On the ranks of the additive and the multiplicative groups of a brace

Abstract

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.