from the construction in Section 3.

In the construction of simple left braces in Section 3 we use, for each pair of distinct primes $p$ and $q$, a ${\mathbb{Z}}/(p)$-vector space $V$ with a non-singular symmetric bilinear form $b$ and an element $f\in \mathrm{O}(V,b)$ of order $q$ such that $f-{\operatorname{id}}$ is bijective. By Remark \[dimension\], in order to discover sharp lower bounds for the numbers $l_z$ used in the main result Theorem \[MainResult\], a natural question is to determine the minimal dimension of such a vector space $V$.

Note that for $q=2$, we can take $V={\mathbb{Z}}/(p)$ and the symmetric bilinear form $b\colon V\times V\longrightarrow {\mathbb{Z}}/(p)$ defined by $b(a,b)=ab$, for all $a,b\in{\mathbb{Z}}/(p)$. In this case, the element $f\in
\mathrm{O}(V,b)$ defined by $f(a)=-a$, for all $a\in {\mathbb{Z}}/(p)$, has order $2$ and $f-{\operatorname{id}}$ is bijective. Thus the minimal dimension, in this case, is $1$.

Another motivation for this study is the following problem stated in [@BCJO Problem 5.2]: for which prime numbers $p,q$ and positive integers $n,m$ there exists a simple left brace of order $p^nq^m$?

In the next remark we explain that there are some obvious restrictions on $n$ and $m$.

If $B$ is a simple left brace of order $p_1^{m_1}p_2^{m_2}$, where $p_1,p_2$ 