On small Sylow numbers of finite groups

TL;DR

This paper investigates the conditions under which the number of Sylow p-subgroups in a finite group is a prime power, showing that if this number is less than p^2, then almost all such numbers are prime powers.

Contribution

It establishes a new criterion linking small Sylow numbers to their being prime powers in finite groups.

Findings

If n_p(G) < p^2, then n_p(G) is almost always a prime power.

The paper provides conditions under which Sylow subgroup counts are prime powers.

Results contribute to understanding the structure of finite groups through Sylow subgroup counts.

Abstract

Let $G$ be a finite group and $n_p(G)$ the number of Sylow $p$-subgroups of $G$. In this paper, we prove if $n_p(G)<p^2$ then almost all numbers $n_p(G)$ are a power of a prime.