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On the non-nilpotent graphs of a group

Year 2017, , 78 - 96, 11.07.2017
https://doi.org/10.24330/ieja.325927

Abstract

 Let $G$ be a group and $nil(G)=\{x \in G \mid \langle x,y \rangle \text{ is nilpotent for all }\\ y \in G\}$.
 Associate a graph $\mathfrak{R}_G$ (called the non-nilpotent graph of $G$) with $G$ as follows: Take $G \setminus nil(G)$ as the vertex set and two vertices are adjacent if they generate a non-nilpotent subgroup. In this paper we study the graph theoretical properties of $\mathfrak{R}_G$. We conjecture that the domination number of the non-nilpotent graph of every finite non-abelian simple group is 2. We also conjecture that if $G$ and $H$ are two non-nilpotent finite groups such that $\mathfrak{R}_G\cong \mathfrak{R}_H$, then $|G| = |H|$. Among other results, we show that the non-nilpotent graph of $D_{10}$ is double-toroidal.
 

References

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  • A. Abdollahi and M. Zarrin, Non-nilpotent graph of a group, Comm. Algebra, 38(12) (2010), 4390-4403.
  • A. Azad, M. A. Iranmanesh, C. E. Praeger and P. Spiga, Abelian coverings of finite general linear groups and an application to their non-commuting graphs, J. Algebraic Combin., 34(4) (2011), 638-710.
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  • M. R. Darafsheh, Groups with the same non-commuting graph, Discrete Appl. Math., 157(4) (2009), 833-837.
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  • L. Pyber, The number of pairwise noncommuting elements and the index of the centre in a nite group, J. London Math. Soc., 35(2) (1987), 287-295.
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  • R. Schmidt, Zentralisatorverbande endlicher gruppen, Rend. Sem. Mat. Univ. Padova, 44 (1970), 97-131.
  • R. M. Solomon and A. J. Woldar, Simple groups are characterized by their non-commuting graphs, J. Group Theory, 16(6) (2013), 793-824.
  • V. P. Sunkov, Periodic group with almost regular involutions, Algebra i Logika, 7(1) (1968), 113-121.
  • D. B. West, Introduction to Graph Theory (Second Edition), PHI Learning Private Limited, New Delhi, 2009.
  • A. T. White, Graphs, Groups and Surfaces, North-Holland Mathematics Studies, 8, American Elsevier Publishing Co., Inc., New York, 1973.
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Year 2017, , 78 - 96, 11.07.2017
https://doi.org/10.24330/ieja.325927

Abstract

References

  • A. Abdollahi, S. Akbari and H. R. Maimani, Non-commuting graph of a group, J. Algebra, 298(2) (2006), 468-492.
  • A. Abdollahi and M. Zarrin, Non-nilpotent graph of a group, Comm. Algebra, 38(12) (2010), 4390-4403.
  • A. Azad, M. A. Iranmanesh, C. E. Praeger and P. Spiga, Abelian coverings of finite general linear groups and an application to their non-commuting graphs, J. Algebraic Combin., 34(4) (2011), 638-710.
  • J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, American Elsevier Publishing Co., Inc., New York, 1976.
  • M. R. Darafsheh, Groups with the same non-commuting graph, Discrete Appl. Math., 157(4) (2009), 833-837.
  • A. K. Das and D. Nongsiang, On the genus of the nilpotent graphs of finite groups, Comm. Algebra, 43(12) (2015), 5282-5290.
  • The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.6.4, 2013 (http://www.gap-system.org).
  • B. Huppert and N. Blackburn, Finite Groups, III, Springer-Verlag, Berlin, 1982.
  • B. H. Neumann, A problem of Paul Erdos on groups, J. Austral. Math. Soc. Ser. A, 21(4) (1976), 467-472.
  • A. Yu. Ol'shanskii, Geometry of De ning Relations in Groups, Kluwer Academic Publishers Group, Dordrecht, 1991.
  • L. Pyber, The number of pairwise noncommuting elements and the index of the centre in a nite group, J. London Math. Soc., 35(2) (1987), 287-295.
  • D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Part 2, Springer-Verlag, New York, 1972.
  • D. J. S. Robinson, A Course in the Theory of Groups, Graduate Texts in Mathematics, 80, Springer-Verlag, New York-Berlin, 1982.
  • D. M. Rocke, p-Groups with abelian centralizers, Proc. London Math. Soc., 30(3) (1975), 55-75.
  • R. Schmidt, Zentralisatorverbande endlicher gruppen, Rend. Sem. Mat. Univ. Padova, 44 (1970), 97-131.
  • R. M. Solomon and A. J. Woldar, Simple groups are characterized by their non-commuting graphs, J. Group Theory, 16(6) (2013), 793-824.
  • V. P. Sunkov, Periodic group with almost regular involutions, Algebra i Logika, 7(1) (1968), 113-121.
  • D. B. West, Introduction to Graph Theory (Second Edition), PHI Learning Private Limited, New Delhi, 2009.
  • A. T. White, Graphs, Groups and Surfaces, North-Holland Mathematics Studies, 8, American Elsevier Publishing Co., Inc., New York, 1973.
  • C. Wickham, Classification of rings with genus one zero-divisor graphs, Comm. Algebra, 36(2) (2008), 325-345.
There are 20 citations in total.

Details

Subjects Mathematical Sciences
Journal Section Articles
Authors

Deiborlang Nongsiang This is me

Promode Kumar Saikia This is me

Publication Date July 11, 2017
Published in Issue Year 2017

Cite

APA Nongsiang, D., & Saikia, P. K. (2017). On the non-nilpotent graphs of a group. International Electronic Journal of Algebra, 22(22), 78-96. https://doi.org/10.24330/ieja.325927
AMA Nongsiang D, Saikia PK. On the non-nilpotent graphs of a group. IEJA. July 2017;22(22):78-96. doi:10.24330/ieja.325927
Chicago Nongsiang, Deiborlang, and Promode Kumar Saikia. “On the Non-Nilpotent Graphs of a Group”. International Electronic Journal of Algebra 22, no. 22 (July 2017): 78-96. https://doi.org/10.24330/ieja.325927.
EndNote Nongsiang D, Saikia PK (July 1, 2017) On the non-nilpotent graphs of a group. International Electronic Journal of Algebra 22 22 78–96.
IEEE D. Nongsiang and P. K. Saikia, “On the non-nilpotent graphs of a group”, IEJA, vol. 22, no. 22, pp. 78–96, 2017, doi: 10.24330/ieja.325927.
ISNAD Nongsiang, Deiborlang - Saikia, Promode Kumar. “On the Non-Nilpotent Graphs of a Group”. International Electronic Journal of Algebra 22/22 (July 2017), 78-96. https://doi.org/10.24330/ieja.325927.
JAMA Nongsiang D, Saikia PK. On the non-nilpotent graphs of a group. IEJA. 2017;22:78–96.
MLA Nongsiang, Deiborlang and Promode Kumar Saikia. “On the Non-Nilpotent Graphs of a Group”. International Electronic Journal of Algebra, vol. 22, no. 22, 2017, pp. 78-96, doi:10.24330/ieja.325927.
Vancouver Nongsiang D, Saikia PK. On the non-nilpotent graphs of a group. IEJA. 2017;22(22):78-96.