Divisibility and type-conditions in generalized nilpotent groups (Q1261144)

The author proves some extension of results on \(\pi\)-divisibility in generalized nilpotent groups, where \(\pi\) is a set of primes. For a hypercentral group \(G\) it is known (Chernikov) that \(G\) is \(\pi\)- divisible whenever \(G/G'\) is \(\pi\)-divisible. The author proves that the commutator subgroup \(G'\) may be replaced by its \(\sigma\)-isolator, when \(G\) is \(\sigma\)-free for any set of primes \(\sigma\). Similarly for a nilpotent group \(G\) it is known (Warfield) that the lower central term \(\Gamma_ k G\) is \(\pi\)-divisible when the upper central factor group \(G/Z_{k-1}G\) is \(\pi\)-divisible. Here it is proved that the same is true for hypercentral groups and torsionfree groups such that every pair of elements generates a nilpotent subgroup. The main investigations successfully extend the theory of element types from abelian groups to generalized nilpotent groups.