TY - JOUR

T1 - On characterization by Gruenberg–Kegel graph of finite simple exceptional groups of Lie type

AU - Maslova, Natalia V.

AU - Panshin, Viktor V.

AU - Staroletov, Alexey M.

N1 - The first author is supported by the Ministry of Science and Higher Education of the Russian Federation, project 075-02-2023-935 for the development of the regional scientific and educational mathematical center “Ural Mathematical Center” (Sect. ). The second author is supported by the Mathematical Center in Akademgorodok under the agreement No. 075-15-2022-281 with the Ministry of Science and Higher Education of the Russian Federation (Sect. ). The third author is supported by RAS Fundamental Research Program, project FWNF-2022-0002 (Sect. ).

PY - 2023/9

Y1 - 2023/9

N2 - The Gruenberg–Kegel graph (or the prime graph) Γ (G) of a finite group G is the graph whose vertex set is the set of prime divisors of |G| and in which two distinct vertices r and s are adjacent if and only if there exists an element of order rs in G. A finite group G is called almost recognizable (by Gruenberg–Kegel graph) if there is only a finite number of pairwise non-isomorphic finite groups having Gruenberg–Kegel graph as G. If G is not almost recognizable, then it is called unrecognizable (by Gruenberg–Kegel graph). Recently Peter J. Cameron and the first author have proved that if a finite group is almost recognizable, then the group is almost simple. Thus, the question of which almost simple groups (in particular, finite simple groups) are almost recognizable is of prime interest. We prove that every finite simple exceptional group of Lie type, which is isomorphic to neither [InlineEquation not available: see fulltext.] with n⩾ 1 nor G2(3) and whose Gruenberg–Kegel graph has at least three connected components, is almost recognizable. Moreover, groups [InlineEquation not available: see fulltext.], where n⩾ 1 , and G2(3) are unrecognizable.

AB - The Gruenberg–Kegel graph (or the prime graph) Γ (G) of a finite group G is the graph whose vertex set is the set of prime divisors of |G| and in which two distinct vertices r and s are adjacent if and only if there exists an element of order rs in G. A finite group G is called almost recognizable (by Gruenberg–Kegel graph) if there is only a finite number of pairwise non-isomorphic finite groups having Gruenberg–Kegel graph as G. If G is not almost recognizable, then it is called unrecognizable (by Gruenberg–Kegel graph). Recently Peter J. Cameron and the first author have proved that if a finite group is almost recognizable, then the group is almost simple. Thus, the question of which almost simple groups (in particular, finite simple groups) are almost recognizable is of prime interest. We prove that every finite simple exceptional group of Lie type, which is isomorphic to neither [InlineEquation not available: see fulltext.] with n⩾ 1 nor G2(3) and whose Gruenberg–Kegel graph has at least three connected components, is almost recognizable. Moreover, groups [InlineEquation not available: see fulltext.], where n⩾ 1 , and G2(3) are unrecognizable.

KW - Exceptional group of Lie type

KW - Finite group

KW - Gruenberg–Kegel graph (prime graph)

KW - Recognition by Gruenberg–Kegel graph

KW - Simple group

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85168680047&origin=inward&txGid=b9029e4f43c0970123aeab60618fa9c5

UR - https://www.mendeley.com/catalogue/55b5fed8-7bde-3848-820e-9c62a9db3b49/

U2 - 10.1007/s40879-023-00672-7

DO - 10.1007/s40879-023-00672-7

M3 - Article

VL - 9

JO - European Journal of Mathematics

JF - European Journal of Mathematics

SN - 2199-675X

IS - 3

M1 - 78

ER -