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

Author:

Gross Fletcher

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.

Publisher

American Mathematical Society (AMS)

Subject

Applied Mathematics,General Mathematics

Reference7 articles.

1. Solvability of groups of odd order;Feit, Walter;Pacific J. Math.,1963

2. On finite groups of exponent 𝑝^{𝑚}𝑞ⁿ;Gross, Fletcher;J. Algebra,1967

3. On the 𝑝-length of 𝑝-soluble groups and reduction theorems for Burnside’s problem;Hall, P.;Proc. London Math. Soc. (3),1956

4. Subnormale Untergruppen und 𝑝-Sylowgruppen;Huppert, Bertram;Acta Sci. Math. (Szeged),1961

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