# Associated Lie ring for a strongly central series

## 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.