Volume 16, Issue 1 (4-2021)                   IJMSI 2021, 16(1): 137-144 | Back to browse issues page

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Let $G$ be a finite group. The main supergraph $mathcal{S}(G)$ is a graph with vertex set $G$ in which two vertices $x$ and $y$ are adjacent if and
only if $o(x) mid o(y)$ or $o(y)mid o(x)$. In this paper, we will show that $Gcong L_{2}(q)$ if and only if $mathcal{S}(G)cong mathcal{S} (L_{2}(q))$, where $q$ is a prime power. This work implies that Thompson's problem holds for the simple group $L_{2}(q)$.
Type of Study: Research paper | Subject: General

1. P. J. Cameron, The power graph of a finite group, II,textit{J. Group Theory}, textbf{13} (2010), 779-783. [DOI:10.1515/jgt.2010.023]
2. I. Chakrabarty, S. Ghosh, M. K. Sen, Undirected power graphs of semigroups, textit{Semigroup Forum}, textbf{78} (2009), 410-426. [DOI:10.1007/s00233-008-9132-y]
3. G. Y. Chen, On Frobenius and $2$-Frobenius group, textit{J.Southwest China Normal Univ}, textbf{20} (1995), 485-487. (in Chinese)
4. G. Y. Chen, Further reflections on Thompson's conjecture,textit{J. Algebra}, textbf{218} (1999), 276-285. [DOI:10.1006/jabr.1998.7839]
5. G. Y. Chen, On Thompson's conjecture, textit{J. Algebra},textbf{185}(1) (1996), 184--193. [DOI:10.1006/jabr.1996.0320]
6. J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R.A. Wilson, textit{Atlas of finite groups}, Clarendon, Oxford, 1985.
7. G. Frobenius, Verallgemeinerung des Sylow'schen Satzes.textit{Berl. Ber}, (1895), 981--993. (In German)
8. A. Hamzeh, A. R. Ashrafi, Automorphism groups of supergraphs of the power graph of a finite group, textit{European J. Combin}, textbf{60} (2017), 82-88. [DOI:10.1016/j.ejc.2016.09.005]
9. B. Huppert, textit{Endliche Gruppen}, I, Springer, Berlin,1967. [DOI:10.1007/978-3-642-64981-3]
10. N. Iiyori, H. Yamaki, Prime graph components of the simple groups of Lie type over the field of even characteristic, textit{J. Algebra}, textbf{155}(2) (1993), 335-343. [DOI:10.1006/jabr.1993.1048]
11. A. Khalili, A. Iranmanesh, A characterization of linear group $L_{2}(p)$, textit{Czechoslovak Math. J}, textbf{64} (139) (2014),459-464. [DOI:10.1007/s10587-014-0112-y]
12. A. Khalili, S. S. Salehi, Some alternating and symmetric groups and related graphs, textit{Beitr Algebra Geom}, textbf{59} (2018),21-24. [DOI:10.1007/s13366-017-0360-8]
13. A. Khalili, S. S. Salehi, Some results on the main supergraph of finite groups, textit{Algebra Discrete Math}, textbf{30}(2), (2020), 172-178. [DOI:10.12958/adm584]
14. A. Khalili, S. S. Salehi, The small Ree group $^{2}G_{2}(3^{2n+1})$ and related graph, textit{Comment. Math. Univ. Carolin}, textbf{59}(3) (2018), 271--276. [DOI:10.14712/1213-7243.2015.255]
15. A. Khalili, S. S. Salehi, Recognizability of finite groups by Suzuki group, textit{Arch. Math., Brno, }textbf{55} (2019),225-228. [DOI:10.5817/AM2019-4-225]
16. V. D. Mazurov, E. I. Khukhro, textit{Unsolved problems in group theory}, The Kourovka Notebook, (English version), ArXiv e-prints,(18), January 2014. Available at http://arxiv.org/abs/1401.0300v6.
17. A. S. Kondtratev, V. D. Mazurove, Recognition of alternating groups of prime degree from their element orders, textit{Sib. Math.J},textbf{41}(2) (2000), 294-302. [DOI:10.1007/BF02674599]
18. J. S. Williams, Prime graph components of finite groups, textit{J. Algebra}, textbf{69}(2) (1981), 487--513. [DOI:10.1016/0021-8693(81)90218-0]