Associated Lie ring for a strongly central series

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Definition

Suppose G is a nilpotent group and G_i form a strongly central series for G:

G = G_1 \ge G_2 \ge G_3 \ge \dots \ge G_k = 1.

where [G_m,G_n] \le G_{m+n} is the condition for being strongly central. The associatied Lie ring to G is defined as follows:

L = \bigoplus_{i=1}^{k-1} G_i/G_{i+1}.

  • The Lie bracket is defined as follows. For \overline{g} \in G_i/G_{i+1} and \overline{h} \in G_j/G_{j+1}, [\overline{g},\overline{h}] is defined as the element of G_{i+j}/G_{i+j+1} giving the coset of [g,h] (here, g,h are representatives of \overline{g}, \overline{h}). The Lie bracket is then extended additively to the whole of L.

The fact that [g,h] \in G_{i+j}, as well as the fact that [\overline{g},\overline{h}] is independent of the choice of representatives, follow from the condition of being strongly central. Checking that the Jacobi identity is satisfied requires some additional work. For full proof, refer: Associated Lie ring for a strongly central series is a Lie ring

For any nilpotent group, the lower central series is strongly central (For full proof, refer: Lower central series is strongly central), and thus has an associated Lie ring. The associated Lie ring for the lower central series is simply termed the associated Lie ring -- in other words, when the strongly central series is not specified, it is assumed to be the lower central series. This Lie ring is also termed the Magnus-Sanov Lie ring.