2021/04/07 by A. Caranti, Caranti, Andrea, Ilaria Del Corso +1
Mathematics · #20D15 20B35 12F10 16T05 20E18 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.2104.03211
openalex publication_date 2021/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In \cite[Theorem 2.5]Bac16 Bachiller proved that if (G, ⋅, ∘) is a brace of order the power of a prime p and the rank of (G,⋅) 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,∘) determines the isomorphism type of (G,⋅). In this paper we complement Bachiller's result in two directions. In Theorem 2.2 we prove that if (G, ⋅, ∘) is a brace of order the power of a prime p, then (G,⋅) has small rank (i.e. < p-1) if and only if (G,∘) 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.