The divisible radical of a group.
The author investigates the notion of ``divisible radical'' for non-Abelian groups. The divisible radical subgroup of an arbitrary group, modeled on the Abelian case, is defined to be its unique maximal divisible normal subgroup, \textit{whenever it exists}. If such a radical for a group \(G\) exists, it will be denoted by \(\delta(G)\). The main theme is to find conditions on a group under which such a radical exists and after the existence guaranteed, properties of the radical are investigated. However, the author provides groups for which such a radical does not exist. The main results are the following. Theorem 1.2. Let \(G\) be any group. a) \(G/\tau(G)\) has a unique maximal, locally nilpotent, divisible normal subgroup, where \(\tau(G)\) is the unique maximal locally finite normal subgroup of \(G\). b) The subgroup of \(G\), generated by all divisible normal subgroups of \(G\) that are hypercentral groups, is locally nilpotent and divisible. c) If \(G\) is locally nilpotent and torsion-free, then \(\delta(G)\) exists, \(\delta(G/\delta(G))=\langle 1\rangle\) and \(\delta(G)=\langle D\leq G:D\text{ is divisible}\rangle\). Theorem 1.3. There exists a non-divisible torsion-free metabelian group that is the product of two of its divisible normal subgroups. In particular \(\delta(G)\) for such a group \(G\) does not exist. Theorem 1.4. For every prime \(p\) there exists a non-divisible metabelian \(p\)-group \(G\) that is the product of two of its divisible normal subgroups. Again \(\delta(G)\) for this group does not exist. A group \(G\) is called `semi-divisible' if \(G=G^m=\langle g^m:g\in G\rangle\) for every positive integer \(m\). Theorem 1.5. There exists a non-divisible, torsion-free, locally nilpotent group \(T\) that is perfect and semi-divisible. In particular \(T/\delta(T)\) is neither hypo Abelian nor hypo (of finite exponent).
scientific article; zbMATH DE number 5637490