Quasirecognition by prime graph of the simple group \(^2F_4(q)\). (Q1046783)

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 \(^2F_4(2^{2m+1})\).