The Class Number Of Same Element Order In Finite Group And The Structure Of Group

In the finite group,group order and element order have great influence to the structure of group.we can get the structure of group and the nature of group,even the classification of same groups(especially simple group) by research group order and element order.In the paper, we research the structure of some groups by two aspects.one is group order other is the class number of same order in finite group. Let G be a finite group, η(G) be the class number of same element order in finite group.How about the structure of group when η(G) to be minimum.This paper is divided into the following four sections to introduce the main content:In section one:we introduce some backgrounds of our research.In section two:we introduce some basic definition and lemmas.In section three:we introduce some important proposition about the class num-ber of same order in element.In section four:we discuss the structure of finite group when the class number of same order to be minimum.