Varieties of groups and normal-subgroup lattices -- a survey (Q1337164)

This is mainly a survey paper dealing with the problem of distinguishing group varieties by means of their congruence varieties (i.e., lattice varieties generated by the (congruence=) normal subgroup lattices of all groups belonging to each of the given group varieties). It follows from a result of \textit{D. W. Barnes} [J. Aust. Math. Soc. 2, 17-34 (1961; Zbl 0115.253)] that if \(G\) is a finite group such that for each prime \(p\) the Sylow \(p\)-subgroups of \(G\) have nilpotency class \(< p\), then the congruence variety of the group variety generated by \(G\) is the same as the congruence variety of abelian groups of exponent dividing the exponent of \(G\). On the other hand, \textit{Cs. Szabó} and the reviewer [Algebra Univers. 33, No. 2, 191-195 (1995)] found a lattice identity distinguishing the congruence variety of groups of exponent 4 and nilpotency class 2 from the congruence variety of abelian groups of exponent 4. As the third main result in this area, the authors formulate a corollary to a theorem of \textit{Ch. Herrmann} and \textit{A. P. Huhn} [Math. Z. 144, 185-194 (1975; Zbl 0316.06006)]: group varieties of distinct finite exponents have different congruence varieties. Here they give a new, elegant proof, using a sequence of new, very simple lattice identities, which might deserve further study.