The torsion type of nilpotent groups
In the theory of torsion-free abelian groups of finite rank the outer type of such groups plays an important role. This type can be defined from certain torsion quotients. In this paper this definition is extended to torsion-free nilpotent groups \(G\) of finite rank. Many properties of this invariant type are exhibited and the differences from the abelian case are pointed out. There is a connection between the torsion-types of \(G/\zeta_{k-1}(G)\) and \(\gamma_ k(G)\), where \(\zeta_ i(G)\) denotes the upper central series and \(\gamma_ i(G)\) the lower central series of \(G\). Moreover, a formula for the torsion type of a product \(UN\) of a subgroup \(U\) and a normal subgroup \(N\) of a nilpotent group of finite rank is given. The special commutator identities that are needed in the proofs for the nilpotent case are perhaps of independent interest.
torsion-free abelian groups of finite rank
scientific article; zbMATH DE number 627426
scientific article