Algebraic sets with fully characteristic radicals

Abstract

We obtain a necessary and sufficient condition for an algebraic set in a group to have a fully characteristic radical. As a result, we see that if the radical of a system of equation \(S\) over a group \(G\) is fully characteristic, then there exists a class \(\mathfrak{X}\) of subgroups of \(G\) such that elements of \(S\) are identities of \(\mathfrak{X}\).