Bi-Skew Braces and Regular Subgroups of the Holomorph

Abstract

L. Childs has defined a skew brace $(G,\cdot ,\circ )$ to be a bi-skew brace if $(G,\circ ,\cdot )$ is also a skew brace, and has given applications of this concept to the equivalent theory of Hopf-Galois structures. The goal of this paper is to deal with bi-skew braces $(G,\cdot ,\circ )$ from the yet equivalent point of view of regular subgroups of the holomorph of $(G,\cdot )$. In particular, we find that certain groups studied by T. Kohl, F. Dalla Volta and the author, and C. Tsang all yield examples of bi-skew braces. Building on a construction of Childs, we also give various methods for constructing further examples of bi-skew braces.

16 pages - accepted for publication in the Journal of Algebra

References in corpus (3)

• Hopf-Galois structures on extensions of degree $p^{2}q$ and skew braces of order $p^{2}q$: the cyclic Sylow $p$-subgroup case
• Bi-skew braces and Hopf Galois structures
• The multiple holomorph of a finitely generated abelian group

Cited by in corpus (12)

• From endomorphisms to bi-skew braces, regular subgroups, the Yang--Baxter equation, and Hopf--Galois structures
• On the connection between Hopf--Galois structures and skew braces
• On bi-skew braces and brace blocks
• Brace blocks from bilinear maps and liftings of endomorphisms
• Relative Rota-Baxter groups and skew left braces
• Finite $p$-groups of class two with a large multiple holomorph
• On the ranks of the additive and the multiplicative groups of a brace
• Transfinite hypercentral iterated wreath product of integral domains
• Affine structures on groups and semi-braces
• The mutually normalizing regular subgroups of the holomorph of a cyclic group of prime power order
• On the triviality and non-triviality of the automorphism group of a skew brace
• A classification of module braces over the ring of $\mathbf{p}$-adic integers