Characterization Of A Finite Group By Its Abelian Subgroups

This paper discusses two problems.The first is to discuss the structure of the finite group G whose set of the number of Abelian subgroups of the same order is{1,3}.The second is to characterize alternating groups An and symmetric groups Sn(n=3,4,5)by their order,orders and the number of their nontrivial Abelian subgroups.The whole dissertation is divided into four chapters.In the first chapter,we introduce the research background.In the second chapter,we introduce some basic concepts and common conclu-sions.In the third chapter,we study the structure of finite groups by the number of Abelian subgroups.It proves that there is no finite group G satisfying that the set of the number of Abelian subgroups of the same order is {1,2}.It is determined the structure of the finite group G whose set of the number of Abelian subgroups of the same order is {1,3}.As a consequence,it is determined the finite groups that the set of the number of subgroups of the same order of G is {1,3}.In the fourth chapter,we use elementary approaches to characterize alternating groups An and symmetric groups Sn.Without using concepts and conclusions of the prime graph and the simple group classification.We characterize the alternating group A5 and the symmetric group S5 by their order,orders and the number of their nontrivial Abelian subgroups.