Brace blocks from bilinear maps and liftings of endomorphisms

Abstract

We extend two constructions of Alan Koch, exhibiting methods to construct brace blocks, that is, families of group operations on a set $G$ such that any two of them induce a skew brace structure on $G$. We construct these operations by using bilinear maps and liftings of endomorphisms of quotient groups with respect to a central subgroup. We provide several examples of the construction, showing that there are brace blocks which consist of distinct operations of any given cardinality. One of the examples we give yields an answer to a question of Cornelius Greither. This example exhibits a sequence of distinct operations on the $p$-adic Heisenberg group $(G,\cdot )$ such that any two operations give a skew brace structure on $G$ and the sequence of operations converges to the original operation "$\cdot$".

19 pages

References in corpus (4)

• Rota-Baxter groups, skew left braces, and the Yang-Baxter equation
• Bi-skew braces and Hopf Galois structures
• Skew braces from Rota--Baxter operators: a cohomological characterisation and some examples
• From endomorphisms to bi-skew braces, regular subgroups, the Yang--Baxter equation, and Hopf--Galois structures

Cited by in corpus (6)

• Skew braces from Rota--Baxter operators: a cohomological characterisation and some examples
• On the connection between Hopf--Galois structures and skew braces
• On bi-skew braces and brace blocks
• Finite $p$-groups of class two with a large multiple holomorph
• Affine structures on groups and semi-braces
• The mutually normalizing regular subgroups of the holomorph of a cyclic group of prime power order