Algebraic sets with fully characteristic radicals

TL;DR

This paper characterizes when algebraic sets in groups have fully characteristic radicals, linking the radical's properties to identities of specific subgroup classes within the group.

Contribution

It provides a necessary and sufficient condition for algebraic sets to have fully characteristic radicals, advancing understanding of algebraic set structures in group theory.

Findings

Radical of a system is fully characteristic iff certain subgroup class identities hold

Characterization of algebraic sets with fully characteristic radicals

Connection between radicals and subgroup identities

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 $\mathfrak{X}$.