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Let $G$ be a finite group and $C_2$ the cyclic group of order 2. Consider the 8 multiplicative operations $(x,y)\mapsto {\left(x^i{y}^j\right)}^k$, where $i,j,k\in \{-1,1\}$. Define a new multiplication on $G\times C_2$ by assigning one of the above 8 multiplications to each quarter $(G\times \{i\left\}\right)\times (G\times \{j\left\}\right)$, for $i,j\in C_2$. If the resulting quasigroup is a Bol loop, it is Moufang. When $G$ is nonabelian then exactly four assignments yield Moufang loops that are not associative; all (anti)isomorphic, known as loops $M(G,2)$.

Let $G$ be a finite group and $C_2$ the cyclic group of order $2$. Consider the $8$ multiplicative operations $(x,y)\mapsto {\left(x^i{y}^j\right)}^k$, where $i$, $j$, $k\in \{-1,1\}$. Define a new multiplication on $G\times C_2$ by assigning one of the above $8$ multiplications to each quarter $(G\times \{i\left\}\right)\times (G\times \{j\left\}\right)$, for $i,j\in C_2$. We describe all situations in which the resulting quasigroup is a Bol loop. This paper also corrects an error in P. Vojtěchovsk’y: On the uniqueness of loops $M(G,2)$.

The existence and uniqueness (up to equivalence defined below) of code loops was first established by R. Griess in [3]. Nevertheless, the explicit construction of code loops remained open until T. Hsu introduced the notion of symplectic cubic spaces and their Frattini extensions, and pointed out how the construction of code loops followed from the (purely combinatorial) result of O. Chein and E. Goodaire contained in [2]. Within this paper, we focus on their combinatorial construction and prove...

A groupoid is alternative if it satisfies the alternative laws $x\left(xy\right)=\left(xx\right)y$ and $x\left(yy\right)=\left(xy\right)y$. These laws induce four partial maps on ${\mathbb{N}}^{+}\times {\mathbb{N}}^{+}$ $$(r,s)\mapsto (2r,s-r),(r-s,2s),(r/2,s+r/2),(r+s/2,s/2),$$ that taken together form a dynamical system. We describe the orbits of this dynamical system, which allows us to show that $n$th powers in a free alternative groupoid on one generator are well-defined if and only if $n\le 5$. We then discuss some number theoretical properties of the orbits, and the existence of alternative loops without two-sided inverses.

Groups are usually axiomatized as algebras with an associative binary operation, a two-sided neutral element, and with two-sided inverses. We show in this note that the same simplicity of axioms can be achieved for some of the most important varieties of loops. In particular, we investigate loops of Bol-Moufang type in the underlying variety of magmas with two-sided inverses, and obtain ``group-like'' equational bases for Moufang, Bol and C-loops. We also discuss the case when the inverses are only...

We derive necessary and sufficient conditions for there to exist a latin square of order $n$ containing two subsquares of order $a$ and $b$ that intersect in a subsquare of order $c$. We also solve the case of two disjoint subsquares. We use these results to show that: (a) A latin square of order $n$ cannot have more than $\frac{n}{m}\left(\genfrac{}{}{0pt}{}{n}{h}\right)/\left(\genfrac{}{}{0pt}{}{m}{h}\right)$ subsquares of order $m$, where $h=\lceil (m+1)/2\rceil$. Indeed, the number of subsquares of order $m$ is bounded by a polynomial of degree at most $\sqrt{2m}+2$ in $n$. (b) For all $n\ge 5$ there exists a loop of order $n$ in which every...

A loop is automorphic if all its inner mappings are automorphisms. We construct the free commutative automorphic $2$-generated loop of nilpotency class $3$. It has dimension $8$ over the integers.