The classification of the finite groups whose abelian subgroups of equal prime power order are conjugate

TL;DR

This paper characterizes finite groups where all abelian p-subgroups of the same order are conjugate, providing a complete structural classification under this condition.

Contribution

It offers a comprehensive classification of finite groups with conjugate abelian p-subgroups of equal order, resolving a structural question in group theory.

Findings

Finite groups with conjugate abelian p-subgroups are fully classified.

The structure of such groups is explicitly described.

The classification applies to all primes dividing the group order.

Abstract

Let $G$ be a finite group and assume $p$ is a prime dividing the order of $G$. Suppose for any such $p$, that every two abelian $p$-subgroups of $G$ of equal order are conjugate. The structure of such a group $G$ has been settled in this article.