Geodesic
Recognition of the Group $E_6(2)$ by Gruenberg-Kegel Graph
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 27 (2021) no. 4, pp. 263-268 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

The Gruenberg-Kegel graph (or the prime graph) of a finite group $G$ is a simple graph $\Gamma(G)$ whose vertices are the prime divisors of the order of $G$, and two distinct vertices $p$ and $q$ are adjacent in $\Gamma(G)$ if and only if $G$ contains an element of order $pq$. A finite group is called recognizable by Gruenberg-Kegel graph if it is uniquely determined up to isomorphism in the class of finite groups by its Gruenberg-Kegel graph. In this paper, we prove that the finite simple exceptional group of Lie type $E_6(2)$ is recognizable by its Gruenberg-Kegel graph.
Keywords: finite group; simple group; exceptional group of Lie type; Gruenberg-Kegel graph (prime graph). 
@article{TIMM_2021_27_4_a17,
     author = {W. Guo and A. S. Kondrat'ev and N. V. Maslova},
     title = {Recognition of the {Group} $E_6(2)$ by {Gruenberg-Kegel} {Graph}},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {263--268},
     year = {2021},
     volume = {27},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2021_27_4_a17/}
}
TY  - JOUR
AU  - W. Guo
AU  - A. S. Kondrat'ev
AU  - N. V. Maslova
TI  - Recognition of the Group $E_6(2)$ by Gruenberg-Kegel Graph
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2021
SP  - 263
EP  - 268
VL  - 27
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/TIMM_2021_27_4_a17/
LA  - en
ID  - TIMM_2021_27_4_a17
ER  - 
%0 Journal Article
%A W. Guo
%A A. S. Kondrat'ev
%A N. V. Maslova
%T Recognition of the Group $E_6(2)$ by Gruenberg-Kegel Graph
%J Trudy Instituta matematiki i mehaniki
%D 2021
%P 263-268
%V 27
%N 4
%U http://geodesic.mathdoc.fr/item/TIMM_2021_27_4_a17/
%G en
%F TIMM_2021_27_4_a17
W. Guo; A. S. Kondrat'ev; N. V. Maslova. Recognition of the Group $E_6(2)$ by Gruenberg-Kegel Graph. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 27 (2021) no. 4, pp. 263-268. http://geodesic.mathdoc.fr/item/TIMM_2021_27_4_a17/

[1] Conway J.H., Curtis R.T., Norton S.P., Parker R.A., Wilson R.A., Atlas of finite groups, Clarendon Press, Oxford, 1985, 252 pp. | Zbl

[2] Gerono G.C., “Note sur la resolution en nombres entiers et positifs del'equation $x^m=y^n+1$”, Nouv. Ann. Math.(2), 9 (1870), 469–471

[3] Gorenstein D., Finite groups, Harper and Row, New York, 1968, 642 pp. | Zbl

[4] Gorenstein D., Lyons R., Solomon R., The classication of the finite simple groups, Number 3. Part I, Math. Surveys Monogr., 40, no. 3, Amer. Math. Soc., Providence, RI, 1998, 420 pp.

[5] Iiyori N., Yamaki H., “Prime graph components of the simple groups of Lie type over the fields of even characteristic”, J. Algebra, 155:2 (1993), 335–343 ; corrigenda, J. Algebra, 181:2 (1996), 659 | DOI | Zbl | DOI | Zbl

[6] Jansen C., Lux K., Parker R., Wilson R., An atlas of Brauer characters, Clarendon Press, Oxford, 1995, 327 pp. | Zbl

[7] Kondrat'ev A.S., “Prime graph components of finite simple groups”, Math. USSR Sb., 67:1 (1990), 235–247 | DOI

[8] Kondrat'ev A.S., Mazurov V.D., “Recognition of alternating groups of prime degree from the orders of their elements”, Siberian. Math. J., 41:2 (2000), 294–302 | DOI

[9] Mazurov V.D., “The set of orders of elements in a finite group”, Algebra and Logic, 33:1 (1994), 49–55 | DOI | Zbl

[10] Vasil'ev A.V., Gorshkov I.B., “On recognition of finite simple groups with connected prime graph”, Siberian Math. J., 50:2 (2009), 233–238 | DOI | Zbl

[11] Vasil'ev A.V., Vdovin E.P., “An adjacency criterion for the prime graph of a finite simple group”, Algebra and Logic, 44:6 (2005), 381–406 | DOI | Zbl

[12] Vasil'ev A.V., Vdovin E.P., “Cocliques of maximal size in the prime graph of a finite simple group”, Algebra and Logic, 50:4 (2011), 291–322 | DOI | Zbl

[13] Williams J. S., “Prime graph components of finite groups”, J. Algebra, 69:2 (1981), 487–513 | DOI | Zbl

[14] Zavarnitsine A.V., “Finite groups with a five-component prime graph”, Siberian Math. J., 54:1 (2013), 40–46 | DOI | Zbl

[15] Zsigmondy K., “Zur Theorie der Potenzreste”, Monatsh. Math. Phys., 3:1 (1892), 265–284 | DOI | Zbl