\(\mathbb Q\)-completions of free solvable groups.
A group \(G\) is said to be \(m\)-rigid if \(G\) contains a normal series of the form \(G=G_1>G_2>\cdots>G_m>G_{m+1}=1\), whose quotients \(G_i/G_{i+1}\) are abelian and are torsion-free as right \(\mathbb Z[G/G_i]\)-modules. Such a series (if it exists) is defined by the group \(G\) uniquely, and the derived length of \(G\) is exactly \(m\). It is proved that every \(m\)-rigid group is embedded in a complete \(m\)-rigid group. Under the specified condition of the embedding, the \(\mathbb Q\)-completion of the group \(G\) in the class of rigid groups is defined uniquely up to \(G\)-isomorphism. It is stated that the centralizer of any element of such \(\mathbb Q\)-completion of a free solvable group, which does not lie in the last nontrivial member of a rigid series of this completion, is isomorphic to the additive group of \(\mathbb Q\).
scientific article; zbMATH DE number 6498556
scientific article