## On the relation between upper central quotients and lower central series of a group

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## 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.## References

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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