The Influence Of The Number Related To The Highest Order Elements On The Group Structure

As we all know,the quantitative characterization of groups has always been an important topic in the field of group theory.The key of quantitative characterization is the number of quantities used.For a long time,people have always expected to use fewer quantities to characterize the property and structure of groups.In the 1980s,J.G.Thompson,the Fields winner,proposed the conjecture that“groups whose numbers of elements of each order are equal have the same solvability”.The conjecture has not been completely solved up to now,but people have found some special conditions for the conjecture from the number of elements with the largest order.At the same time,Professor Shi Wujie proposed the conjecture that“finite simple groups can be characterized by the group order and set of element orders".The conjecture was completely solved in 2009,but how to weaken the condition of conjecture has become a new topic.Since 2012,the group order and the largest element order have been used to characterize a series of finite groups,and weaken the condition of Shi Wujie's conjecture locally.These works show that element with largest order play a special role in characterizing the properties and structures of groups.In this paper,the influence of the numbers related to the element with largest order on the group structure is discussed.Firstly,the influence of group order and the largest element order on group structure is studied.And the automorphism group of simpleK₄-groups is characterized by group order and largest element order.Secondly,the largest element order,the number of the elements with largest order and the set of order of centralizer of elements with the largestest order,i.e.the 1st ONC-degree of a finite group is studied to clarify the influence of the structure of a finite group,and the automorphism group of simpleK₃-groups and simpleK₄-groups are characterized by the 1st ONC-degree.