Abstract
It was a fundamental discovery of Mal’cev that for groups, the property of being linear of (fixed) degree n is of ‘finite character’. From the metamathematical point of view, his observation was that this property can be expressed in a suitable first-order language; algebraically, what it means is that a group G is linear of degree n if and only if for every finite subset S of G there exists a degree-n linerar representation p of (S) which separates S, i.e., such that |p(S)| = |S|. This sounds rather like saying that G is locally residually (linear of degree n), but is in fact stronger: a direct product of infinitely many elementary abelian groups of distinct prime exponents and unbounded ranks is residually linear of degree 1 but has no faithful linear representation over any field. If we make the additional assumption that G is finitely generated, however, then results like Mal’cev’s obtain under the weaker hypothesis; this observation is due to [Wilson 1991 b ]. The following proof is based on an idea that we learned from J.D. Dixon.
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© 2003 Birkhäuser Verlag
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Lubotzky, A., Segal, D. (2003). Linearity Conditions for Infinite Groups. In: Subgroup Growth. Progress in Mathematics, vol 212. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-8965-0_25
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DOI: https://doi.org/10.1007/978-3-0348-8965-0_25
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0348-9846-1
Online ISBN: 978-3-0348-8965-0
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