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FAITHFUL REPRESENTATIONS OF FREE PRODUCTS

Published online by Cambridge University Press:  01 August 1997

R. JOHN H. MINTY
Affiliation:
School of Mathematical Sciences, Queen Mary and Westfield College, Mile End Road, London E1 4NS

Abstract

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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 GαF.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.

Type
Research Article
Copyright
The London Mathematical Society 1997