c-Nilpotent Multiplier of Finite p-Groups

  • Peyman NiroomandEmail author
  • Farangis Johari
  • Mohsen Parvizi


The aim of this work is to find some exact sequences on the c-nilpotent multiplier of a group G. We also give an upper bound for the c-nilpotent multiplier of finite p-groups and give the explicit structure of groups whose take the upper bound. Finally, we will get the exact structure of the c-nilpotent multiplier and determine c-capable groups in the class of extra-special and generalized extra-special p-groups. It lets us to have a vast improvement over the last results on this topic.


c-Nilpotent multiplier Capability p-Groups 

Mathematics Subject Classification

Primary 20C25 Secondary 20D15 



  1. 1.
    Baer, R.: Representations of groups as quotient groups, I, II, and III. Trans Amer. Math. Soc. 54, 295–419 (1945)zbMATHGoogle Scholar
  2. 2.
    Berkovich, Y.G.: On the order of the commutator subgroups and the Schur multiplier of a finite \(p\)-group. J. Algebra 144, 269–272 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berkovich, Y.G., Zvonimir, J.: Groups of Prime Power Order. Vol. 2. de Gruyter Expositions in Mathematics, 47. Walter de Gruyter GmbH & Co. KG, Berlin (2008)CrossRefGoogle Scholar
  4. 4.
    Beyl, F.R., Felgner, U., Schmid, P.: On groups occurring as center factor groups. J. Algebra 61, 161–177 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brown, R., Johnson, D.L., Robertson, E.F.: Some computations of non-abelian tensor products of groups. J. Algebra 111, 177–202 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brown, R., Loday, J.-L.: Van Kampen theorems for diagrams of spaces. Topology 26, 311–335 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Burns, J., Ellis, G.: On the nilpotent multipliers of a group. Math. Z. 226, 405–428 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Burns, J., Ellis, G.: Inequalities for Baer invariants of finite groups. Canad. Math. Bull. 41(4), 385–391 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ellis, G.: Tensor products and \(q\)-crossed modules. J. Lond. Math. Soc. 51(2), 243–258 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ellis, G., McDermott, A.: Tensor products of prime-power groups. J. Pure Appl. Algebra 132(2), 119–128 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ellis, G.: On groups with a finite nilpotent upper central quotient. Arch. Math. 70, 89–96 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ellis, G.: On the Schur multiplier of \(p\)-groups. Commun. Algebra 9, 4173–4177 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ellis, G., Wiegold, J.: A bound on the Schur multiplier of a prime power group. Bull. Aust. Math. Soc. 60, 191–196 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ellis, G.: On the relation between upper central quotient and lower central series of a group. A.M.S. Soc. 353, 4219–4234 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fröhlich, A.: Baer invariants of algebras. Trans. Am. Math. Soc. 109, 221–244 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hall, M.: The Theory of Groups. MacMillan Company, NewYork (1959)zbMATHGoogle Scholar
  17. 17.
    Hall, P.: The classificatin of prime-power groups. J. Reine Angew. Math. 182, 130–141 (1940)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Hall, P.: Verbal and marginal subgroups. J. Reine Angew. Math. 182, 156–157 (1940)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Heineken, H., Nikolova, D.: Class two nilpotent capable groups. Bull. Aust. Math. Soc. 54, 347–352 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Huppert, B., Blackburn, N.: Finite Groups II. Springer, Berlin (1982)zbMATHGoogle Scholar
  21. 21.
    Jafari, S.H., Saeedi, F., Khamseh, E.: Characterization of finite \(p\)-groups by their non-abelian tensor square. Commun. Algebra 41, 1954–1963 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Jones, M.R.: Multiplicutors of \(p\)-groups. Math. Z. 127, 165–166 (1972)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Jones, M.R.: Some inequalities for the multiplicator of a finite group. Proc. Am. Math. Soc. 39, 450–456 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Karpilovsky, G.: The Schur Mulptiplier, N.S. 2, London Math. Soc. Monogr. Oxford University Press, Oxford (1987)Google Scholar
  25. 25.
    Leedham-Green, C.R., MacKay, S.: Baer invariants, isologism, varietal laws and homology. Acta Math. 137, 99–150 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    MacDonald, J.L.: Group derived functors. J. Algebra 10, 448–477 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Mashayekhy, B., Sanati, M.A.: On the order of nilpotent multipliers of finite \(p\)-groups. Commun. Algebra 33(7), 2079–2087 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Mashayekhy, B., Mohammadzadeh, F.: Some inequalities for nilpotent multipliers of powerful \(p\)-groups. Bull. Iran. Math. Soc. 33(2), 61–71 (2007)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Moghaddam, M.R.R.: The Baer invariant of a direct product. Arch. Math. 33, 504–511 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Moghaddam, M.R.R.: Some inequalities for Baer invariant of a finite group. Bull. Iran. Math. Soc. 9, 5–10 (1981)MathSciNetGoogle Scholar
  31. 31.
    Moghaddam, M.R.R.: On the Schur-Baer property. J. Aust. Math. Soc. Ser. A 31, 343–361 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Moghaddam, M.R.R., Kayvanfar, S.: A new notion derived from varieties of groups. Algebra Colloq. 4(1), 1–11 (1997)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Niroomand, P.: On the order of Schur multiplier of non-abelian \(p\)-groups. J. Algebra 322, 4479–4482 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Niroomand, P.: The Schur multiplier of \(p\)-groups with large derived subgroup. Arch. Math. 95, 101–103 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Niroomand, P., Parvizi, M.: A remark on the capability of finite \(p\)-groups. J. Adv. Res. Pure Math. 5(4), 91–95 (2013)MathSciNetGoogle Scholar
  36. 36.
    Niroomand, P., Parvizi, M.: On the \(2\)-nilpotent multiplier of finite \(p\)-groups. Glasg. Math. J. 57, 201–210 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Stancu, R.: Almost all generalized extra-special \(p\)-groups are resistant. J. Algebra 249, 120–126 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Zhou, X.: On the order of Schur multipliers of finite \(p\)-groups. Commun. Algebra 1, 1–8 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Xu, M.Y., An, L.J., Zhang, Q.H.: Finite \(p\)-groups all of whose non-abelian proper subgroups are generated by two elements. J. Algebra 319, 3603–3620 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2019

Authors and Affiliations

  1. 1.School of Mathematics and Computer ScienceDamghan UniversityDamghanIran
  2. 2.Department of Pure MathematicsFerdowsi University of MashhadMashhadIran

Personalised recommendations