One of the characteristics of modern algebra is the development of new
tools and concepts for exploring classes of algebraic systems, whereas
the research on individual algebraic systems (e. g., groups, rings, Lie
algebras, etc. ) continues along traditional lines. The early work on
classes of alge- bras was concerned with showing that one class X of
algebraic systems is actually contained in another class F. Modern
research into the theory of classes was initiated in the 1930's by
Birkhoff's work [1] on general varieties of algebras, and Neumann's
work [1] on varieties of groups. A. I. Mal'cev made fundamental
contributions to this modern development. ln his re- ports [1, 3] of
1963 and 1966 to The Fourth All-Union Mathematics Con- ference and to
another international mathematics congress, striking the- ories of
classes of algebraic systems were presented. These were later included
in his book [5]. International interest in the theory of formations of
finite groups was aroused, and rapidly heated up, during this time,
thanks to the work of Gaschiitz [8] in 1963, and the work of Carter
and Hawkes [1] in 1967. The major topics considered were saturated
formations, Fitting classes, and Schunck classes. A class of groups is
called a formation if it is closed with respect to homomorphic images
and subdirect products. A formation is called saturated provided that G
E F whenever Gjip(G) E F.