Premium
Faithful Representations of Free Products
Author(s) -
Minty R. John H.
Publication year - 1997
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610797005231
Subject(s) - mathematics , free product , combinatorics , commutative ring , degree (music) , product (mathematics) , identity (music) , group (periodic table) , zero divisor , intersection (aeronautics) , field (mathematics) , discrete mathematics , commutative property , pure mathematics , chemistry , physics , geometry , organic chemistry , acoustics , engineering , aerospace engineering
In 1940 Nisnevic̆ 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 ⩾0. If for each α the group G α has a faithful representation of degree n over F α then the free product* α∈Λ G 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 ⩽GL( n , F ) is a family of subgroups for which there exists Z ⩽GL( n , F ) such that for all α the intersection G α ∩ F .1 n = Z , then the free product of the groups * Z G 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.