Some Remarks on the Varieties of Pairs of Groups

Document Type : Research Paper

Author

Department of Mathematics, Neyshabur Branch, Islamic Azad University, Neyshabur, Iran; Tel.: +123-45-678910 Fax: +123-45-678910

Abstract

Let V be a variety of groups defined by a set v of laws .Let (N,G) be a pair of groups in which N is a normal subgroup of G. we define the lower and upper V-marginal series of the pair (N,G) and prove some results on the varieties nilpotent pais of groups. Moreover, we extend some properties of the Baer-invariant and isologism of a pairs of groups.

Keywords


  1. G. Ellis, The Schur multiplier of a pair of groups, Appl. Categ. Structures 6(3), 355-371 (1998).
  2. A. Gholami, Z. Mohammad Abadi and S. Heidarian, Some generalized varietal properties of a pair of groups, Southeast Asian Bull. Math 36, 301-308 (2012).
  3. M. Hassanzadeh, A. Pourmirzaei and S. Kayvanfar, On the nilpotency of a pair of groups, Southeast Asian Bull. Math 37, 67-77 (2013).
  4. S. Heidarian, A. Gholami and Z. Mohammad Abadi, The structure of ν-isologic pairs of groups, Filomat 26 (1), 67-79 (2011).
  5. C.R. Leedham-Green and S. Mackay, Baer- Invariants, Isologism, Varietal Lows and Homology, Acta Math 137, 99-150 (1976).
  6. S. Lotfi and M. Taheri, Some relations between isologic and varietal perfect groups, Hindawi Journal of Mathematics, Volume, ID5093983, 4pages (2016).
  7. M.R.R. Moghaddam and A.R. Salemkar, Varietal isologism and covering groups, Arch. Math. 75 (1) 8-15 (2000).
  8. M.R.R. Moghaddam, A.R. Salemkar and H.M. Sanny, Some inequalities for the Baerinvariant of a finite groups, Indag. Math . 18(1), 73-82 (2007).
  9. M.R.R. Moghaddam, A.R. Salemkar and P. Niroomand, Covering groups respect to two varieties of groups, Southeast Asian Bull. Math. 31, 313-320 (2007).
  10. H. Neuman, Varieties of groups, Speringer- Verlag Berlin Heidelberg, New York, 1967.
  11. A. Onsory and M. Araskhan, The Generalization of the Stalling’s Theorem, Journal of Generalized Lie Theory and Applications, 3 pages, 2015.
  12. M. R. Rismanchian and M. Araskhan, Generalized Baer-Invariant of a pair of groups and the direct limit, Journal of Sciences, Islamic Republic of Iran, 22 (2), 159-164 (2011).
  13. M. R. Rismanchian and M. Araskhan, Generalized Baer-Invariant of a Pair of Groups and Marginal Extension, Journal of Sciences, Islamic Republic of Iran, 23 (3), 251-256 (2012).
  14. M. R. Rismanchian and M. Araskhan, Some properties on the Baer-invariant of a pair of groups and νG-marginal series, Turk. J. Math. 37, 259-266 (2013).
  15. S. M. Taheri and F. Shahini, Varietal nilpotent groups, World Applied Sciences Journal. 12(12), 2314-2316 (2011)