The lower central series {γ α (G)} of a group G, where α varies over all ordinal numbers, is defined by setting γ1(G) = G, γα+1(G) = [G, γ α (G)] and for a limit ordinal τ, γ τ (G) = ∩α<τ γ α (G), where for subsets H, K of G, [H, K] denotes the subgroup of G generated by all commutators [h, k] := h−1hk = h−1k−1 hk, h ∈ H, k ∈ K. The group G is said to be transfinitely nilpotent if γ α (G) = 1 for some ordinal α, or simply nilpotent if α is a finite ordinal. In particular, if γ ω (G) = 1, where ω is the least infinite ordinal, then G is said to be residually nilpotent. In this Chapter we present various constructions and methods for studying the residual nilpotence of groups. Our aim is primarily to investigate the transfinite terms of the lower central series. The results discussed here can be viewed as an attempt to develop a ‘limit theory’ for lower central series.
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© 2009 Springer-Verlag Berlin Heidelberg
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(2009). Lower Central Series. In: Lower Central and Dimension Series of Groups. Lecture Notes in Mathematics, vol 1952. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85818-8_1
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DOI: https://doi.org/10.1007/978-3-540-85818-8_1
Publisher Name: Springer, Berlin, Heidelberg
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