Representations of free metabelian $\mathcal {D}_\pi$-groups

Abstract:

For $\pi$ a set of primes, a ${\mathcal {D}_\pi }$-group is a group $G$ with the property that, for every element $g$ in $G$ and every prime $p$ in $\pi ,g$ has a unique $p$th root in $G$. Two faithful representations of free metabelian ${\mathcal {D}_\pi }$-groups are established: the first representation is inside a suitable power series algebra and shows that free metabelian ${\mathcal {D}_\pi }$-groups are residually torsion-free nilpotent; the second is in terms of two-by-two matrices and is analogous to W. Magnus’ representation of free metabelian groups using two-by-two matrices. In a subsequent paper [12], these representations will be used to derive several properties of free metabelian ${\mathcal {D}_\pi }$-groups.