When Is the Commutant of a Bol Loop a Subloop?
نویسنده
چکیده
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.
منابع مشابه
N 05 Commutants of Bol Loops of Odd Order
In this note we show that the commutant of a Bol loop of odd order is a subloop.
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