On the universality of systems of words in permutation groups (Q794757)

Let \(G\) be a group, \(x_ 1,...,x_ n\) variables, \(y_ 1,...,y_ m\) parameters, and \(w_ i=w_ i(x_ 1,...,x_ n;y_ 1,...,y_ m)\) (\(i\in I)\) group words in these variables and parameters. Then \(W=\{w_ i;\quad i\in I\}\) is called \(G\)-universal if for all \(y_ 1,...,y_ m\in G\) there exist \(x_ 1,...,x_ n\in G\) such that for all \(i\in I,\quad w_ i(x_ 1,...,x_ n;y_ 1,...,y_ m)=e.\) Two groups \(G\) and \(H\) are called equationally equivalent if for any finite set W of words \(w_ i\) as above, \(W\) is \(G\)-universal iff \(W\) is \(H\)-universal, i.e. \(G\) and \(H\) cannot be distinguished by the solvability of a system of finitely many equations in variables and parameters. In the classes of infinite symmetric groups, their normal subgroups, and their factor groups, the authors determine those groups which are equationally equivalent.