Riv. Mat. Univ. Parma, Vol. 13, No. 1, 2022

Andrea Caranti [a] and Ilaria Del Corso [b]

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

Pages: 31-46
Received: 7 Aprile 2021
Accepted in revised form: 8 July 2021
Mathematics Subject Classification: 16T25, 12F10, 20D15, 20B35, 20E18.
Keywords: Braces, skew braces, Yang-Baxter equation, holomorph, regular subgroups, Hopf-Galois structures.
Authors address:
[a]: Dipartimento di Matematica, Università degli Studi di Trento, via Sommarive 14, I-38123 Trento, Italy
[b]: Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy

Dedicated to Roberto Dvornicich on the occasion of his seventieth birthday

The authors are members of INdAM-GNSAGA. The authors gratefully acknowledge support from the Departments of Mathematics of the Universities of Pisa and Trento. The second author has performed this activity in the framework of the PRIN 2017, title "Geometric, algebraic and analytic methods in arithmetic".

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Abstract: In [Bac16 , Theorem 2.5] 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 2.3 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.

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