# A Question About Combinatorial Groups (Jie Wu)

Let x1,...,xn be letters and let w(x1,...,xn) denote a word in the free group F(x1,...,xn). Let 1 denote the identity of a group. The group G(n) is defined combinatorially by
• generators: x1,...,xn;
• relations:
(1) the product element x1x2...xn;
(2) the words w(x1,...,xn) that satisfy:
w(x1,...,xi-1,1,xi+1,...,xn)=1 for each i=1,2,...,n.

Problems:
1. Determine the order of the center Z(G(n)).
2. Determine the 2-primary component of Z(G(n)).

Remarks:
• [ J. Wu] Z(G(n)) is isomorphic to the n-th homotopy group of the 2-sphere and so Z(G(n)) is isomorphic to the n-th homotopy group of the 3-sphere for n>2 by the Hopf fibration.
• By using results on \pi*(S3), one gets
1. Z(G(n)) is finite for n>3.
2. [Selick] the p-primary component of Z(G(n)) has exponent p for p>2.
3. [James-Cohen] the 2-primary component of Z(G(n)) has exponent 4.
4. [Toda] Z(G(n)) is known for n<23.
5. [Curtis-Mahowald] the 2-primary component of Z(G(n)) is known up to n=64 or so.
6. Here are some examples (from Toda's book, composition methods in homotopy groups of spheres):
7. Z(G(2))=Z(G(3))=Z,
Z(G(4))=Z/2Z,
Z(G(5))=Z/2Z,
Z(G(6))=Z/4Z +Z/3Z,
Z(G(7))=Z/2Z,
Z(G(8))=Z/2Z,
Z(G(9))=Z/3Z,
Z(G(10))=Z/3Z+Z/5Z,
Z(G(11))=Z/2Z,
Z(G(12))=Z/2Z+Z/2Z,
Z(G(13))=Z/4Z+Z/2Z+Z/3Z,
Z(G(14))=Z/4Z+Z/2Z+Z/2Z+Z/3Z+Z/7Z,
Z(G(15))=Z/2Z+Z/2Z,
Z(G(16))=Z/2Z+Z/3Z,
Z(G(17))=Z/2Z+Z/3Z+Z/5Z,
Z(G(18))=Z/2Z+Z/3Z+Z/5Z,
Z(G(19))=Z/2Z+Z/2Z+Z/3Z,
Z(G(20))=Z/4Z+Z/2Z+Z/2Z+Z/3Z,
Z(G(21))=Z/4Z+Z/2Z+Z/2Z +Z/3Z,
Z(G(22))=Z/4Z+Z/2Z+Z/3Z+Z/11Z,

Observe that in this table the p-torsion component of Z(G(n)) is either Z/pZ or 0 for p>2. Also Z/4Z+Z/2Z+Z/2Z is the maximal 2-component of Z(G(n)) in this table.

The calculations given by Curtis and Mahowald also suggest that Z(G(n)) is very small in some sense.

• Z(G(n)) is UNKNOWN for general n.

• Conjecture: There exists a positive integer k such that the order of the p-torsion component of Z(G(n)) is less than or equal to pk for any n>3.