Verbal quandles with one parameter

Abstract

We find all words $W(x,y,z)$ in the free group $F(x,y,z)$, such that for every group $G$ and an element $c\in G$ the algebraic system $(G,{\ast }_{W,c})$ with the binary operation ${\ast }_{W,c}$ given by $a{\ast }_{W,c}b=W(a,b,c)$ for $a,b\in G$ is a quandle. Such quandles are called verbal quandles with one parameter.