Lithuanian Mathematical Journal

, Volume 56, Issue 4, pp 492–502 | Cite as

The Influence of the Complete Nonexterior Square Graph on some Infinite Groups



The notion of nonabelian exterior square may be formulated for a pro-p-group G (p prime), getting the complete nonabelian exterior square\( G\widehat{\varLambda}G \) of G. We introduce the complete nonexterior square graph\( {\widehat{\varGamma}}_G \) of G, investigating finiteness conditions on G from restrictions on \( {\widehat{\varGamma}}_G \) and viceversa. This graph has the set of vertices G − (G), where (G) is the set of all elements of G commuting with respect to the operator \( \widehat{\varLambda} \), and two vertices x and y are joined by an edge if \( x\widehat{\varLambda}y\ne 1 \). Studying \( {\widehat{\varGamma}}_G \), we find the well-known noncommuting graph as a subgraph. Moreover, we show results on the structure of G and introduce a new class of groups, which originates naturally when G is infinite but \( {\widehat{\varGamma}}_G \) is finite.


complete nonabelian exterior square noncommuting graph pro-p-groups capable groups exterior degree 


05C25 20E18 20J05 05C63 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    F.N. Abd Manaf, A. Erfanian, F.G. Russo, and N.H. Sarmin, On the exterior degree of the wreath product of finite Abelian groups, Bull. Malays. Math. Sci. Soc. (2), 37:25–36, 2014.Google Scholar
  2. 2.
    A. Abdollahi, S. Akbari, and H.R. Maimani, Non-commuting graph of a group, J. Algebra, 298:468–492, 2006.MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    B. Bollobás, Modern Graph Theory, Springer, Berlin, 1998.CrossRefMATHGoogle Scholar
  4. 4.
    R. Brown, P. Higgins, and R. Sivera, Nonabelian Algebraic Topology. Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids, EMS Tracts Math., Vol. 15, EMS, Zürich, 2011.Google Scholar
  5. 5.
    R. Brown and J.L. Loday, Van Kampen theorems for diagrams of spaces, Topology, 26:311–335, 1987.MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    G. Ellis, Tensor products and q-crossed modules, J. Lond. Math. Soc., 2:241–258, 1995.MathSciNetMATHGoogle Scholar
  7. 7.
    A. Erfanian, P. Niroomand, M. Parvizi, and B. Tolue, Non-exterior square graph, Filomat, to appear.Google Scholar
  8. 8.
    A. Erfanian and R. Rezaei, On the commutativity degree of compact groups, Arch. Math., 93:201–212, 2009.MathSciNetMATHGoogle Scholar
  9. 9.
    K.H. Hofmann and S.A. Morris, The Structure of Compact Groups, de Gruyter, Berlin, 2006.CrossRefMATHGoogle Scholar
  10. 10.
    K.H. Hofmann and F.G. Russo, The probability that x and y commute in a compact group, Math. Proc. Camb. Philos. Soc., 153:557–571, 2012.MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    P. Moravec, On the Schur multipliers of finite p-groups of given coclass, Isr. J. Math., 185:189–205, 2011.MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    P. Niroomand, R. Rezaei, and F.G. Russo, Commuting powers and exterior degree of finite groups, J. Korean Math. Soc., 49:855–865, 2012.MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    P. Niroomand and F.G. Russo, A note on the exterior centralizer, Arch. Math., 93:505–512, 2009.MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    D.E. Otera, F.G. Russo, and C. Tanasi, Some algebraic and topological properties of the nonabelian tensor product, Bull. Korean Math. Soc., 50:1069–1077, 2013.MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    R. Rezaei and F.G. Russo, Commuting elements with respect to the operator Λ in infinite groups, Bull. Korean Math. Soc., 53:1353–1362, 2016.CrossRefMATHGoogle Scholar
  16. 16.
    D.J.S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Vol. 1, Springer, Berlin, 1972.CrossRefMATHGoogle Scholar
  17. 17.
    J. Rotman, An Introduction to Algebraic Topology, Springer, Berlin, 1988.CrossRefMATHGoogle Scholar
  18. 18.
    F.G. Russo, Problems of connectivity between the Sylow graph, the prime graph and the non-commuting graph of a group, Adv. Pure Math., 2:391–396, 2012.CrossRefGoogle Scholar
  19. 19.
    F.G. Russo, The topology of the nonabelian tensor product of profinite groups, Bull. Korean Math. Soc., 53:751–763, 2016.MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    A. Shalev, Profinite groups with restricted centralizers, Proc. Am. Math. Soc., 122:1279–1284, 1994.MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    R. Solomon and A. Woldar, Simple groups are characterized by their non-commuting graphs, J. Group Theory, 16:793–824, 2013.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Applied MathematicsUniversity of Cape TownCape TownSouth Africa

Personalised recommendations