  In this note, we give short proofs of the well-known results that the
exponent of the Schur multiplier $\M$ divides the exponent of $\G$ for finite
$\p$-groups of maximal class and potent $\p$-groups. Moreover, we prove the
same for a finite $\p$-group $\G$ satisfying $\G^{\p^2}\subset
\gamma_{\p}(\G)$, and for $3$-groups of class $5$. We do this by proving a
general lemma, and show that these three classes of groups satisfy the
hypothesis of our lemma.
