ABSTRACT
It is well known, and easy to prove, that, for a group G, the algebra K[G] is right noetherian if and only if it is left noetherian. This is due to the antiautomorphism of K[G] determined by g → g −1 for g ∊ G. Moreover, in this case, G has the a.c.c. on subgroups; see [203], § 10.2. This seems to be the only general result on noetherian group algebras. On the other hand, a classical result of Higman asserts that the group algebra of a polycyclic-by-finite group is noetherian. A very deep structure, and representation theory of these algebras have recently been developed. For a survey of this topic, we refer to [204]. This motivated the results in Chapter 11 dealing with subsemigroups of polycyclic-by-finite groups. We note that no known examples of noetherian group algebras arise from groups that are not polycyclic-by-finite.
