OFFSET
2,2
COMMENTS
The Fibonacci group F(r,n) has presentation <a_1,a_2,...,a_n|a_1*a_2*...*a_r=a_{r+1},...>, where there are n relations, obtained from the first relation by applying the permutation (1,2,,n) to the subscripts and reducing subscripts mod n. Then T(n,k) = |F(n,k)|.
T(7,5) was not known in 1998 (Chalk).
REFERENCES
Campbell, Colin M.; and Gill, David M. On the infiniteness of the Fibonacci group F(5,7). Algebra Colloq. 3 (1996), no. 3, 283-284.
D. L. Johnson, Presentation of Groups, Cambridge, 1976, see table p. 182.
Mednykh, Alexander; and Vesnin, Andrei; On the Fibonacci groups, the Turk's head links and hyperbolic 3-manifolds, in Groups-Korea '94 (Pusan), 231-239, de Gruyter, Berlin, 1995.
Nikolova, Daniela B., The Fibonacci groups - four years later, in Semigroups (Kunming, 1995), 251-255, Springer, Singapore, 1998.
Nikolova, D. B.; and Robertson, E. F., One more infinite Fibonacci group. C. R. Acad. Bulgare Sci. 46 (1993), no. 3, 13-15.
Thomas, Richard M., The Fibonacci groups revisited, in Groups - St. Andrews 1989, Vol. 2, 445-454, London Math. Soc. Lecture Note Ser., 160, Cambridge Univ. Press, Cambridge, 1991.
LINKS
Brunner, A. M., The determination of Fibonacci groups, Bull. Austral. Math. Soc. 11 (1974), 11-14.
A. M. Brunner, On groups of Fibonacci type, Proc. Edinburgh Math. Soc. (2) 20 (1976/77), no. 3, 211-213.
C. M. Campbell and P. P. Campbell, Search techniques and epimorphisms between certain groups and Fibonacci groups, Irish Math. Soc. Bull. No. 56 (2005), 21-28.
Chalk, Christopher P., Fibonacci groups with aspherical presentations, Comm. Algebra 26 (1998), no. 5, 1511-1546.
C. P. Chalk and D. L. Johnson, The Fibonacci groups II, Proc. Roy. Soc. Edinburgh Sect. A 77 (1977), no. 12, 79-86.
Helling, H.; Kim, A. C.; and Mennicke, J. L.; A geometric study of Fibonacci groups, J. Lie Theory 8 (1998), no. 1, 1-23.
Derek F. Holt, An alternative proof that the Fibonacci group F(2,9) is infinite, Experiment. Math. 4 (1995), no. 2, 97-100.
David J. Seal, The orders of the Fibonacci groups, Proc. Roy. Soc. Edinburgh, Sect. A 92 (1982), no. 3-4, 181-192.
A. Szczepanski, The Euclidean representations of the Fibonacci groups, Quart. J. Math. 52 (2001), 385-389.
EXAMPLE
The array begins:
k = 1 2 3 4 5 6 7 8 9 10 ...
----------------------------------------------------------
n=1: 0 0 0 0 0 0 0 0 0 0 ...
n=2: 1 1 8 5 11 0 29 0 0 0 ...
n=3: 2 8 2 0 22 1512 0 0 0 0 ...
n=4: 3 3 63 3 0 0 0 0 ? 0 ...
n=5: 4 24 0 624 4 0 0 0 0 0 ...
n=6: 5 5 5 125 7775 5 0 0 0 0 ...
n=7: 6 48 342 0 ? 7^6-1 6 0 0 0 ...
n=8: 7 7 0 7 ? 0 8^7-1 7 0 0 ...
n=9: 8 80 8 6560 0 0 0 9^8-1 8 0 ...
n=10 9 9 999 4905 9 ? ? 0 10^9-1 9 ...
...
For example, T(2,5) = 11, since the presentation <a,b,c,d,e | ab=c, bc=d, cd=e, de=a, ea=b> defines the cyclic group of order 11. This example is due to John Conway.
This table is based on those in Johnson (1976) and Thomas (1989), supplemented by values from Chalk (1998). We have ignored the n=1 row when reading the table by antidiagonals.
CROSSREFS
KEYWORD
AUTHOR
N. J. A. Sloane, Dec 29 2011
STATUS
approved