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

Ellis, Graham

doi:10.1090/S0002-9947-01-02812-4

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