Algebraic linear groups as whole groups of automorphisms and the closeness of their verbal subgroups

Ju. I. Merzljakov

Algebra i logika, Tome 6 (1967) no. 1, pp. 83-94

Résumé

Every algebraic linear group over a field $K$ of the characteristic $0$ is rationally isomorphic to a group of all automorphisms of some universal algebra $\mathrm{V}^\phi$ which arises from a finite-dimensional vector space $\mathrm{V}$ over $K$ by adding to it some finite collection $\Phi$ of polylinear operations on $\mathrm{V}$ (see [3], pp. 305–306). For an arbitrary word $V$ the verbal subgroup $V(G)$ of any algebraic linear group $G$ over the universal domain is closed and has a finite width.