M.R. Zinov’eva. On finite simple groups of exceptional Lie type over fields of different characteristics with coinciding prime graphs ... С. 147-160

Suppose that $G$ is a finite group, $\pi(G)$ is the set of prime divisors of its order, and $\omega(G)$ is the set of orders of its elements. A graph with the following adjacency relation is defined on $\pi(G)$: different vertices $r$ and $s$ from $\pi(G)$ are adjacent if and only if $rs\in \omega(G)$. This graph is called the {\it Gruenberg--Kegel graph} or the  prime graph of $G$ and is denoted by $GK(G)$. In A.V. Vasil'ev's Question 16.26 from the "Kourovka Notebook", it is required to describe all pairs of nonisomorphic finite simple nonabelian groups with identical Gruenberg-Kegel graphs. M. Hagie (2003) and M.A. Zvezdina (2013) gave such a description in the case where one of the groups coincides with a sporadic group and an alternating group, respectively. The author  (2014) solved this question for pairs finite simple groups of Lie type over fields of the same characteristic. In the present paper we prove the following theorem.

 Theorem. Let $G$ be a finite simple group of exceptional Lie type over a field with $q$ elements, and let $G_1$ be a finite simple group of Lie type over a field with $q$ elements nonisomorphic to $G$, where $q$ and $q_1$ are coprime. If $GK(G)=GK(G_1)$, then one of the following statements holds:

$(1)$ $\{G,G_1\}=\{G_2(3),A_1(13)\}$;

$(2)$ $\{G,G_1\}=\{{^2}F_4(2)',A_3(3)\}$;

$(3)$ $\{G,G_1\}=\{{^3}D_4(q),A_2(q_1)\}$, where $(q_1-1)_3\neq 3$ and $q_1+1\neq 2^{k_1}$;

$(4)$ $\{G,G_1\}=\{{^3}D_4(q),A_4^{\pm}(q_1)\}$, where $(q_1\mp1)_5\neq 5$;

$(5)$ $\{G,G_1\}=\{G_2(q),G_2(q_1)\}$, where $q$ and $q_1$ are not powers of the number~3;

$(6)$ $\{G,G_1\}$ is one of the pairs $\{F_4(q),F_4(q_1)\}$, $\{{^3}D_4(q),{^3}D_4(q_1)\}$, and $\{E_8(q),E_8(q_1)\}$.

 The existence of pairs of groups in statements (3)-(6) is unknown.

Keywords: finite simple exceptional group of Lie type, spectrum, prime graph

Received April 3, 2020

Revised May 11, 2020

Accepted May 25, 2020

Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 20-01-00456) and  by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).

Marianna Rifkhatovna Zinov’eva, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Ekaterinburg, 620083 Russia, e-mail: zinovieva-mr@yandex.ru

REFERENCES

1.   Unsolved problems in group theory. The Kourovka notebook, ed. V.D. Mazurov, 18th ed., Novosibirsk: Inst. Mat. SO RAN Publ., 2014, 253 p.

2.   Hagie M. The prime graph of a sporadic simple group. Comm. Algebra, 2003, vol. 31, no. 9, pp. 4405–4424. doi: 10.1081/AGB-120022800 

3.   Zvezdina M. A. On nonabelian simple groups having the same prime graph as an alternating group. Sib. Math. J., 2013, vol. 54, no. 1, pp. 47–55. doi: 10.1134/S0037446613010072 

4.   Zinov’eva M.R. Finite simple groups of Lie type over a field of the same characteristic with the same prime graph. Tr. Inst. Mat. Mekh., 2014, vol. 20, no. 2, pp. 168–183 (in Russian).

5.   Zinov’eva M.R. On Finite simple classical groups over fields of different characteristics with coinciding prime graphs. Proc. Steklov Inst. Math., 2017, vol. 297, suppl. 1, pp. 223–239. doi: 10.1134/S0081543817050248 

6.   Kondrat’ev A.S. Prime graph components of finite simple groups. Math. USSR-Sb., 1990, vol. 67, no. 1, pp. 235–247. doi: 10.1070/SM1990v067n01ABEH001363 

7.   Williams J.S. Prime graph components of finite groups. J. Algebra, 1981, vol. 69, no. 2, pp. 487–513. doi: 10.1016/0021-8693(81)90218-0 

8.   Vasiliev A.V., Vdovin E.P. An Adjacency criterion for the prime graph of a finite simple group. Algebra and Logic, 2005, vol. 44, no. 6, pp. 381–406. doi: 10.1007/s10469-005-0037-5 

9.   Vasiliev A.V., Vdovin E.P. Cocliques of maximal size in the prime graph of a finite simple group. Algebra and Logic, 2011, vol. 50, no. 4, pp. 291–322. doi: 10.1007/s10469-011-9143-8 

10.   Zsigmondy K. Zur Theorie der Potenzreste. Monatsh. Math. Phys., 1892, vol. 3, no. 1, pp. 265–284. doi: 10.1007/BF01692444 

11.   Gerono G.C. Note sur la resolution en nombres entiers et positifs de l’equation $x^m=y^n+1$. Nouv. Ann. Math. (2), 1870, vol. 9, pp. 469–471.

12.   Bugeaud Y., Mihăilescu P. On the Nagell-Ljunggren equation $\displaystyle\frac{x^n-1}{x-1}=y^q$. Math. Scand., 2007, vol. 101, no. 2, pp. 177–183. doi: 10.7146/math.scand.a-15038 

13.   Zavarnitsine A.V. Recognition of the simple groups $L_3(q)$ by element orders. J. Group Theory, 2004, vol. 7, no. 1, pp. 81–97. doi: 10.1515/jgth.2003.044 

14.   Zavarnitsine A.V. Finite simple groups with narrow prime spectrum. Sib. Elektron. Mat. Izv., 2009, vol. 6, pp. 1–12.

Cite this article as: M.R. Zinov’eva. On finite simple groups of exceptional Lie type over fields of different characteristics with coinciding prime graphs, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2020, vol. 26, no. 2, pp. 147–160; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2021, Vol. 313, Suppl. 1, pp. S228–S240.