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# Article

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Keywords:
main supergraph; simple Ree group; Thompson's problem
Summary:
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 $G\cong ^{2}G_{2}(3^{2n+1})$ if and only if $\mathcal{S}(G)\cong \mathcal{S}(^{2}G_{2}(3^{2n+1}))$. As a main consequence of our result we conclude that Thompson's problem is true for the small Ree group $^{2}G_{2}(3^{2n+1})$.
References:
[1] Asboei A. K., Amiri S. S. S.: Some alternating and symmetric groups and related graphs. Beitr. Algebra Geom. 59 (2018), no. 1, 21–24. DOI 10.1007/s13366-017-0360-8 | MR 3761396
[2] Asboei A. K., Amiri S. S. S.: Some results on the main supergraph of finite groups. accepted in Algebra Discrete Math.
[3] Cameron P. J.: The power graph of a finite group II. J. Group Theory 13 (2010), no. 6, 779–783. DOI 10.1515/jgt.2010.023 | MR 2736156
[4] Chakrabarty I., Ghosh S., Sen M. K.: Undirected power graphs of semigroups. Semigroup Forum 78 (2009), no. 3, 410–426. DOI 10.1007/s00233-008-9132-y | MR 2511776
[5] Chen G. Y.: On structure of Frobenius group and $2$-Frobenius group. J. Southwest China Normal Univ. 20 (1995), no. 5, 485–487 (Chinese).
[6] Ebrahimzadeh B., Iranmanesh A., Parvizi Mosaed H.: A new characterization of Ree group $^{2}G_2(q)$ by the order of group and the number of elements. Int. J. Group Theory 6 (2017), no. 4, 1–6. MR 3695074
[7] Frobenius G.: Verallgemeinerung des Sylow'schen Satzes. Berl. Ber. (1895), 981–993 (German).
[8] Hamzeh A., Ashrafi A. R.: Automorphism groups of supergraphs of the power graph of a finite group. European J. Combin. 60 (2017), 82–88. DOI 10.1016/j.ejc.2016.09.005 | MR 3567537
[9] Mazurov V. D., Khukhro E. I.: Unsolved Problems in Group Theory. Kourovka Notebook, Novosibirsk, Inst. Mat. Sibirsk. Otdel. Akad., 2006. MR 2263886
[10] Shi W.-J.: A characterization of $U_{3}(2^{n})$ by their element orders. Xinan Shifan Daxue Xuebao Ziran Kexue Ban 25 (2000), no. 4, 353–360. MR 1784865
[11] Ward H. N.: On Ree's series of simple groups. Trans. Amer. Math. Soc. 121 (1966), 62–89. MR 0197587
[12] Weisner L.: On the number of elements of a group which have a power in a given conjugate set. Bull. Amer. Math. Soc. 31 (1925), no. 9–10, 492–496. DOI 10.1090/S0002-9904-1925-04087-2 | MR 1561103
[13] Williams J. S.: Prime graph components of finite groups. J. Algebra 69 (1981), no. 2, 487–513. DOI 10.1016/0021-8693(81)90218-0 | MR 0617092 | Zbl 0471.20013
[14] Wilson R. A.: The Finite Simple Groups. Graduate Texts in Mathematics, 251, Springer, London, 2009. DOI 10.1007/978-1-84800-988-2 | MR 2562037
[15] Zhang Q., Shi W., Shen R.: Quasirecognition by prime graph of the simple groups $G_{2}(q)$ and $^{2}B_{2}(q)$. J. Algebra Appl. 10 (2011), no. 2, 309–317. DOI 10.1142/S0219498811004598 | MR 2795740

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