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A282530
Number of finite FRUTE loops of order n up to isomorphism.
0
0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 27, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0
OFFSET
1,8
COMMENTS
For a groupoid Q and x in Q, define the right (left) translation map R_x: Q->Q by yR_x=yx (L_x: Q->Q by yL_x=xy). A loop is a groupoid Q with neutral element 1 in which all translations are bijections in Q. A loop Q is called a FRUTE loop if it satisfies the identity (x.xy)z=(y.xz)x for all x, y, z in Q. The smallest associative non-commutative finite FRUTE loop is of order 8, the quaternion group having 8 elements.
LINKS
T. G. Jaiyeola, A. A. Adeniregun and M. A. Asiru, Finite FRUTE loops, Journal of Algebra and its Applications, 16:2(2017), 10 pages.
EXAMPLE
a(8)=2 since there are 2 FRUTE loops of order 8, one of which is the quaternion group of order 8 and a(16)=6 since there are 6 FRUTE loops of order 16.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Muniru A Asiru, Feb 17 2017
STATUS
approved