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Generalized group identities in linear groups. (English. Russian original) Zbl 0574.20032
Math. USSR, Sb. 51, 33-46 (1985); translation from Mat. Sb., Nov. Ser. 123(165), No. 1, 35-49 (1984).
Let G be a subgroup of \(GL_ n\)(K) for some algebraically closed field K, and let \(a_ 1,...,a_{m+1}\) be elements of \(GL_ n(K)\). An expression \[ h(T_ 1,...,T_ k)=a_ 1T_{i_ 1}^{\ell_ 1}a_ 2...a_ mT_{i_ m}^{\ell_ m}a_{m+1} \] with \(l_ i\in {\mathbb{Z}}\) is called a generalized group monomial if the conditions \(i_ j=i_{j+1}\), \(\ell_ j\ell_{j+1}<0\) imply that \(a_{j+1}\) does not commute with at least one element of G. A generalized group identity \(h(T_ 1,...,T_ k)=1\) is said to hold in G if \(h(b_ 1,...,b_ k)=1\) for all \(b_ 1,...,b_ k\) in G. The main theorem of the paper says that there exist no generalized identities in G if and only if \(G\supset SL_ n(K)\).
Reviewer: S.I.Gel’fand

20G15 Linear algebraic groups over arbitrary fields
20F05 Generators, relations, and presentations of groups
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