More on a question of M. Newman on isomorphic subgroups of solvable groups

TL;DR

This paper investigates conditions under which isomorphic subgroups of finite solvable groups share the property of maximality, providing new affirmative results and structural restrictions on potential counterexamples.

Contribution

It extends previous work by showing the affirmative answer holds except in specific prime power index cases and details structural constraints on minimal counterexamples.

Findings

Affirmative answer for most cases of the question.

Restrictions on the structure of minimal counterexamples.

Identification of prime index cases where the answer may fail.

Abstract

M.Newman has asked if it is the case that whenever H and K are isomorphic subgroups of a finite solvable group G with H maximal, then K is also maximal. This question was considered in a paper of I.M. Isaacs and the second author, where (among other things) the answer was shown to be affirmative if H has an Abelian Sylow 2-subgroup. Here, we show that the answer is affirmative unless the index of H is a power of a prime less than 5 and we obtain further restrictions on the structure of a purported minimal counterexample.