A New Characterization of Certain Symmetric and Alternating Groups

Abstract

The degree pattern of a finite group G has been introduced in[Algebra Colloquium,2005,12(3):431-442]and denoted by D(G).The group G is called k-fold OD-characterizable if there exist exactly k non-isomorphic groups H satisfying conditions |H|=|G|and D(H)=D(G).In addition,a 1-fold OD-characterizable group is simply called OD-characterizable.The following simple groups are uniquely determined by their orders and degree patterns:all sporadic simple groups,the alternating groups Ap(p≥5 is a twin prime) and some simple groups of Lie type.In this problem,those groups with connected prime graphs are somewhat much difficult to be solved.In this paper,we continue this investigation.In particular,we show that the symmetric groups S_(81) and S_(82) are 3-fold OD-characterizable.We also show that the alternating groups A_(130) and A_(140) are OD-characterizable.It is worth mentioning that the latter gives a positive answer to a conjecture in[Frontiers of Mathematics in China,2009,4(4):669-680].