# The Theory of Classes of Groups

Part of the Mathematics and Its Applications book series (MAIA, volume 505)

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Part of the Mathematics and Its Applications book series (MAIA, volume 505)

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.

Algebraic structure Group theory Lattice Mathematica algebra finite group

- DOI https://doi.org/10.1007/978-94-011-4054-6
- Copyright Information Kluwer Academic Publishers and Science Press 2000
- Publisher Name Springer, Dordrecht
- eBook Packages Springer Book Archive
- Print ISBN 978-94-010-5785-1
- Online ISBN 978-94-011-4054-6
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