FINITE-FINITARY GROUPS OF AUTOMORPHISMS

In this paper we attempt to describe the structure of groups G of automorphisms of an abelian group M with the property that M(g - 1) is finite for every element g of G. These groups are closely related to the finitary linear groups over finite fields. The abelian case is critical for our work and the core result of this paper is the following. An abelian group A is isomorphic to a group G as above with M torsion if and only if A is torsion and has a residually-finite subgroup B with A/B a direct sum of cyclic groups.

Finitarily linear wreath products

We consider faithful finitary linear representations of (generalized) wreath products A wrΩH of groups A by H over (potentially) infinite-dimensional vector spaces, having previously considered completely reducible such representations in an earlier paper. The simpler the structure of A the more complex, it seems, these representations can become. If A has no non-trivial abelian normal subgroups, the conditions we present are both necessary and sufficient. They imply, for example, that for such an A, if there exists such a representation of the standard wreath product A wr H of infinite dimension, then there already exists one of finite dimension.