The Model Theory of Fc-Groups

Abstract

Following R. Baer and B. H. Neumann a group G is called and F-C Group if for each g ɛ G the conjugacy class { χ -1 gχ;ɛ ɛ G } of g is finite. In her celebrated paper W. Szmielew solved the problems of decidability and internal characterization of elementary equivalence for the class of abelian groups. The model theory of abelian groups is by now well established. For the next slightly larger class, i.e. the class of nilpotent groups (even nil-2) only a few sporadic results are known. We found it more promising to look at another generalization of commutativity, namely the class of FC-groups. We have the following results: (1) For all nɛω : the theory of { G ; [ G : Z( G )] ≤ n } is decidable. (2) The theory of { G ; [ G : Z( G )] 0 } is undecidable. (3) The theory of all FC-groups and the theory of all BFC-groups are both undecidable. (4) We classified all stable, ω -stable and ℵ 0 - cate gorical groups G for which [ G :Z( G )] is finite. (5) We determined the Δ 2 - theory of all periodic FC-groups. (6) We have the surprising result, that the Hypercenter of every FC-groups of finite exponent is first-order definable without parameters. It seems to us that it is possible in the near future to get elementary invariants for periodic FC-groups.