  Let $G$ be a finite $p$-group of order $p^{n}$ with
$|M(G)|=p^{\frac{n(n-1)}{2}-t},$ where $M(G)$ is the Schur multiplier of $G$.
Ya.G. Berkovich, X. Zhou, and G. Ellis have determined the structure of $G$
when $t=0,1,2,3$. In this paper, we are going to find some structures for an
abelian $p$-group $G$ with conditions on the exponents of $G, M(G),$ and
$S_2M(G)$, where $S_2M(G)$ is the metabelian multiplier of $G$.
