# Classifying uniformly generated groups

## TL;DR

This paper classifies finite uniformly generated groups, which are characterized by minimal generating chains, revealing their connection to finite projective geometries without relying on simple group classification.

## Contribution

It provides a classification of uniformly generated groups independently of simple group classification, linking them to finite projective geometries.

## Key findings

- Classified all uniformly generated groups.
- Connected uniformly generated groups to finite projective geometries.
- Avoided using simple group classification in the proof.

## Abstract

A finite group $G$ is called *uniformly generated*, if whenever there is a (strictly ascending) chain of subgroups $1<\langle x_1\rangle<\langle x_1,x_2\rangle <\cdots<\langle x_1,x_2,\dots,x_d\rangle=G$, then $d$ is the minimal number of generators of $G$. Our main result classifies the uniformly generated groups without using the simple group classification. These groups are related to finite projective geometries by a result of Iwasawa on subgroup lattices.