Quasirecognition by prime graph of the groups \(^2 D_{2n}(q)\) where \(q<10^5\) (Q1649136)

Summary: Let \(G\) be a finite group. The prime graph \(\Gamma (G)\) of \(G\) is defined as follows: The set of vertices of \(\Gamma (G)\) is the set of prime divisors of \(| G|\) and two distinct vertices \(p\) and \(p'\) are connected in \(\Gamma (G)\), whenever \(G\) contains an element of order \(pp'\). A non-abelian simple group \(P\) is called recognizable by prime graph if for any finite group \(G\) with \(\Gamma (G)=\Gamma (P)\), \(G\) has a composition factor isomorphic to \(P\). It is been proved that finite simple groups \(^2 D_n(q)\), where \(n \neq 4 k\), are quasirecognizable by prime graph. Now in this paper we discuss the quasirecognizability by prime graph of the simple groups \(^2 D_{2 k}(q)\), where \(k \geq 9\) and \(q\) is a prime power less than \(10^5\).