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-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.