| A set of the number of the elements of the same order in a finite group G is called the same-order type of G. The same-order type is widely researched, groups with the length of same-order type 2 and 3 is proved to be the nilpotent group and the solvable group respectively. This paper studies the classification of the groups of some given special same-order types and this paper has been divided into four chapters.The first chapter gives us some quantity problems and the research background of same-order type in group theory and introduces some basic concepts and results in group theory.In the second chapter, we have considered the classification of same-order types with length 4 and obtain the result that if the same-order of a group G is {1,pq,4p,9q} for p,q primes, if and only if G is isomorphic to the alternating group A5.In the third chapter we research on the classification of nonabelian simple group with same-order type of length 4 and such groups are isomorphic to the alternating group A5.In the last chapter we discuss the classification of groups with the same-order types of power of 2 and such groups are isomorphic to a subgroup of Z2m+1×Zr1×Zr2× …×Zri which m is an positive integer and ri=2eei +1 is an Fermat prime number and ei≤m for i=1,2…,t. |
