On finitely generated soluble groups with no large wreath product sections. (English) Zbl 0537.20013

Kropholler, P. H.

Proc. Lond. Math. Soc., III. Ser. 49, 155-169 (1984).

The following important result is established. A finitely generated soluble group G is a minimax group if and only if for no prime p is there a section of G which is isomorphic with the standard wreath product of two cyclic groups of order p. This generalizes results of A. S. Kirkinskij and the reviewer, and serves to emphasize the central position of minimax groups in the theory of finitely generated soluble groups.
The proof is cohomological and depends on certain embedding properties in low dimensions of cohomology groups of soluble minimax groups with coefficients in modules that are direct limits.
An interesting minor result is the observation that every soluble minimax group is a (finite) product of cyclic subgroups. It would be interesting to know if the converse holds: is every soluble product of cyclic groups a minimax group?

Reviewer: D.J.S.Robinson

MSC:

20F16	Solvable groups, supersolvable groups
20J05	Homological methods in group theory
20E15	Chains and lattices of subgroups, subnormal subgroups

Keywords:

finitely generated soluble group; section; wreath product; cohomology groups; soluble minimax groups; product of cyclic subgroups