Associated Lie ring of free group equals free Lie ring on abelianization

History

The result is attributed to Magnus and Witt and the isomorphism of the theorem is often termed the Magnus-Witt isomorphism.

Statement

Suppose $F$ is a free group. Then, the associated Lie ring of $F$ is isomorphic to the free Lie ring on the abelianization of $F$. The isomorphism is in fact an isomorphism of graded Lie rings if we remember the gradations of the constructions.

References

• On five well-known commutator identities by Graham Ellis, Journal of the Australian Mathematical Society, Volume 54, Page 1 - 19(Year 1993): ^{More info}, Page 2. This paper proves a more general formulation.