The. Groups Order And The Order Of The Highest Order Elements With The Group Structure

It is a well-known topic to characterize a finite simple group by using two quantities, the order of G and πe(G) in the past30years, where πe(G) denotes the set of orders of elements in G. Professor W. J. Shi, the famous expert who put forward this topic, has done much work for it and proved that almost all simple groups can be uniquely determined by such two quantities. His study was spoken highly of by professor J.G.Thompson, an Fields medalist. In1992, this topic was listed in the book Unsolved Problems in Group Theory as a famous conjecture (the problem12.39). And in2009, this topic has been finished by V. D. Mazurov, et al.One can see that there should be enough quantities in quantitative characteri-zation of finite simple groups, and of course the less the number of quantities is, the better. We now will try to characterize some finite groups by using less quantities. In this paper, we first characterize some finite simple groups and symmetric groups, by using the order of a group and the largest element order.(For some special cases, we use the second largest element order). And then we use only three characters of the element with largest order (the order, the number and the centernizer of it) to characterize some simple groups. The paper consists of4chapters and the main results are in chapters3and4.In chapter3, we study the relationship between the structure of a finite group and its order and largest element order. And it is proved that sporadic simple groups, some simple groups of Lie Type, some alternating groups, some symmetric groups, simple K3-groups, and most simple K4-groups can be determined by their order and the largest element order.In chapter4, we study the relationship between the structure of a finite group and the three characters of the element with largest order (the order, the num-ber and the centernizer of it). And characterize simple K3-groups, sporadic simple groups, some simple groups of Lie Type, some simple K4-groups, and some alter-nating groups by using the three characters.