Recognition Of Some Finite Simple Groups D_n（3） And S₄（q）

In this paper, we study the recognition problems of somefinite simple groupssuch as Dn (3),where n5is an odd integer or n p1for a prime p3, andS4(q),where3q50is a prime power.In particular, we discuss OD-characterization ofS4(q)and prime graph quasirecognition of some Dn (3).At first, we make a review about recognitions of some finite simple groups, such asprime graph recognition, prime graph quasirecognition and OD-characterization inrecent years. The researchs about these articles and the relationship of recognitioninspire the idear of the author.The main results and innovations are as follows:1. Through contrasting prime graph, OD-characterization, non-commuting graphrecognition and double order recognition, we find that there is a relation betweenthem.Futhermore, we prove that Dn (3), where n9is an odd integer,isquasirecognizable by its prime graph，is also recognizable by its non-commuting graphand double order.2. We give some upper boundaries and lower boundaries for t (r, S)in order toprove that Dn (3), where n9is an odd integer, is quasirecognizable by its prime graph.Thereby, AAM’s conjecture is true for Dn (3), where n5is an odd integer or n p1for a prime p3.Thus Dn (3)is an example of an infinite series of finite simple groupsrecognizable by their non-commuting graphs, whose prime graphs are connected forsome n.3. Through analyzingthe connectivity of prime graph ofS4(q), we adopt theunified method to prove thatS4(q), where3q50is a prime power, is OD-characterizable. The innovation is using unified method to prove that a series of finitesimple groups are OD-characterizable.