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: 3146
Received: 7 Aprile 2021
Accepted in revised form: 8 July 2021
Mathematics Subject Classification: 16T25, 12F10, 20D15, 20B35, 20E18.
Keywords: Braces, skew braces, YangBaxter equation, holomorph, regular subgroups, HopfGalois structures.
Authors address:
[a]: Dipartimento di Matematica, Università degli Studi di Trento, via Sommarive 14, I38123 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 INdAMGNSAGA. 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".
Full Text (PDF)
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 \(p1\), 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. \(< p1\)) if and only if
\((G,\circ)\) has small rank. We also provide examples of
groups of rank \(p1\) 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), 18721881.
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,
HopfGalois structures of squarefree degree,
J. Algebra 559 (2020), 5886.
MR4093704
 [Bac16]

D. Bachiller,
Counterexample to a conjecture about braces,
J. Algebra 453 (2016), 160176.
MR3465351
 [Ber00]

Y. Berkovich,
On subgroups of finite \(p\)groups,
J. Algebra 224 (2000), 198240.
MR1739577
 [Ber02]

Y. Berkovich,
On subgroups and epimorphic images of finite \(p\)groups,
J. Algebra 248 (2002), 472553.
MR1882110
 [Ber05]

Y. Berkovich,
Alternate proofs of two theorems of Philip Hall on finite \(p\)groups, and some related results,
J. Algebra 294 (2005), 463477.
MR2183360
 [Byo96]

N. P. Byott,
Uniqueness of Hopf Galois structure for separable field extensions,
Comm. Algebra 24 (1996), 32173228.
MR1402555
 [Byo04]

N. P. Byott,
HopfGalois structures on Galois field extensions of degree \(pq\),
J. Pure Appl. Algebra 188 (2004), 4557.
MR2030805
 [Byo13]

N. P. Byott,
Nilpotent and abelian HopfGalois structures on field extensions,
J. Algebra 381 (2013), 131139.
MR3030514
 [Byo15]

N. P. Byott,
Solubility criteria for HopfGalois structures,
New York J. Math. 21 (2015), 883903.
MR3425626
 [Car20]

A. Caranti,
Biskew braces and regular subgroups of the holomorph,
J. Algebra 562 (2020), 647665.
MR4130907
 [CCDC20]

E. Campedel, A. Caranti and I. Del Corso,
HopfGalois 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), 11651210.
MR4089566
 [Chi89]

L. N. Childs,
On the Hopf Galois theory for separable field extensions,
Comm. Algebra 17 (1989), 809825.
MR990979
 [Chi05]

L. N. Childs,
Elementary abelian Hopf Galois structures and polynomial formal groups,
J. Algebra 283 (2005), 292316.
MR2102084
 [Chi19]

L. N. Childs,
Biskew braces and Hopf Galois structures,
New York J. Math. 25 (2019), 574588.
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,
SpringerVerlag, BerlinNew York, 1969.
MR0260724
 [ESS99]

P. Etingof, T. Schedler and A. Soloviev,
Settheoretical solutions to the quantum YangBaxter equation,
Duke Math. J. 100 (1999), 169209.
MR1722951
 [FCC12]

S. C. Featherstonhaugh, A. Caranti and L. N. Childs,
Abelian Hopf Galois structures on primepower Galois field extensions,
Trans. Amer. Math. Soc. 364 (2012), 36753684.
MR2901229
 [GP87]

C. Greither and B. Pareigis,
Hopf Galois theory for separable field extensions,
J. Algebra 106 (1987), 239258.
MR0878476
 [GV17]

L. Guarnieri and L. Vendramin,
Skew braces and the YangBaxter equation,
Math. Comp. 86 (2017), 25192534.
MR3647970
 [Hal34]

P. Hall,
A Contribution to the Theory of Groups of PrimePower Order,
Proc. London Math. Soc. (2) 36 (1934), 2995.
MR1575964
 [Koh98]

T. Kohl,
Classification of the Hopf Galois structures on prime power radical extensions,
J. Algebra 207 (1998), 525546.
MR1644203
 [Nas19]

T. Nasybullov,
Connections between properties of the additive and the multiplicative groups of a twosided skew brace,
J. Algebra 540 (2019), 156167.
MR4003478
 [NZ18]

K. Nejabati Zenouz,
On HopfGalois 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 YangBaxter equation,
J. Algebra 307 (2007), 153170.
MR2278047
 [SV18]

A. Smoktunowicz and L. Vendramin,
On skew braces (with an appendix by N. Byott and L. Vendramin),
J. Comb. Algebra 2 (2018), 4786.
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), 253265.
MR4077413
 [Tsa19]

C. Tsang,
Nonexistence of HopfGalois structures and bijective crossed homomorphisms,
J. Pure Appl. Algebra 223 (2019), 28042821.
MR3912948
Home Riv.Mat.Univ.Parma