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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Author: John F. Ledlie
Journal: Trans. Amer. Math. Soc. 153 (1971), 307-346
MSC: Primary 20.40
MathSciNet review: 0276341
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0276341-4
PII: S 0002-9947(1971)0276341-4
Keywords: Metabelian group, $ \mathcal{D}$-group, unique roots, power series algebra representation, matrix representation
Article copyright: © Copyright 1971 American Mathematical Society