Associated Lie ring for a strongly central series
where is the condition for being strongly central. The associatied Lie ring to is defined as follows:
- As an Abelian group, it is the associated direct sum for the series:
- The Lie bracket is defined as follows. For and , is defined as the element of giving the coset of (here, are representatives of ). The Lie bracket is then extended additively to the whole of .
The fact that , as well as the fact that 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.