Symmetric skew braces and brace systems

Abstract For a skew left brace ( G , rspace="4.2pt">? ? stretchy="false">) {(G,\cdot\,,\circ)} , the map ? : ? Aut ? {\lambda:(G,\circ)\to\operatorname{Aut}(G,\cdot\,)} a ? {a\mapsto\lambda_{a},} where ? b = - 1 ? {\lambda_{a}(b)=a^{-1}\cdot(a\circ b)} for all ? {a,b\in G} is group homomorphism. Then ? can also be viewed as from {(G,\cdot\,)} to {\operatorname{Aut}(G,\cdot\,)} which, in general, may not A will called ?-anti-homomorphic (?-homomorphic) if {\lambda:(G,\cdot\,)\to\operatorname{Aut}(G,\cdot\,)} an anti-homomorphism (a homomorphism). We mainly study such braces. device method constructing class of binary operations on given set so that with any two constitutes ?-homomorphic symmetric brace. Most constructions braces dealt literature fall framework our construction. then carry out various specific infinite groups.