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On the Relation between Upper Central Quotients and Lower Central Series of a Group

Graham Ellis
Transactions of the American Mathematical Society
Vol. 353, No. 10 (Oct., 2001), pp. 4219-4234
Stable URL: http://www.jstor.org/stable/2693793
Page Count: 16
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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 ZcH. In this article we study the influence of the quotient group G = H/N on the lower central subgroup γc+1H. In particular, for any finite group G we give bounds on the order and exponent of γc+lH. For G equal to a dihedral group, or quaternion group, or extra-special group we list all possible groups that can arise as γc+1H. Our proofs involve: (i) the Baer invariants of G, (ii) the Schur multiplier 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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