It's easy to see that the descending central series of a group induces a graded Lie algebra .(see for example Serre's Harvard lectures or Magnus-Solitar book). I think in general this can be complicated, but this should be well-known:



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> What is the structure of this Lie algebra for the symmetric group?
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Lie algabra of symmetric group
For any $n>1$, the lower central series for the symmetric group is $S\_n > A\_n\geq A\_n \geq A\_n \geq \cdots$, so the Lie ring formed by the sum of successive quotients is the group $\mathbb{Z}/2\mathbb{Z}$, equipped with the Lie bracket that is identically zero.


If you want to gain intuition for this construction with finite groups, I suggest you consider nilpotent groups, since their lower central series actually reach the trivial group. For example, many $p$-groups will yield nonabelian Lie algebras over $\mathbb{F}\_p$.


