Quasirecognition by prime graph of the simple group \(^2F_4(q)\). (English) Zbl 1181.20012
The prime graph \(\Gamma(G)\) of a finite group \(G\) is constructed as follows. Its vertex set consists of the set containing the prime divisors of the order of \(G\) and two distinct primes \(p\) and \(q\) are joined by an edge if and only if \(G\) contains an element of order \(pq\).
In this paper it is shown that \(\Gamma(G)=\Gamma(^2F_4(2^{2m+1}))\) implies that \(G\) has a unique non-Abelian composition factor isomorphic to \(^2F_4(2^{2m+1})\); hence \(m\geq 1\).
Moreover, if \(G\) is some finite group whose order is equal to that of \(^2F_4(2^{2m+1})\) and for which \(\Gamma(G)=\Gamma(^2 F_4(2^{2m+1}))\) also holds, then \(G\) must be isomorphic to \(^2F_4(2^{2m+1})\).
In this paper it is shown that \(\Gamma(G)=\Gamma(^2F_4(2^{2m+1}))\) implies that \(G\) has a unique non-Abelian composition factor isomorphic to \(^2F_4(2^{2m+1})\); hence \(m\geq 1\).
Moreover, if \(G\) is some finite group whose order is equal to that of \(^2F_4(2^{2m+1})\) and for which \(\Gamma(G)=\Gamma(^2 F_4(2^{2m+1}))\) also holds, then \(G\) must be isomorphic to \(^2F_4(2^{2m+1})\).
Reviewer: R. W. van der Waall (Huizen)
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.) |
