关于一些交错单群的新刻画

A New Characterization of Some Alternating Groups

设πe(G)表示群G中元素阶的集合,k1(G),k2(G)分别表示G中最高阶元素的阶和次高阶元素的阶。V.D.Mazurov等人2009年证明了用元素阶集合πe(G)和群的阶G刻画有限单群。本文试图用更少的数量刻画交错单群,并证明了:1)设G为有限群,M为交错单群An(n=5,6,7,9,10,11,13),则G≌M当且仅当|G|=|M|,且k1(G)=k1(M);2)设G为有限群,M为交错单群An(n=8,12),则G≌M当且仅当|G|=|M|,且ki(G)=ki(M),i=1,2。

Let G be a finite group, πe（G） denote the set of orders of elements in G, k1 （G） denote the largest element order of G, and k2 （G）, the second largest element order. 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 past 30 years. In 2009, this topic has been finished by V. D. Mazurov, et al. This paper will try to characterize some alternating groups by using less quantities and proves that： 1） Let G be a finite group, M he alternating group A, （n= 5, 6, 7, 9, 10, 11, 13）. Then G=M if and only if [ G ] = [ M ] , and k1 （G） =k1（M） ; 2） Let G be a finite group, M be alternating group A,（n=8, 12）. Then G=Mif and only if [ G ] = [ M ] , and ki（G） =i（M）, where i=1, 2.