Varieties of algebras are equationally defined classes of algebras, or "primitive classes" in MAL'CEV'S terminology. They made their first explicit appearance in the 1930's, in Garrett BIRKHOFF'S paper on "The structure of abstract algebras" and B. H. NEUMANN'S paper "Identical relations in groups I". For quite some time after this, there is little published evidence that the subject remained alive. In fact, however, as part of "universal algebra", it aroused great interest amongst those who had access, directly or indirectly, to PHILIP HALL'S lectures given at Cambridge late in the 1940's. More recently, category theory has provided a general setting since varieties, suitably interpreted, are very special examples of categories. Whether their relevance to category theory goes beyond this, I do not know. And I doubt that the category theoretical approach to varieties will be more than a fringe benefit to group theory. Whether or not my doubts have substance, the present volume owes its existence not to the fact that varieties fit into a vastly more general pattern, but to the benefit group theory has derived from the classification of groups by varietal properties. It is this aspect of the study of varieties that seems to have caused its reappearance in the literature in the 1950's.

Inhalt

1. The Basic Facts.- 1. Preliminaries.- 2. Words, Laws, Verbal Subgroups.- 3. Relatively Free Groups.- 4. Varieties.- 5. Varieties as Closed Classes of Groups.- 6. The n-Generator Groups and the n-Variable Laws of a Variety.- 7. Discrimination and Residual Properties.- 8. Verbal Products.- 2. Product Varieties.- 1. The Algebra of Varieties.- 2. Wreath Products and Discrimination.- 3. The Uniqueness of Factorization.- 4. Some Classes of Indecomposable Varieties.- 5. Product Varieties Generated by a Finitely Generated Group.- 6. Residual Properties of the Free Groups of Product Varieties.- 3. Nilpotent Varieties.- 1. Summary of Properties of Nilpotent Groups.- 2. Residual Properties.- 3. A Lemma on Words with an Application to Free Products.- 4. The Laws of a Nilpotent Variety and Related Topics.- 5. Generating Groups of Finite Rank.- 6. The Variety of All Metabelian Nilpotent Groups of Class c.- 4. Miscellaneous Properties of Relatively Free Groups.- 1. Remarks on Automorphisms and the Hopf Property.- 2. Free Subgroups of Free Groups.- 3. Theorems Like Auslander and Lyndon's: the Schreier Property.- 4. The Splitting Property; Direct Decomposability.- 5. The Laws of Finite Groups.- 1. Critical Groups and Cross Varieties.- 2. The Theorem of Oates and Powell.- 3. Critical Groups and Subvarieties.- 4. Critical p-Groups and Locally Finite Varieties; a Summary of Developments.- References.- Author Index.