Generalized prime graphs and the structure of finite groups.

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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.