Symmetric skew braces and brace systems

Abstract

Abstract For a skew left brace ( G , ⋅ , ∘ ) {(G,\cdot\,,\circ)} , the map λ : ( G , ∘ ) → Aut ⁡ ( G , ⋅ ) {\lambda:(G,\circ)\to\operatorname{Aut}(G,\cdot\,)} , a ↦ λ a , {a\mapsto\lambda_{a},} where λ a ⁢ ( b ) = a - 1 ⋅ ( a ∘ b ) {\lambda_{a}(b)=a^{-1}\cdot(a\circ b)} for all a , b ∈ G {a,b\in G} , is a group homomorphism. Then λ can also be viewed as a map from ( G , ⋅ ) {(G,\cdot\,)} to Aut ⁡ ( G , ⋅ ) {\operatorname{Aut}(G,\cdot\,)} , which, in general, may not be a homomorphism. A skew left brace will be called λ-anti-homomorphic (λ-homomorphic) if λ : ( G , ⋅ ) → Aut ⁡ ( G , ⋅ ) {\lambda:(G,\cdot\,)\to\operatorname{Aut}(G,\cdot\,)} is an anti-homomorphism (a homomorphism). We mainly study such skew left braces. We device a method for constructing a class of binary operations on a given set so that the set with any two such operations constitutes a λ-homomorphic symmetric skew brace. Most of the constructions of symmetric skew braces dealt with in the literature fall in the framework of our construction. We then carry out various such constructions on specific infinite groups.