Generalized prime graphs and the structure of finite groups.

设P是群性质，G是有限群. G的与P相关联的广义素图Γ，或简称G的P-图，是顶点集为π（G）的图。Γ的两个顶点p，q被一条边连接当且仅当存在G的一个阶可被pq整除的P-子群。P-图的结构与G的P-子群格L（G，P）的结构有着深刻的联系。本文从P-图的角度研究了P-子群格（群）的结构。定义了一个新的代数UD（Q），并研究了UD（L（G，P））.我们还发现了G的结构与UD（L（G，P））的结构之间的一些关系，并完全确定了对某些群的UD（Q）的结构.

Let P be a group property and let G be a finite group. The generalized prime graph Γ of G associated with P, or simply theP-graph of G, is a graph whose vertex set is π(G). Two vertices p,q of Γare joined by an edge if and only if there exists a P-subgroup of G whose order is divisible by pq. There are deep relations between the structure of P-graphs and the one of P-subgroup lattices L(G,P) of G. In our research we study the structure of P-subgroup lattice (or group) from point of view of P-graphs. We define a new algebra UD(Q) for a quiver Q and we study UD(L(G,P)). We also find some relations between the structure of G and the one of UD(L(G,P)) and determine the structure of UD(Q) for some quiver Q completely.