On the analog category of finite groups

TL;DR

This paper investigates the analog category of finite groups, revealing its proportionality to the largest Sylow subgroup size and demonstrating that the universal upper bound is not tight.

Contribution

It introduces a new perspective on the analog category of finite groups and shows the upper bound based on group order is far from optimal.

Findings

Analog category size is proportional to the largest Sylow subgroup.

Universal upper bound by group order is not tight.

Provides insights into the structure of finite groups.

Abstract

We show that the analog category of a finite group is essentially proportional to the size of its largest Sylow subgroup. We conclude that the universal upper bound given by the order of the group is very far from optimal.