the orthogonal groups over finite fields of characteristic relatively prime to $p$. The existence and limitations concerning such elements are discussed in Section 4. A problem on prime braces, recently proposed in [@KSV], Question 4.3, is solved in Section 5. Finally, in Section 6, we construct another new family of finite simple left braces using the technique of Section 3.

Finite left braces with abelian multiplicative Sylow subgroups {#agroup}
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A classical result of P. Hall states that every finite solvable group $G$ has a Sylow system, that is a family of Sylow subgroups $\mathcal{S}=\{ S_1,\dots ,S_n\}$ of $G$, one for each prime dividing the order of $G$, such that $S_iS_j=S_jS_i$ for all $i,j\in
\{1, \dots, n\}$ (see [@KS pages 137–139]). The system normalizer of $G$ associated with $\mathcal{S}$ is $M(G)=\bigcap_{i=1}^n N_G(S_i)$. If the solvable group $G$ is a normal subgroup of a non necessarily solvable finite group $L$, the system normalizer of $G$ relative to $L$ associated to $\mathcal{S}$ is $M_L(G)=\bigcap_{i=1}^n N_L(S_i)$. System normalizers and relative system normalizers were introduced by P. Hall in [@hall].

Recall that the multiplicative group $(B,\cdot )$ of any finite left brace $(B,+,\cdot )$ is solvable. But not all finite solvable groups are isomorphic 