Constructible Matrix Groups

Abstract

We prove that the additive group of a ring K is constructible if the group GL ₂ (K) is constructible. It is stated that under one extra condition on K, the constructibility of GL ₂ (K) implies that K is constructible as a module over its subring L generated by all invertible elements of the ring L; this is true, in particular, if K coincides with L, for instance, if K is a field or a group ring of an Abelian group with the specified property. We construct an example of a commutative associative ring K with 1 such that its multiplicative group K* is constructible but its additive group is not. It is shown that for a constructible group G represented by matrices over a field, the factors w.r.t. members of the upper central series are also constructible. It is proved that a free product of constructible groups is again constructible, and conditions are specified under which relevant statements hold of free products with amalgamated subgroup; this is true, in particular, for the case where an amalgamated subgroup is finite. Also we give an example of a constructible group GL ₂ (K) with a non-constructible ring GL. Similar results are valid for the case where the group SL ₂ (K) is treated in place of GL ₂ (K) .