Maximal subgroups of finite soluble groups in general position

TL;DR

This paper explores the relationship between the maximum size of independent families of maximal subgroups and the maximum size of irredundant generating sequences in finite soluble groups, establishing conditions for equality and constructing examples with large differences.

Contribution

It proves that MaxDim(G) equals m(G) when the derived subgroup is nilpotent and constructs soluble groups with arbitrarily large differences between MaxDim(G) and m(G).

Findings

MaxDim(G)=m(G) if the derived subgroup is nilpotent.

MaxDim(G)-m(G) can be arbitrarily large for certain soluble groups.

Constructed examples with Fitting length 2 and large MaxDim(G).

Abstract

For a finite group $G$ we investigate the difference between the maximum size MaxDim$(G)$ of an "independent" family of maximal subgroups of $G$ and maximum size $m(G)$ of an irredundant sequence of generators of $G$. We prove that MaxDim$(G)=m(G)$ if the derived subgroup of $G$ is nilpotent. However MaxDim$(G)-m(G)$ can be arbitrarily large: for any odd prime $p,$ we construct a finite soluble group with Fitting length 2 satisfying $m(G)=3$ and MaxDim$(G)=p.$