FAITHFUL REPRESENTATIONS OF FREE PRODUCTS

Abstract

In 1940 Nisnevič published the following theorem [3]. Let (G_α)_α∈Λ be a family of groups indexed by some set Λ and (F_α)_α∈Λ a family of fields of the same characteristic p[ges ]0. If for each α the group G_α has a faithful representation of degree n over F_α then the free product *_α∈ΛG_α has a faithful representation of degree n+1 over some field of characteristic p. In [6] Wehrfritz extended this idea. If (G_α)_α∈Λ [les ]GL(n, F) is a family of subgroups for which there exists Z[les ]GL(n, F) such that for all α the intersection G_α∩F.1_n=Z, then the free product of the groups *_ZG_α with Z amalgamated via the identity map is isomorphic to a linear group of degree n over some purely transcendental extension of F.

Initially, the purpose of this paper was to generalize these results from the linear to the skew-linear case, that is, to groups isomorphic to subgroups of GL(n, D_α) where the D_α are division rings. In fact, many of the results can be generalized to rings which, although not necessarily commutative, contain no zero-divisors. We have the following.