ON THE DIRECT PRODUCT OF
NEARLY S-PERMUTABLE SUBGROUPS
Bilal N. Al-Hasanat1, Awni F. Al-Dababseh2,
Khaled A. Al-Sharo3, Baheej R. Al-Shuraifeen4
1,2,4Department of Mathematics
Al Hussein Bin Talal University
Ma'an, JORDAN
3 Department of Mathematics
Al al-Bayt University
Mafraq, JORDAN

Abstract

A subgroup $H$ is said to be S-permutable in $G$ if it permutes with all Sylow subgroups of $G$. A subgroup $H$ of $G$ is called nearly S-permutable in $G$ if for every prime $p$ such that $\gcd(p,\vert H\vert)=1$ and for every subgroup $K$ of $G$ containing $H$, the normalizer $N_K(H)$ contains some Sylow $p$-subgroup of $K$. The main aims of this article is to classify the family of all nearly S-permutable subgroups for certain groups and study the direct product of their subgroups. Moreover, we prove that the direct product of certain nearly S-permutable subgroups is necessary nearly S-permutable.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 35
Issue: 2
Year: 2022

DOI: 10.12732/ijam.v35i2.9

Download Section



Download the full text of article from here.

You will need Adobe Acrobat reader. For more information and free download of the reader, please follow this link.

References

  1. [1] R.A. Ikram, On Certain Subgroup Lattices of Finite Groups, Master’s Thesis, Al Al-Bayt University (2020).
  2. [2] K.A. Al-Sharo, On nearly s-permutable subgroups of finite groups, Comm. Algebra, 40, No 1 (2012), 315–326.
  3. [3] B.N. Al-Hasanat, A. Aldabaseh and A. Alissah, Permutable subgroups of groups of order 16, International Journal of Applied Mathematics, 32, No 1 (2019), 101–110, DOI: 10.12732/ijam.v32i1.10.
  4. [4] D.S. Dummet and R.M. Foot, Abstract Algrbra, John Wiley and Sons, Inc., New York (2004).
  5. [5] J.B. Fraleigh, A First Course in Abstract Algebra, Pearson Education, India (2003).
  6. [6] I.N. Herstein, Abstract Algebra, John Wiley and Sons, Inc., Hoboken, NJ., 3rd Ed. (1995).
  7. [7] M. Isaacs, Finite Group Theory, AMS, Rhode Island (2008).
  8. [8] T.W. Judson, Abstract Algebra: Theory and Applications, TN: PWS Publishing Company (2010).
  9. [9] O.H. Kegel, Sylow-gruppen und subnormalteiler endlicher gruppen, Mathematische Zeitschrift, 78, No 1 (1962), 205–221.
  10. [10] K.M. Al-Jamal, K.A. Al-Sharo, A.T. Abu Ghani, Finite groups in which nearly s-permutability is a transitive relation, International Journal of Mathematics and Computer Science, 2 (2019), 493–499.
  11. [11] Z.Y. Karataz and M.R. Dixon, Groups with all subgroups permutable or soluble, International Journal of Group Theory, 2, No 1 (2013), 37–43.
  12. [12] O. Ore, Contributions to the theory of groups of finite order, Duke Math. J., 5, No 2 (1939), 431–460.
  13. [13] D.J. Robinson, A Course in the Theory of Groups, Springer Science & Business Media (1996).