# zbMATH — the first resource for mathematics

Quasirecognition of a class of finite simple groups by the set of element orders. (Russian, English) Zbl 1035.20012
Sib. Mat. Zh. 44, No. 2, 241-255 (2003); translation in Sib. Math. J. 44, No. 2, 195-207 (2003).
Given a finite group $$G$$, denote by $$\omega(G)$$ the set of its element orders. A finite group $$G$$ is called recognizable by $$\omega(G)$$ if every finite group $$H$$ with $$\omega(H)=\omega(G)$$ is isomorphic to $$G$$. The authors investigate a weaker property. A finite group $$G$$ is called quasirecognizable by $$\omega(G)$$ if every finite group $$H$$ with $$\omega(H)=\omega(G)$$ has a section isomorphic to $$G$$.
The Gruenberg-Kegel graph $$GK(G)$$ of $$G$$ is the graph in which the vertices are the prime divisors of $$| G|$$ and two distinct vertices $$p$$ and $$q$$ are joined if and only if $$pq\in\omega(G)$$. The authors [Ukr. Mat. Zh. 54, No. 7, 998-1003 (2002; Zbl 1024.20014)] proved that if $$L$$ is a simple finite group for which the Gruenberg-Kegel graph has at least 4 connected components then $$L$$ is quasirecognizable by $$\omega(L)$$. In the article under review, they extend their result as follows: If $$L$$ is a simple finite group for which the Gruenberg-Kegel graph has at least 3 connected components then $$L$$ is quasirecognizable by $$\omega(L)$$ except for the alternating group $$A_6$$.

##### MSC:
 20D05 Finite simple groups and their classification 20D06 Simple groups: alternating groups and groups of Lie type 20D60 Arithmetic and combinatorial problems involving abstract finite groups 05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
Full Text: