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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On the relation between upper central quotients and lower central series of a group
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by Graham Ellis PDF
Trans. Amer. Math. Soc. 353 (2001), 4219-4234 Request permission

Abstract:

Let $H$ be a group with a normal subgroup $N$ contained in the upper central subgroup $Z_cH$. In this article we study the influence of the quotient group $G=H/N$ on the lower central subgroup $\gamma _{c+1}H$. In particular, for any finite group $G$ we give bounds on the order and exponent of $\gamma _{c+1}H$. For $G$ equal to a dihedral group, or quaternion group, or extra-special group we list all possible groups that can arise as $\gamma _{c+1}H$. Our proofs involve: (i) the Baer invariants of $G$, (ii) the Schur multiplier $\mathcal {M}(L,G)$ of $G$ relative to a normal subgroup $L$, and (iii) the nonabelian tensor product of groups. Some results on the nonabelian tensor product may be of independent interest.
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Additional Information
  • Graham Ellis
  • Affiliation: Max-Planck-Institut für Mathematik, Gottfried-Claren-Strasse 26, Bonn, Germany
  • Address at time of publication: Department of Mathematics, National University of Ireland, Galway, Ireland
  • Email: graham.ellis@nuigalway.ie
  • Received by editor(s): February 12, 1999
  • Published electronically: June 6, 2001
  • © Copyright 2001 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 353 (2001), 4219-4234
  • MSC (2000): Primary 20F14, 20F12
  • DOI: https://doi.org/10.1090/S0002-9947-01-02812-4
  • MathSciNet review: 1837229