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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

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