N-lower central series

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Definition

Suppose G is a group and n is an integer. The n-lower central series of G is defined as the following descending series H_n, n \in \mathbb{N}:

  • The first member of the series is G, i.e., H_1 = G.
  • For any n, H_{n+1} is the subgroup of G generated by all n-commutators [x,y]_n = (xy)^ny^{-n}x^{-n} where x \in H_n, y \in G.