We describe, in a constructive way, a family of commutative loops of odd order, $n\ge 7$, which have no nontrivial subloops and whose multiplication group is isomorphic to the alternating group ${𝒜}_n$.

We review results for the embedding of orthogonal partial Latin squares in orthogonal Latin squares, comparing and contrasting these with results for embedding partial Latin squares in Latin squares. We also present a new construction that uses the existence of a set of $t$ mutually orthogonal Latin squares of order $n$ to construct a set of $2t$ mutually orthogonal Latin squares of order $n^t$.

Our purpose is to determine the complete set of mutually orthogonal squares of order d, which are not necessary Latin. In this article, we introduce the concept of supersquare of order d, which is defined with the help of its generating subgroup in $${𝔽}_d\times {𝔽}_d$$ . We present a method of construction of the mutually orthogonal supersquares. Further, we investigate the orthogonality of extraordinary supersquares, a special family of squares, whose generating subgroups are extraordinary. The extraordinary subgroups...

We give a short account of the construction and properties of left neofields. Most useful in practice seem to be neofields based on the cyclic group and particularly those having an additional divisibility property, called D-neofields. We shall give examples of applications to the construction of orthogonal latin squares, to the design of tournaments balanced for residual effects and to cryptography.

Let $G(\circ )$ and $G(*)$ be two groups of finite order $n$, and suppose that they share a normal subgroup $S$ such that $u\circ v=u*v$ if $u\in S$ or $v\in S$. Cases when $G/S$ is cyclic or dihedral and when $u\circ v\ne u*v$ for exactly $n^2/4$ pairs $(u,v)\in G\times G$ have been shown to be of crucial importance when studying pairs of 2-groups with the latter property. In such cases one can describe two general constructions how to get all possible $G(*)$ from a given $G=G(\circ )$. The constructions, denoted by $G[\alpha ,h]$ and $G[\beta ,\gamma ,h]$, respectively, depend on a coset $\alpha$ (or two cosets $\beta$ and $\gamma$) modulo $S$, and on an...