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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
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$.

References [Enhancements On Off] (What's this?)

  • [1] Dock Sang Rim, Modules over finite groups, Ann. of Math. (2) 69 (1959), 700-712. MR 21 #3474. MR 0104721 (21:3474)

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Keywords: Abelianization, commutator subgroup, covering space, homology of groups
Article copyright: © Copyright 1973 American Mathematical Society

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