When is the commutant of a Bol loop a subloop?

Abstract

A left Bol loop is a loop satisfying x ( y ( x z ) ) = ( x ( y x ) ) z x(y(xz)) = (x(yx))z . The commutant of a loop is the set of elements which commute with all elements of the loop. In a finite Bol loop of odd order or of order 2 k 2k , k 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 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 K such that K K is in the left and middle nuclei of the resulting loop, we show how to construct classes of Bol loops with a non-subloop commutant. In particular, we obtain all Bol loops of order 16 16 with a non-subloop commutant.