An affirmative answer to a conjecture related to the solvability of groups

TL;DR

This paper proves that finite groups with at most p^2 Sylow p-subgroups for each odd prime p are always solvable, confirming a conjecture and advancing understanding of group structure.

Contribution

It provides a positive resolution to a conjecture about the solvability of finite groups based on Sylow p-subgroup counts.

Findings

Finite groups with ≤ p^2 Sylow p-subgroups for odd p are solvable.

Confirms the conjecture posed in prior work.

Enhances understanding of the relationship between Sylow subgroups and group solvability.

Abstract

In this paper, we show that each finite group $G$ containing at most $p^2$ Sylow $p$-subgroups for each odd prime number $p$, is a solvable group. In fact, we give a positive answer to the conjecture in \cite{Rob}.