摘要
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 Op(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 Op(G) = 1. Then the p-length lp(G) < plr(G); if in addition plr(G) = 1p(G) and p > 5 is odd, then plr(G) = 0 or 1.
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 Op(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 Op(G) = 1. Then the p-length lp(G) < plr(G); if in addition plr(G) = 1p(G) and p > 5 is odd, then plr(G) = 0 or 1.
