On varieties generated by a finitely generated group

On varieties of groups generated by wreath products of abelian groups

“…Thus B contains a subgroup D isomorphic with an infinite direct power of cycle C p k . D is a discriminating group in the variety A p k [BNNN64 ]. Therefore by Lemma 2.2 the group B itself is a discriminating group for the variety A p k and so var (A Wr B) = A p u · A p k .…”

“…Definition 2.1 (see [BNNN64 ]). The set D of groups is said to be discriminating, if for arbitrary finite word set V with the property that, for each w ∈ V there exists a homomorphism δ w of a free group F n into some group of D, such that δ w (w) = 1, there exist a group D ∈ D and a single homomorphism δ of F n into D, such that δ(w) = 1 for all w ∈ V .…”

Verbal wreath products and certain product varieties of groups

“…When we say that a word in F "involves" the variable x it we shall mean that the word contains x t or xj 1 …”

“…Let <^ be the endomorphism of F k which fixes the elements of S t and maps all other free generators in {x 1 is a basis for a direct product of n" isomorphic copies of C a , considered as a vector space of dimension n"* over GF(q).…”

“…Recently A. L. Smel'kin [14] proved that a product variety 1 USS is generated by a finite group if and only if U is nilpotent, S3 is abelian, and the exponents of U and S3 are coprime. Alternatively, by the theorem of Oates and Powell [13], we may say that a Cross variety is decomposable if and only if it is of the above form.…”