Michael K. Kinyon

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In the spirit of Glauberman’s fundamental work in B-loops and Moufang loops [17] [18], we prove Cauchy and strong Lagrange theorems for Bol loops of odd order. We also establish necessary conditions for the existence of a simple Bol loop of odd order, conditions which should be useful in the development of a Feit-Thompson theorem for Bol loops. Bol loops(More)
We show that the skew-symmetrized product on every Leibniz algebra E can be realized on a reductive complement to a subalgebra in a Lie algebra. As a consequence, we construct a nonassociative multiplication on E which, when E is a Lie algebra, is derived from the integrated adjoint representation. We apply this construction to realize the bracket(More)
Let Q be a conjugacy closed loop, and N(Q) its nucleus. Then Z(N(Q)) contains all associators of elements of Q. If in addition Q is diassociative (i.e., an extra loop), then all these associators have order 2. If Q is power-associative and |Q| is finite and relatively prime to 6, then Q is a group. If Q is a finite non-associative extra loop, then 16 | |Q|.
We study conjugacy closed loops (CC-loops) and power-associative CC-loops (PACC-loops). If Q is a PACC-loop with nucleus N , then Q/N is an abelian group of exponent 12; if in addition Q is finite, then |Q| is divisible by 16 or by 27. There are eight nonassociative PACC-loops of order 16, three of which are not extra loops. There are eight nonassociative(More)
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(More)