𝑝-solvable groups with few automorphism classes of subgroups of order 𝑝

Abstract

This paper investigates the relationship between the p-length, l p ( G ) {l_p}(G) , of the finite p-solvable group G and the number, a p ( G ) {a_p}(G) , of orbits in which the subgroups of order p are permuted by the automorphism group of G. If p > 2 p > 2 and a p ( G ) ≦ 2 {a_p}(G) \leqq 2 , it is shown that l p ( G ) ≦ a p ( G ) {l_p}(G) \leqq {a_p}(G) . If p = 2 p = 2 and a 2 ( G ) = 1 {a_2}(G) = 1 , it is proved that either l p ( G ) ≦ a p ( G ) {l_p}(G) \leqq {a_p}(G) or G / O 2 ′ ( G ) G/{O_{2’}}(G) is a specific group of order 48.