The shape of solvable groups with odd order

TL;DR

This paper establishes bounds on the minimal composition length of solvable groups with a given solvable length, focusing on groups of odd order and providing explicit exponential inequalities.

Contribution

It introduces new bounds on the minimal composition length for solvable groups with specified solvable length, especially for odd order groups.

Findings

Bounds on composition length for solvable groups with length d

Explicit inequalities for odd order groups

Improved understanding of group structure complexity

Abstract

We prove that the minimal composition length, $c$, of a solvable group with solvable length $d$ satisfies $9^{(d-3)/9}< c< 9^{(d+1)/5}$, and the minimal composition length, $c^o$, of a group with odd order and solvable length $d$ satisfies $7^{(d-2)/5}< c^o< 2^d$.