Proceedings of the Steklov Institute of Mathematics | 2021
On Finite Simple Groups of Exceptional Lie Type over Fields of Different Characteristics with Coinciding Prime Graphs
Abstract
Suppose that\xa0 $$G$$ πΊ is a finite group, $$\\pi(G)$$ π πΊ is the set of prime divisors of its order, and $$\\omega(G)$$ π πΊ is the set of orders of its elements. A graph with the following adjacency relation is defined on\xa0 $$\\pi(G)$$ π πΊ : different vertices\xa0 $$r$$ π and\xa0 $$s$$ π from\xa0 $$\\pi(G)$$ π πΊ are adjacent if and only if\xa0 $$rs\\in\\omega(G)$$ π π π πΊ . This graph is called the GruenbergβKegel graph or the prime graph of\xa0 $$G$$ πΊ and is denoted by\xa0 $$GK(G)$$ πΊ πΎ πΊ . In A.V.\u2009Vasilβevβs Question\xa016.26 from The Kourovka Notebook , it is required to describe all pairs of nonisomorphic finite simple nonabelian groups with identical GruenbergβKegel graphs. M.\xa0Hagie (2003) and M.A.\xa0Zvezdina (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 of finite simple groups of Lie type over fields of the same characteristic. In the present paper, we prove the following theorem.\xa0\xa0 Theorem. \xa0Let\xa0 $$G$$ πΊ be a finite simple group of exceptional Lie type over a field with\xa0 $$q$$ π elements, and let\xa0 $$G_{1}$$ subscript πΊ 1 be a finite simple group of Lie type over a field with\xa0 $$q$$ π elements nonisomorphic to\xa0 $$G$$ πΊ , where\xa0 $$q$$ π and\xa0 $$q_{1}$$ subscript π 1 are coprime. If $$GK(G)=GK(G_{1})$$ πΊ πΎ πΊ πΊ πΎ subscript πΊ 1 , then one of the following holds: $$(1)\\ \\{G,G_{1}\\}=\\{G_{2}(3),A_{1}(13)\\}$$ 1 πΊ subscript πΊ 1 subscript πΊ 2 3 subscript π΄ 1 13 ;\xa0\xa0 $$(2)\\ \\{G,G_{1}\\}=\\{{{}^{2}}F_{4}(2)^{\\prime},A_{3}(3)\\}$$ 2 πΊ subscript πΊ 1 superscript subscript πΉ 4 2 superscript 2 β² subscript π΄ 3 3 ;\xa0\xa0 $$(3)\\ \\{G,G_{1}\\}=\\{{{}^{3}}D_{4}(q),A_{2}(q_{1})\\}$$ 3 πΊ subscript πΊ 1 superscript subscript π· 4 3 π subscript π΄ 2 subscript π 1 , where $$(q_{1}-1)_{3}\\neq 3$$ subscript subscript π 1 1 3 3 and $$q_{1}+1\\neq 2^{k_{1}}$$ subscript π 1 1 superscript 2 subscript π 1 ;\xa0\xa0 $$(4)\\ \\{G,G_{1}\\}=\\{{{}^{3}}D_{4}(q),A_{4}^{\\pm}(q_{1})\\}$$ 4 πΊ subscript πΊ 1 superscript subscript π· 4 3 π superscript subscript π΄ 4 plus-or-minus subscript π 1 , where $$(q_{1}\\mp 1)_{5}\\neq 5$$ subscript minus-or-plus subscript π 1 1 5 5 ;\xa0\xa0 $$(5)\\ \\{G,G_{1}\\}=\\{G_{2}(q),G_{2}(q_{1})\\}$$ 5 πΊ subscript πΊ 1 subscript πΊ 2 π subscript πΊ 2 subscript π 1 , where\xa0 $$q$$ π and\xa0 $$q_{1}$$ subscript π 1 are not powers of\xa03;\xa0\xa0 $$(6)\\ \\{G,G_{1}\\}$$ 6 πΊ subscript πΊ 1 is one of the pairs $$\\{F_{4}(q),F_{4}(q_{1})\\}$$ subscript πΉ 4 π subscript πΉ 4 subscript π 1 , $$\\{{{}^{3}}D_{4}(q),{{}^{3}}D_{4}(q_{1})\\}$$ superscript subscript π· 4 3 π superscript subscript π· 4 3 subscript π 1 , and $$\\{E_{8}(q),E_{8}(q_{1})\\}$$ subscript πΈ 8 π subscript πΈ 8 subscript π 1 .\xa0\xa0The existence of pairs of groups in statements (3)β(6) is unknown.