元的阶之集与群的性质

The Set of Element Orders and the Property of Group

设G是一个有限群,记ω(G)为G的每个元素的阶的集合,称为谱.记τ为自然数集N的子集,满足1∈τ,且若m∈τ,则m的正因子s∈τ,称τ为合理子集.得到了当ω(G)的任意一个合理子集τ都满足h(τ)≥1时群G的性质,其中h(τ)为满足ω(H)=τ的群H的同构类的个数.

Let Gbe a finite group,and denote byω（G）the spectrum of G,i.e.,the set of its element orders.Denote byτthe reasonable subset of the set of natural numbers Nthat satisfies 1∈τand if m∈τ, then s∈τ,where s is a positive factor of m.We get the property of group Gwhen every reasonable subset τω（G）that satisfies h（τ）≥1,where h（τ）stands for the number of isomorphism classes of finite group H with ω（H）=τ.