Verbal products of groups
The author is concerned with a variety \(V\) of groups of exponent zero (namely he focuses on infinite groups that are close to free groups in this variety). The main result is the following theorem: If \(\kappa\) is a regular uncountable cardinal, then there exist \(2^ \kappa\) mutually nonisomorphic groups \(\{F_ i : i < 2^ \kappa\}\) that are strongly \(\kappa\)-free in the variety \(\mathcal F\) of all groups, such that they all have rank \(\kappa\) and, for each \(i < 2^ \kappa\), the following is valid: \(\forall \overline{\varphi} \in \text{End} (F_ i / F_ i') (\exists) \ell \in \mathbb{Z} \forall x \in F_ i/ F_ i'\) (\(\overline {\varphi} (x) = x^ \ell\)). As a corollary it is obtained that if \(\kappa\) is a regular uncountable cardinal and \(V\) an arbitrary variety of groups of exponent zero whose free groups are approximated by solvable groups, then there exist \(2^ \kappa\) mutually nonisomorphic strongly \(\kappa\)-free groups \(\{G_ i : i < 2^ \kappa\}\) in \(V\) of cardinality \(\kappa\) such that each of them is indecomposable into the \(V\)-product of two nontrivial \(V\)-factors.
scientific article; zbMATH DE number 561012