Group nilpotency from a graph point of view

Document Type : Ischia Group Theory 2022

Authors

1 Dipartimento di Matematica e Applicazioni, Università di Milano-Bicocca, Milano, Italy

2 Dipartimento di Matematica, Università di Padova, Padova, Italy

3 Dipartimento di Matematica, Università di Salerno, Salerno, Italy

Abstract

Let $\Gamma_G$ denote a graph associated with a group $G$. A compelling question about finite groups asks whether or not a finite group $H$ must be nilpotent provided $\Gamma_H$ is isomorphic to $\Gamma_G$ for a finite nilpotent group $G$. In the present work we analyze the problem for different graphs that one can associate with a finite group, both reporting on existing answers and contributing to new ones.

Keywords

Main Subjects


[24] A. Lucchini, Finite groups with the same join graph as a finite nilpotent group, Glasg. Math. J., 63 (2021) 640–650.
[25] A. Lucchini and A. Maróti, On the clique number of the generating graph of a finite group, Proc. Amer. Math. Soc., 137 (2009) 3207–3217.
[26] A. Lucchini and A. Maróti, Some results and questions related to the generating graph of a finite group, Ischia Group Theory 2008, World Sci. Publ., Hackensack, NJ (2009) 183–208.
[27] A. Lucchini, A. Maróti, and C. Roney-Dougal, On the generating graph of a simple group, J. Austral. Math. Soc., 103 (2017) 91–103.
[28] A. Lucchini and D. Nemmi, On the connectivity of the non-generating graph, Arch. Math. (Basel), 118 (2022) 563–576.
[29] M. Mirzargar and R. Scapellato, Finite groups with the same power graph, Comm. Algebra, 50 (2022) 1400–1406.
[30] A. R. Moghaddamfar, About noncommuting graphs, Siberian Math. J., 47 (2006) 911–914.
[31] B. H. Neumann, A problem of Paul Erdős on groups, J. Austral. Math. Soc. Ser. A, 21 (1976) 467–472.
[32] D. J. S. Robinson, A course in the theory of groups, second ed., Grad. Texts in Math. 80, Springer, New York, (1996).
[33] R. Schmidt, Zentralisatorverbände endlicher gruppen, Rend. Sem. Mat. Univ. Padova, 44 (1970) 97–131.
[34] S. Zahirovic, I. Bosnjak and R. Madarasz, A study of enhanced power graphs of finite groups, J. Algebra Appl., 19 (2020) 20 pp.
[24] A. Lucchini, Finite groups with the same join graph as a finite nilpotent group, Glasg. Math. J., 63 (2021) 640–650.
[25] A. Lucchini and A. Maróti, On the clique number of the generating graph of a finite group, Proc. Amer. Math. Soc., 137 (2009) 3207–3217.
[26] A. Lucchini and A. Maróti, Some results and questions related to the generating graph of a finite group, Ischia Group Theory 2008, World Sci. Publ., Hackensack, NJ (2009) 183–208.
[27] A. Lucchini, A. Maróti, and C. Roney-Dougal, On the generating graph of a simple group, J. Austral. Math. Soc., 103 (2017) 91–103.
[28] A. Lucchini and D. Nemmi, On the connectivity of the non-generating graph, Arch. Math. (Basel), 118 (2022) 563–576.
[29] M. Mirzargar and R. Scapellato, Finite groups with the same power graph, Comm. Algebra, 50 (2022) 1400–1406.
[30] A. R. Moghaddamfar, About noncommuting graphs, Siberian Math. J., 47 (2006) 911–914.
[31] B. H. Neumann, A problem of Paul Erdős on groups, J. Austral. Math. Soc. Ser. A, 21 (1976) 467–472.
[32] D. J. S. Robinson, A course in the theory of groups, second ed., Grad. Texts in Math. 80, Springer, New York, (1996).
[33] R. Schmidt, Zentralisatorverbände endlicher gruppen, Rend. Sem. Mat. Univ. Padova, 44 (1970) 97–131.
[34] S. Zahirovic, I. Bosnjak and R. Madarasz, A study of enhanced power graphs of finite groups, J. Algebra Appl., 19 (2020) 20 pp.

Articles in Press, Corrected Proof
Available Online from 21 September 2023
  • Receive Date: 07 March 2023
  • Revise Date: 17 September 2023
  • Accept Date: 21 September 2023
  • Published Online: 21 September 2023