Advertisement

Siberian Mathematical Journal

, Volume 60, Issue 1, pp 41–55 | Cite as

Arithmetic Graphs and Classes of Finite Groups

  • A. F. VasilyevEmail author
  • V. I. MurashkaEmail author
Article
  • 4 Downloads

Abstract

An arithmetic graph function is a mapping associating to a finite group G the graph whose vertices are the divisors of |G|. We formulate and study the problem of recognizing hereditary saturated formations by arithmetic graph functions, and solve it for some arithmetic graph functions.

Keywords

finite group directed graph recognition by graph arithmetic graph function hereditary saturated formation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Vasilyev A. V. and Vdovin E. P., “An adjacency criterion for the prime graph of a finite simple group,” Algebra and Logic, vol. 44, No. 6, 381–406 (2005).MathSciNetCrossRefGoogle Scholar
  2. 2.
    Zavarnitsine A. V., “Recognition of finite groups by the prime graph,” Algebra and Logic, vol. 45, No. 4, 220–231 (2006).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ballester–Bolinches A., Cossey J., and Esteban–Romero R., “Graphs and classes of finite groups,” Note Mat., vol. 33, No. 1, 89–94 (2013).MathSciNetzbMATHGoogle Scholar
  4. 4.
    Ballester–Bolinches A. and Cossey J., “Graphs, partitions and classes of groups,” Monatsh. Math., vol. 166, No. 3–4, 309–318 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kazarin L. S., Martinez–Pastor A., and Perez–Ramos M. D., “On the Sylow graph of a group and Sylow normalizers,” Israel J. Math., no. 186, 251–271 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Russo F. G., “Problems of connectivity between the Sylow graph, the prime graph and the non–commuting graph of a group,” Adv. Pure Math., no. 2, 391–396 (2012).CrossRefGoogle Scholar
  7. 7.
    Cayley A., “Desiderata and suggestions. 2. The theory of groups: graphical representation,” Amer. J. Math., vol. 1, No. 2, 174–176 (1878).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cooperman G., Finkelstein L., and Sarawagi N., “Applications of Cayley graphs,” in: Applied Algebra, Algebraic Algorithms and Error–Correcting Codes, Berlin; Heidelberg: Springer–Verlag, 1991, 367–378 (Lect. Notes Comput. Sci.; vol. 508).CrossRefGoogle Scholar
  9. 9.
    Neumann B., “A problem of Paul Erdös on groups,” J. Austral. Math. Soc., no. 21, 467–472 (1976).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Williams J. S., “Prime graph components of finite groups,” J. Algebra, no. 69, 487–513 (1981).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kondratiev A. S., “Prime graph components of finite simple groups,” Math. USSR–Sb., vol. 67, No. 1, 235–247 (1990).MathSciNetGoogle Scholar
  12. 12.
    Mazurov V. D., “Recognition of finite groups by a set of orders of their elements,” Algebra and Logic, vol. 37, No. 6, 371–379 (1998).MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hawkes T., “On the class of the Sylow tower groups,” Math. Z., Bd 105, 393–398 (1968).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    D’Aniello A., De Vivo C., and Giordano G., “Lattice formations and Sylow normalizers: a conjecture,” Atti Semin. Mat. Fis. Univ. Modena, no. 55, 107–112 (2007).MathSciNetzbMATHGoogle Scholar
  15. 15.
    Kamornikov S. F. and Sel’kin V. M., Subgroup Functors and Classes of Finite Groups [Russian], Belarusskaya Nauka, Minsk (2003).Google Scholar
  16. 16.
    Doerk K. and Hawkes T., Finite Soluble Groups, Walter de Gruyter, Berlin and New York (1992).CrossRefzbMATHGoogle Scholar
  17. 17.
    Shemetkov L. A., Formations of Finite Groups [Russian], Nauka, Moscow (1978).zbMATHGoogle Scholar
  18. 18.
    Distel R., Graph Theory, Springer–Verlag, Heidelberg and New York (2005).Google Scholar
  19. 19.
    Griess R. L. and Schmid P., “The Frattini module,” Arch. Math., vol. 30, 256–266 (1978).MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gaschütz W., “Über modulare Darstellungen endlicher Gruppen, die von freien Gruppen induziert werden,” Math. Z., Bd 60, 274–286 (1954).CrossRefzbMATHGoogle Scholar
  21. 21.
    Skiba A. N., Algebra of Formations [Russian], Belarusskaya Nauka, Minsk (1997).zbMATHGoogle Scholar
  22. 22.
    Ballester–Bolinches A. and Perez–Ramos M. D., “Two questions of L. A. Shemetkov on critical groups,” J. Algebra, no. 179, 905–917 (1996).MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    D’Aniello A. “Saturated formations closed under Sylow normalizers,” Commun. Algebra, vol. 33, 2801–2805 (2005).MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Huppert B., Endlicher Gruppen. I, Springer, Berlin, Heidelberg, and New York (1967).CrossRefGoogle Scholar
  25. 25.
    Suzuki M., “On a class of doubly transitive groups,” Ann. Math., vol. 75, 105–145 (1962).MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kegel O. H., “Zur Struktur mehrfach factorisierbarer endlicher Gruppen,” Math. Z., vol. 87, no. 1, 42–48 (1965).Google Scholar
  27. 27.
    Kazarin L. S., “Factorizations of finite groups by solvable subgroups,” Ukrainian Math. J., vol. 43, No. 7–8, 883–886 (1991).MathSciNetzbMATHGoogle Scholar
  28. 28.
    Vasilyev A. F., “On the enumeration problem of local formations with a given property,” Voprosy Algebry, no. 3, 3–11 (1987).Google Scholar
  29. 29.
    Pennington E., “Trifactorisable groups,” Bull. Austral. Math. Soc., vol. 8, 461–469 (1973).MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Monakhov V. S., “Finite groups with a given set of Schmidt subgroups,” Math. Notes, vol. 58, No. 5, 1183–1186 (1995).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Francisk Skorina Gomel State UniversityGomelBelarus

Personalised recommendations