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
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.
[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".

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.

References
[AB20a]
E. Acri and M. Bonatto, Skew braces of size $$pq$$, Comm. Algebra 48 (2020), 1872-1881. MR4085764
[AB20b]
E. Acri and M. Bonatto, Skew braces of size $$p^2q$$, arXiv:1912.11889, preprint, 2020. DOI
[AB20c]
A. A. Alabdali and N. P. Byott, Hopf-Galois structures of squarefree degree, J. Algebra 559 (2020), 58-86. MR4093704
[Bac16]
D. Bachiller, Counterexample to a conjecture about braces, J. Algebra 453 (2016), 160-176. MR3465351
[Ber00]
Y. Berkovich, On subgroups of finite $$p$$-groups, J. Algebra 224 (2000), 198-240. MR1739577
[Ber02]
Y. Berkovich, On subgroups and epimorphic images of finite $$p$$-groups, J. Algebra 248 (2002), 472-553. MR1882110
[Ber05]
Y. Berkovich, Alternate proofs of two theorems of Philip Hall on finite $$p$$-groups, and some related results, J. Algebra 294 (2005), 463-477. MR2183360
[Byo96]
N. P. Byott, Uniqueness of Hopf Galois structure for separable field extensions, Comm. Algebra 24 (1996), 3217-3228. MR1402555
[Byo04]
N. P. Byott, Hopf-Galois structures on Galois field extensions of degree $$pq$$, J. Pure Appl. Algebra 188 (2004), 45-57. MR2030805
[Byo13]
N. P. Byott, Nilpotent and abelian Hopf-Galois structures on field extensions, J. Algebra 381 (2013), 131-139. MR3030514
[Byo15]
N. P. Byott, Solubility criteria for Hopf-Galois structures, New York J. Math. 21 (2015), 883-903. MR3425626
[Car20]
A. Caranti, Bi-skew braces and regular subgroups of the holomorph, J. Algebra 562 (2020), 647-665. MR4130907
[CCDC20]
E. Campedel, A. Caranti and I. Del Corso, Hopf-Galois structures on extensions of degree $$p^2q$$ and skew braces of order $$p^2 q$$: the cyclic Sylow $$p$$-subgroup case, J. Algebra 556 (2020), 1165-1210. MR4089566
[Chi89]
L. N. Childs, On the Hopf Galois theory for separable field extensions, Comm. Algebra 17 (1989), 809-825. MR990979
[Chi05]
L. N. Childs, Elementary abelian Hopf Galois structures and polynomial formal groups, J. Algebra 283 (2005), 292-316. MR2102084
[Chi19]
L. N. Childs, Bi-skew braces and Hopf Galois structures, New York J. Math. 25 (2019), 574-588. MR3982254
[Cre20]
T. Crespo, Automatic realization of Hopf Galois structures, J. Algebra Appl. 21 (2022), Paper No. 2250030, 9 pp. MR4381288
[CS69]
S. U. Chase and M. E. Sweedler, Hopf algebras and Galois theory, Lecture Notes in Mathematics, 97, Springer-Verlag, Berlin-New York, 1969. MR0260724
[ESS99]
P. Etingof, T. Schedler and A. Soloviev, Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J. 100 (1999), 169-209. MR1722951
[FCC12]
S. C. Featherstonhaugh, A. Caranti and L. N. Childs, Abelian Hopf Galois structures on prime-power Galois field extensions, Trans. Amer. Math. Soc. 364 (2012), 3675-3684. MR2901229
[GP87]
C. Greither and B. Pareigis, Hopf Galois theory for separable field extensions, J. Algebra 106 (1987), 239-258. MR0878476
[GV17]
L. Guarnieri and L. Vendramin, Skew braces and the Yang-Baxter equation, Math. Comp. 86 (2017), 2519-2534. MR3647970
[Hal34]
P. Hall, A Contribution to the Theory of Groups of Prime-Power Order, Proc. London Math. Soc. (2) 36 (1934), 29-95. MR1575964
[Koh98]
T. Kohl, Classification of the Hopf Galois structures on prime power radical extensions, J. Algebra 207 (1998), 525-546. MR1644203
[Nas19]
T. Nasybullov, Connections between properties of the additive and the multiplicative groups of a two-sided skew brace, J. Algebra 540 (2019), 156-167. MR4003478
[NZ18]
K. Nejabati Zenouz, On Hopf-Galois Structures and Skew Braces of Order $$p^3$$, PhD thesis, The University of Exeter, 2018. handle
[Rum07]
W. Rump, Braces, radical rings, and the quantum Yang-Baxter equation, J. Algebra 307 (2007), 153-170. MR2278047
[SV18]
A. Smoktunowicz and L. Vendramin, On skew braces (with an appendix by N. Byott and L. Vendramin), J. Comb. Algebra 2 (2018), 47-86. MR3763907
[TQ20]
C. Tsang and C. Qin, On the solvability of regular subgroups in the holomorph of a finite solvable group, Internat. J. Algebra Comput. 30 (2020), 253-265. MR4077413
[Tsa19]
C. Tsang, Non-existence of Hopf-Galois structures and bijective crossed homomorphisms, J. Pure Appl. Algebra 223 (2019), 2804-2821. MR3912948

Home Riv.Mat.Univ.Parma