Ascending Subgroups of Irreducible Finitary Linear Group

Let K a field and V a vector space over K. Let FGLK(V ) be the finitary linear group of V over K, namely FGLK(V ) = {g ∈ GLK(V )| [V,g] has finite K-dimension }. Subgroups of FGLK(V ) are called finitary groups. Recently a good amount of work has been done towards a classification of the locally finite, finitary groups (see [1, 2, 7]). On the otherhand very little is known without the assumption of locally finiteness. This paper is meant as a contribution to the general theory of finitary groups. Throughout this paper G is a subgroup of FGLK(V ). Suppose that G is irreducible and H an ascending subgroup of G, that is there exist a well ordered set I and subgroups Hi, i ∈ I, of G including H and G such that HiCHi+1 and if i is a limit ordinal in I, then Hi = ∪j

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