Characterization Of Finite Groups By Orders

The order of a group or an element of a group is the most fundamental concept in group theory. But it plays an important role in the quantitative structure of groups. In the past 25 years, a lot of results on this topic were obtained.Letπe(G) denote the set of all orders of elements in group G. W.J. Shi put forward the following conjecture(cf.[1]):Shi's Conjecture Let G be a group and H a finite simple group. Then G= H if and only if (a)πe(G)=πe(H), and (b)|G|=|H|.This conjecture has been proved recently(cf.[1]-[11]). A problem related to this conjecture is the following conjecture made by J.G. Thompson. For each finite group G and each integer d> 1, Let G(d)={x∈G|xd=1}.Definition G1 and G2 are of the same order type iff|G1(d)|=| G2(d)|, d=1,2,Thompson's Conjecture Suppose G1, G2 are finite groups of the same order type. If G1 is solvable, then G2 is also solvable.Because the above Shi's Conjecture is correct, we can get the following proposi-tion:Proposition Suppose that G1, G2 are finite groups of the same order type. Suppose that G1 is solvable, then G2 is not a nonabelian simple group.As all the simple groups can be characterized by the set of their element orders and group orders, it is a meaning topic to find out the nonsimple groups those can be characterized by the set of their element orders and group orders. In [12], it is proved that all the symmetry groups can be characterized by the set of their element orders and group orders. In this paper, we continue this topic, and prove that some groups with certain group orders and all automorphism groups of the sporadic simple groups which can also be characterized by the set of their element orders and group orders. The paper consists of the five following sections:In Section 1, we introduce some backgrounds of our research.In Section 2, we introduce some symbols, basic concepts used in the paper.In Section 3, we study the groups with order 2qp by the set of their element orders and group orders.In Section 4, we study the groups with order 23p by the set of their element orders and group orders.In Section 5, we study all automorphism groups of the sporadic simple groups by the set of their element orders and group orders.The author prove that all groups with order 2gp, six series groups with order 23p and all automorphism groups of the sporadic simple groups can be characterized by the set of their element orders and group orders.