In 1940 Nisnevič published the following theorem
[3]. Let
(Gα)α∈Λ be a family
of
groups indexed by some set Λ and
(Fα)α∈Λ a family
of
fields of the same
characteristic p[ges ]0. If for each α the group
Gα has a faithful representation of degree
n over Fα then the free product
*α∈ΛGα has
a
faithful representation of degree n+1
over some field of characteristic p. In [6]
Wehrfritz extended this idea. If
(Gα)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
Gα∩F.1n=Z,
then the free product of the groups
*ZGα 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, Dα)
where the Dα 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.