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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The commutator subgroup made abelian


Author: Joel M. Cohen
Journal: Proc. Amer. Math. Soc. 38 (1973), 507-508
MSC: Primary 55A10; Secondary 20F35
DOI: https://doi.org/10.1090/S0002-9939-1973-0315692-9
MathSciNet review: 0315692
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Abstract: A theorem on covering spaces is proved which yields the following information about a group $ \pi $, its commutator subgroup $ \pi '$ and their abelianizations: If $ {\pi ^{ab}} \cong {Z_{{p^n}}}$, a cyclic group of order a power of the prime $ p$, then $ \pi {'^{ab}} = p\pi {'^{ab}}$. Hence if $ \pi $ is also finitely generated, then $ \pi {'^{ab}}$ is finite of order prime to $ p$.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0315692-9
Keywords: Abelianization, commutator subgroup, covering space, homology of groups
Article copyright: © Copyright 1973 American Mathematical Society

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