Minimal non-Iwasawa finite groups

TL;DR

This paper characterizes minimal non-Iwasawa finite groups, explores their modular cases, and investigates their subgroup commutativity degrees, including a family of non-nilpotent examples with degrees approaching one.

Contribution

It provides a detailed structure of minimal non-Iwasawa finite groups and links them to subgroup commutativity degrees, including new examples with specific properties.

Findings

Classification of minimal non-Iwasawa finite groups

Identification of modular minimal non-Iwasawa groups

Construction of a family with subgroup commutativity degree approaching 1

Abstract

In this note, we describe first the structure of minimal non-Iwasawa finite groups. Then we determine the minimal non-Iwasawa finite groups which are modular. Also, we find connections between minimal non-Iwasawa finite groups and the subgroup commutativity degree, and we give an example of a family of non-nilpotent modular finite groups $G_n$, $n\in\mathbb{N}$, whose subgroup commutativity degree tends to $1$ as $n$ tends to infinity.