  In \cite[Theorem 2.5]{Bac16} Bachiller proved that if $(G, \cdot, \circ)$ is
a brace of order the power of a prime $p$ and the rank of $(G,\cdot)$ is
smaller than $p-1$, then the order of any element is the same in the additive
and multiplicative group. This means that in this case the isomorphism type of
$(G,\circ)$ determines the isomorphism type of $(G,\cdot)$. In this paper we
complement Bachiller's result in two directions. In Theorem 2.2 we prove that
if $(G, \cdot, \circ)$ is a brace of order the power of a prime $p$, then
$(G,\cdot)$ has small rank (i.e. $< p-1$) if and only if $(G,\circ)$ has small
rank. We also provide examples of groups of rank $p-1$ in which elements of
arbitrarily large order in the additive group become of prime order in the
multiplicative group. When the rank is larger, orders may increase.
