OFFSET
1,1
COMMENTS
Let [n]={1,..,n}. A number n is groupless iff there is no binary operation . on [n] such that G=([n],.) is a group, and . extends the partial graph of multiplication on [n], i.e., whenever i,j and their usual product i*j are in [n], then i.j=i*j. If n is not groupless, a witness G is sometimes called an FLP group.
The term "groupless" was coined by Thomas Chartier.
LINKS
Andrés Eduardo Caicedo, Thomas A. C. Chartier, Péter Pál Pach, Coloring the n-smooth numbers with n colors, arXiv:1902.00446 [math.NT], 2019.
K. A. Chandler, Groups formed by redefining multiplication, Canad. Math. Bull. Vol. 31 (4), (1988), 419-423.
Th. A. Ch. Chartier, Coloring problems, MS Thesis in Mathematics, Boise State University, December, 2011.
R. Forcade and A. Pollington, What is special about 195? Groups, n-th-power maps and a problem of Graham, in Proceedings of the First Conference of the Canadian Number Theory Association, Banff, 1988, R.A. Mollin, ed., Walter de Gruyter, Berlin, 1990, 147-155.
CROSSREFS
KEYWORD
nonn
AUTHOR
Andres E. Caicedo, Jan 19 2012
STATUS
approved