are R-Regular”, Bull. Malays. Math. Sci. Soc., 36:1 (2013), 131–142  mathscinet  zmath  isi
    49. A. V. Vasil'ev, A. M. Staroletov, “Recognizability of groups $G_2(q)$ by spectrum”, Algebra and Logic, 52:1 (2013), 1–14  mathnet  crossref  mathscinet  zmath  isi
    50. I. B. Gorshkov, “Recognizability of alternating groups by spectrum”, Algebra and Logic, 52:1 (2013), 41–45  mathnet  crossref  mathscinet  zmath  isi
    51. Akhilaghi Z., Khatami M., Khosravi B., “On the Prime Graph of D-N(Q) Where Q Is an Element of (2,5)”, Quaest. Math., 36:4 (2013), 517–535  crossref  mathscinet  isi  scopus
    52. A. S. Kondrat'ev, “Recognizability of groups $E_7(2)$ and $E_7(3)$ by prime graph”, Proc. Steklov Inst. Math. (Suppl.), 289, suppl. 1 (2015), 139–145  mathnet  crossref  mathscinet  isi  elib
    53. A. V. Vasil'ev, A. M. Staroletov, “Almost recognizability by spectrum of simple exceptional groups of Lie type”, Algebra and Logic, 53:6 (2015), 433–449  mathnet  crossref  mathscinet  isi
    54. I. B. Gorshkov, “Recognizability of symmetric groups by spectrum”, Algebra and Logic, 53:6 (2015), 450–457  mathnet  crossref  mathscinet  isi
    55. Zhang Q.L. Wang J.H. Liu W.J., “a Characterization of the Symmetric Group of Prime Degree”, Bull. Iran Math. Soc., 40:2 (2014), 473–480  mathscinet  zmath  isi  elib
    56. Huang X., Qu X., “Dependence System and Its Structure”, 2014 Tenth International Conference on Computational Intelligence and Security (Cis), IEEE, 2014, 326–329  crossref  isi  scopus
    57. A. V. Vasil'ev, M. A. Grechkoseeva, “Recognition by spectrum for simple classical groups in characteristic $2$”, Siberian Math. J., 56:6 (2015), 1009–1018  mathnet  crossref  crossref  mathscinet  isi  elib
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