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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 $G$ ofequal order are conjugate. The structure of such a group $G$ has been settledin this article.

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