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Journal of the London Mathematical Society 1997 56(1):137-148; doi:10.1112/S0024610797005231
© 1997 by London Mathematical Society
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© The London Mathematical Society

Faithful Representations of Free Products

R. John H. Minty

School of Mathematical Sciences, Queen Mary and Westfield College Mile End Road, London E1 4NS

Received 4 June 1995. Revision received 10 May 1996.

In 1940 Nisnevic published the following theorem [3]. Let (G{alpha}){alpha}isin{Lambda} be a family of groups indexed by some set {Lambda} and (F{alpha}){alpha}isin{Lambda} a family of fields of the same characteristic p≥0. If for each {alpha} the group G{alpha} has a faithful representation of degree n over F{alpha} then the free product*{alpha}isin{Lambda} G{alpha} has a faithful representation of degree n+1 over some field of characteristic p. In [6] Wehrfritz extended this idea. If (G{alpha}){alpha}isin{Lambda} ≤GL(n, F) is a family of subgroups for which there exists Z≤GL(n, F) such that for all {alpha} the intersection G{alpha}{cap}F.1n=Z, then the free product of the groups *ZG{alpha} 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{alpha}) where the D{alpha} 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.

Department of Mathematics and Computer Science, University of the West Indies, Mona Campus, Mona, Kingston 7, Jamaica


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