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- TUVAL FOGUEL, MICHAEL K. KINYON
- 2006

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)

A loop whose inner mappings are automorphisms is an automorphic loop (or A-loop). We characterize commutative (A-)loops with middle nucleus of index 2 and solve the isomorphism problem. Using this characterization and certain central extensions based on trilinear forms, we construct several classes of commutative A-loops of order a power of 2. We initiate… (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|.

An A-loop is a loop in which every inner mapping is an automorphism. A problem which had been open since 1956 is settled by showing that every diassociative A-loop is Moufang.

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 loop is (right) automorphic if all its (right) inner mappings are automorphisms. Using the classification of primitive groups of small degrees, we show that there is no non-associative simple commutative automorphic loop of order less than 2, and no non-associative simple automorphic loop of order less than 2500. We obtain numerous examples of… (More)

In this note we show that the commutant of a Bol loop of odd order is a subloop.

- Dean Alvis, Michael K. Kinyon
- The American Mathematical Monthly
- 2001

The panstochastic analogue of Birkhoff's Theorem on doubly-stochastic matrices is proved in the case n = 5. It is shown that this analogue fails when n > 1, n = 5.

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)