阶对有限群的刻画

Characterization of Finite Groups by Order

令πe(G)表示G中元的阶之集。对于所有有限单群,已证明其均可由元阶集及群阶进行刻画。即设G为群,H为有限单群,则当GH且仅当(1)πe(G)=πe(H);(2)G=H。本文继续这一研究,对两类有限非单群进行讨论。首先在不使用2qp阶群的分类的前提下证明了所有阶为2qp(q<p为不同的奇素数)的群可仅用元阶集和群阶加以刻画,然后利用23p阶群的分类证明了有6类23p(p为奇素数)阶群也可由元阶集和群阶唯一确定。

The concepts of the order of a group and its element orders are the most fundamental in group theory.They play an important role in the quantitative structure of groups.It is interesting to find out which groups those can be characterized by their element orders and group orders.Let πe（G） denote the set of all orders of elements in group G.It has been proved recently that all the simple groups can be characterized by the set of element orders and the order of group.Let G be a group and H a finite simple group.Then G if and only if 1） πe（G） = πe（H） ,and 2） G = H .In this paper,we continue the discussion of two series finite nonsimple groups.We proved that G can be characterized by πe（G） and G without using their constructions,where G are groups with order 2qp,q pare odd prime numbers.Then we proved that G can be characterized by πe（G） and G by using their constructions,where G are six series groups with order23p ,p is an odd prime number.