Associated Lie ring for a strongly central series

Definition

Suppose $G$ is a nilpotent group and $G_{i}$ form a strongly central series for $G$:

$G=G_{1}\geq G_{2}\geq G_{3}\geq \dots \geq G_{k}=1$.

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

• As an Abelian group, it is the associated direct sum for the series:

$L=\bigoplus _{i=1}^{k-1}G_{i}/G_{i+1}$.

• 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.