### Groupoids with non-associative triples on the diagonal

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If $Q$ is a quasigroup that is free in the class of all quasigroups which are isotopic to an Abelian group, then its multiplication group $\mathrm{M}ltQ$ is a Frobenius group. Conversely, if $\mathrm{M}ltQ$ is a Frobenius group, $Q$ a quasigroup, then $Q$ has to be isotopic to an Abelian group. If $Q$ is, in addition, finite, then it must be a central quasigroup (a $T$-quasigroup).

We show how to generate all spherical latin trades by elementary moves from a base set. If the base set consists only of a single trade of size four and the moves are applied only to one of the mates, then three elementary moves are needed. If the base set consists of all bicyclic trades (indecomposable latin trades with only two rows) and the moves are applied to both mates, then one move suffices. Many statements of the paper pertain to all latin trades, not only to spherical ones.

This paper completely solves the isomorphism problem for Moufang loops $Q=GC$ where $G\u22b4Q$ is a noncommutative group with cyclic subgroup of index two and $\left|Z\right(G\left)\right|\le 2$, $C$ is cyclic, $G\cap C=1$, and $Q$ is finite of order coprime to three.

Let $G=G(\xb7)$ be a commutative groupoid such that $\{(a,b,c)\in {G}^{3}$; $a\xb7bc\ne ab\xb7c\}=\{(a,b,c)\in {G}^{3}$; $a=b\ne c$ or $a\ne b=c\}$. Then $G$ is determined uniquely up to isomorphism and if it is finite, then $card\left(G\right)={2}^{i}$ for an integer $i\ge 0$.

Let $Q$ be a diassociative A-loop which is centrally nilpotent of class 2 and which is not a group. Then the factor over the centre cannot be an elementary abelian 2-group.

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...

A loop $Q$ is said to be left conjugacy closed (LCC) if the set $\{{L}_{x};x\in Q\}$ is closed under conjugation. Let $Q$ be such a loop, let $\mathcal{L}$ and $\mathcal{R}$ be the left and right multiplication groups of $Q$, respectively, and let $InnQ$ be its inner mapping group. Then there exists a homomorphism $\mathcal{L}\to InnQ$ determined by ${L}_{x}\mapsto {R}_{x}^{-1}{L}_{x}$, and the orbits of $[\mathcal{L},\mathcal{R}]$ coincide with the cosets of $A\left(Q\right)$, the associator subloop of $Q$. All LCC loops of prime order are abelian groups.

We investigate loops defined upon the product ${\mathbb{Z}}_{m}\times {\mathbb{Z}}_{k}$ by the formula $(a,i)(b,j)=((a+b)/(1+t{f}^{i}\left(0\right){f}^{j}\left(0\right)),i+j)$, where $f\left(x\right)=(sx+1)/(tx+1)$, for appropriate parameters $s,t\in {\mathbb{Z}}_{m}^{*}$. Each such loop is coupled to a 2-cocycle (in the group-theoretical sense) and this connection makes it possible to prove that the loop possesses a metacyclic inner mapping group. If $s=1$, then the loop is an A-loop. Questions of isotopism and isomorphism are considered in detail.

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