# Free Differential Calculus, IV. The Quotient Groups of the Lower Central Series

@article{Chen1958FreeDC, title={Free Differential Calculus, IV. The Quotient Groups of the Lower Central Series}, author={K. T. Chen and Ralph Hartzler Fox and Roger Lyndon}, journal={Annals of Mathematics}, year={1958}, volume={68}, pages={81} }

The quotient groups Qn(G) =GnGn+i of the lower central series G = G1 D G, D G, D * * of a finitely generated group G are finitely generated abelian groups. Our object is to develop an algorithm for the calculation of Qn from any given finite presentation of G. As a preliminary step, the special case of a free group X is considered. It is known [2], [7] that, for a free group X of rank q, the group Qn(X) is a free abelian group whose rank is the Witt number 0b,(q), and a basis for QJ(X) has been… Expand

#### 350 Citations

Dimension and lower central subgroups

- Mathematics
- 1979

In [7] Passi proved that for a finite p-group, p # 2, one has Gj = D4. We generalize this to say that G, = D, as long as n = p + 1. We also generalize a theorem of Quillen [B], using entirely… Expand

Sjogren's theorem on dimension subgroups

- Mathematics
- 1987

Let G be a group, ZG its integral group ring, and g its augmentation ideal. The nth dimension subgroup Dn(G) of G (over Z) is defined by Dn(G) = G N (1 + gn). It is well known and not difficult to… Expand

Symmetries of free and right-angled Artin groups

- Mathematics
- 2012

The objects of study in this thesis are automorphism groups of free and right-angled Artin groups. Right-angled Artin groups are defined by a presentation where the only relations are commutators of… Expand

Milnor’s invariants and the completions of link modules

- Mathematics
- 1984

Let L be a tame link of k > 2 components in S3, H the abelianization of its group 7TJ(S3 L), and IH the augmentation ideal of the integral group ring ZH. The IH-adic completions of the Alexander… Expand

Torsion-groups of abelian coverings of links

- Mathematics
- 1982

If M is an abelian branched covering of S3 along a link L, the order of HI(M) can be expressed in terms of (i) the Alexander polynomials of L and of its sublinks, and (ii) a "redundancy" function… Expand

Combinatorics of the Free Lie Algebra and the Symmetric Group

- Mathematics
- 1990

Given an alphabet A = {a1, a2 … an}, the smallest subspace of the linear space of A-words, which contains the words of length one and is closed under the bracket operation [f, g] = fg – fg, is… Expand

THE LOWER CENTRAL AND DERIVED SERIES OF THE BRAID GROUPS OF THE FINITELY-PUNCTURED SPHERE

- Mathematics
- 2009

Motivated in part by the study of Fadell–Neuwirth short exact sequences, we determine the lower central and derived series for the braid groups of the finitely-punctured sphere. For n ≥ 1, the class… Expand

The length of conjugators in solvable groups and lattices of semisimple Lie groups

- Mathematics
- 2012

The conjugacy length function of a group Γ determines, for a given a pair of conjugate elements u,v ∈ Γ, an upper bound for the shortest γ in Γ such that uγ = γv, relative to the lengths of u and v.… Expand

The Cohomology of canonical quotients of free groups and Lyndon words

- Mathematics
- 2016

For a prime number $p$ and a free profinite group $S$, let $S^{(n,p)}$ be the $n$th term of its lower $p$-central filtration, and $S^{[n,p]}$ the corresponding quotient. Using tools from the… Expand

Anti-commutative Gröbner-Shirshov basis of a free Lie algebra

- Mathematics
- 2009

The concept of Hall words was first introduced by P. Hall in 1933 in his investigation on groups of prime power order. Then M. Hall in 1950 showed that the Hall words form a basis of a free Lie… Expand

#### References

SHOWING 1-5 OF 5 REFERENCES

FOX, Free d i f e ren f ia l calculus [I

- FDC I,
- 1953

A basis for free Lie rirzgs arzd higher corn?nutators i n free groups

- Proc. Am. Math. Soc
- 1950