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Faithful representations of free products

Minty, John 1997. Faithful representations of free products. Journal of the London Mathematical Society 56 (1) , pp. 137-148. 10.1112/S0024610797005231

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In 1940 Nisnevic̆ 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≥0. If for each α the group Gα has a faithful representation of degree n over Fα then the free product*α∈Λ Gα has a faithful representation of degree n+1 over some field of characteristic p. In [6] Wehrfritz extended this idea. If (Gα)α∈Λ ≤GL(n, F) is a family of subgroups for which there exists Z≤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.

Item Type: Article
Date Type: Publication
Status: Published
Schools: Mathematics
Subjects: Q Science > QA Mathematics
Publisher: London Mathematical Society
ISSN: 0024-6107
Last Modified: 25 Oct 2016 02:52

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