Recognition Of Finite Simple Groups By Spectrum

It is a classic and important subject to study the structure of finite groups by spectrum (i.e., the set of element orders), such as Cauchy theorem and Gruenberg-Kegel theorem. Many scholars have studied the relationship about the spectrum and the structure of finite groups, and given many important results. Shi Wujie proved that Alt₅ is recognizable, and this result opened a wide way for investigations of recognizability of groups. In this paper, we characterize a series of simple groups by spectrum. It consists of the following four chapters:In chapter 1, we introduce some symbols and basic concepts that we usually use in the paper.In chapter 2, we study the adjacency criterions for prime graph of simple groups which are obtained by A.V.Vasiliev and E.P.Vdovin, and point out some incorrect and misleading adjacency criterions, then remedy them. And we also obtain an important corollary.In chapter 3, we study the conjecture of A.S.Kondratiev on the recognizability of simple groups with disconnected prime graph. And we recognize a series of simple groups of Lie type, D_n(q) and C_n(q), over a field of characteristic p.In chapter 4, we study the recognizability of some simple groups with connected prime graph, which completely answers the question provided by V.D.Mazurov. And by spectrum and the property of simple group, we characterize all alternating simple groups. When solving the recognizablity of simple groups, we also find an interesting conclusion that the smallest nonabelian simple group divided by prime p is A₁(p) or Alt₅.