On the p—Local Rank of Finite Groups

In this paper,we shall mainly study the p-solvable finite group in terms of p-local rank,and a group theoretic characterization will be given of finite p-solvabel groups with p-local rank two.Theorem A Let G be a finite p-solvable group with p-local rank plr(G)=2 and Op(G)=1.If P is a Sylow p-subgrounp of G,then P has a normal subgroup Q such that P/Q is cyclic or a generalized quaternion 2-group and the p-rank of Q is at most two.Theorem B Let G be a finite p-solvable group with Op(G)=1.Then the p-length lp(G)≤plr(G);if in addition plr(G)=lp (G) and p≥5 is odd,then plr(G)=0 or 1.

On the <Emphasis Type="Italic">p</Emphasis>-Local Rank of Finite Groups

In this paper, we shall mainly study the p-solvable finite group in terms of p-local rank, and a group theoretic characterization will be given of finite p-solvable groups with p-local rank two. Theorem A Let G be a finite p-solvable group with p-local rank plr(G) = 2 and O _p (G) = 1. If P is a Sylow p-subgroup of G, then P has a normal subgroup Q such that P/Q is cyclic or a generalized quaternion 2-group and the p-rank of Q is at most two. Theorem B Let G be a finite p-solvable group with O _p (G) = 1. Then the p-length l _p (G) < plr(G); if in addition plr(G) = 1 _p (G) and p > 5 is odd, then plr(G) = 0 or 1. Supported by Cheung Kong Scholars Program, National 973 Project and RFDP