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On verbal factors of almost free groups of varieties. (English. Russian original) Zbl 0792.20030

Math. USSR, Sb. 73, No. 2, 501-516 (1992); translation from Mat. Sb. 182, No. 8, 1140-1157 (1991).
Let \(\mathfrak V\) be a variety of groups of zero exponent in which all free groups are approximated by solvable groups, \(k\) be an uncountable regular cardinal number which is not weakly compact. A group \(G \in {\mathfrak V}\) of cardinality \(k\) is called an almost \(\mathfrak V\)-free group if each subgroup of rank less then \(k\) is contained in some \(\mathfrak V\)-free subgroup of \(G\). A group \(G \in {\mathfrak V}\) is called \(k\)-separable in \(\mathfrak V\) if for any infinite set \(A \subset G\), \(| A| < k\) there exists a verbal \(\mathfrak V\)-factor \(A_ 1\) of \(G\) (i.e. \(G\) is a \(\mathfrak V\)-free product of \(A_ 1\) and some group \(B\)) such that \(| A| = | A_ 1|\), \(A \subseteq A_ 1\), and \(A_ 1\) is a \(\mathfrak V\)-free group. It is clear that the class of \(k\)-separable groups in \(\mathfrak V\) is a subclass of the class of almost \(\mathfrak V\)-free groups.
The main result of this work is as follows. Under the assumption of the axiom of constructibility of set theory, it is proved that there are \(2^ k\) in pairs non-isomorphic \(k\)-separable groups in \(\mathfrak V\) of cardinality \(k\), each of which is not decomposable in a \(\mathfrak V\)-free product of two \(k\)-separable subgroups of cardinality \(k\). It is obvious that these groups are not \(\mathfrak V\)-free. The author notes that if \(k\) is a singular cardinal number then a \(k\)-separable group in \(\mathfrak V\) of cardinality \(k\) is \(\mathfrak V\)-free.

MSC:

20E10 Quasivarieties and varieties of groups
20E22 Extensions, wreath products, and other compositions of groups
20A15 Applications of logic to group theory
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
20E07 Subgroup theorems; subgroup growth
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