Journal Title/Source
Transactions of the American Mathematical Society
Publication Date
2008
Volume
360
Page Numbers
2393–2408
Document Type
Journal Article
Department
Mathematics and Computer Science
Abstract
A left Bol loop is a loop satisfying x(y(xz)) = (x(yx))z. The commutant of a loop is the set of all elements which commute with all elements of the loop. In a finite Bol loop of odd order or of order 2k, k odd, the commutant is a subloop. We investigate conditions under which the commutant of a Bol loop is not a subloop. In a finite Bol loop of order relatively prime to 3, the commutant generates an abelian group of order dividing the order of the loop. This generalizes a well-known result for Moufang loops. After describing all extensions of a loop K such that K is in the left and middle nuclei of the resulting loop, we show how to construct classes of Bol loops with non-subloop commutant. In particular, we obtain all Bol loops of order 16 with non-subloop commutant.
Recommended Citation
Phillips, J.D., "WHEN IS THE COMMUTANT OF A BOL LOOP A SUBLOOP?" (2008). Journal Articles. 528.
https://commons.nmu.edu/facwork_journalarticles/528