We discuss central automorphisms of partial linear spaces, particularly those with three points per line. When these automorphisms have order two and their products are restricted to have odd order, we are in the situation of Glauberman's Z∗-theorem. This sheds light on the structure of various coordinatizing loops, particularly Bol and Moufang loops.

Let F be a family of finite loops closed under subloops and factor loops. Then every loop in F has the strong Lagrange property if and only if every simple loop in F has the weak Lagrange property. We exhibit several such families, and indicate how the Lagrange property enters into the problem of existence of finite simple loops. The two most important open problems in loop theory, namely the e...

In this brief note, we exhibit a shortest single product and neutral element axiom for commutative Moufang loops of exponent 3 that was found with the aid of the automated theorem-prover Prover9.

Moufang’s theorem states that if Q is a Moufang loop with elements x, y and z such that x ·yz = xy · z, then these three elements generate a subgroup of Q. The paper contains a new proof of this theorem that is shorter and more transparent than the standardly used proof of Bruck. A loop is a binary structure with a unit such that the equations ax = b and ya = b have unique solutions x = a\b and...

In this dissertation, we consider an extension of the discrete logarithm problem to the case of a semigroup acting on a finite set: the Semigroup Action Problem (SAP). New protocols and one-way trapdoor functions based on the difficulty of such problems are proposed. Several instances are studied both from a conceptual and cryptographic point of view. We discuss the application of existing gene...

Norton and Stein associated a number with each idempotent quasigroup or diagonalized Latin square of given finite order n, showing that it is congruent mod 2 to the triangular number T (n). In this paper, we use a graph-theoretic approach to extend their invariant to an arbitrary finite quasigroup. We call it the cycle number, and identify it as the number of connected components in a certain g...

In this paper we classify finite special Moufang sets M(U, τ ) of odd characteristic. The characteristic 2 case was handled in another paper by De Medts and the author. We prove, using elementary means that U is elementary abelian. Then we show that M(U, τ ) is the unique Moufang set whose little projective group is PSL2(|U |). The emphasis of this paper is on obtaining elementary proofs. Secti...