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June 26, 2019, at 07:06 PM EST  MathWiki / MathWiki / Loops 

Loops
Proposition. An associative quasigroup is a groupoid.
Definition. Let $L$ be a loop. If for all $x,y,z\in L$, $(x\cdot yx)z=x(y\cdot xz)$, then $L$ is called a Bolloop.
$$\alpha(x)\circ\beta(y)=\gamma(x\cdot y)\quad \mbox{for all }x,y\in L,$$ then $L$ and $L'$ are said to be isotopic. In this case, the triple $(\alpha,\beta,\gamma)$ is called an isotopic between $L$ and $L'$. We call $(\alpha,\beta,\gamma)$ an isomorphism if $\alpha=\beta=\gamma$, and, in this case, $L$ and $L'$ are said to be ismorphic. Example. Let $G$ be a group and $H$ a subgroup of $G$. Let $T$ be a complete set of left coset representatives (a transversal). Define a multiplication $*$ on $T$ such that $(x*y)H=xyH$. Then $(T,*)$ is a loop.
Example. If $T$ is a transversal of a subgroup $H$ in $G$ and $T$ is a twisted subgroup, then $T$ is a Bolloop. References [FU00] T. Foguel and A. A. Ungar. Involutory decomposition of groups into twisted subgroups and subgroups. J. Group Theory 3 (2000), 2746. [FU01] T. Foguel and A. A. Ungar. Cyrogroups and the decomposition of groups into twisted subgroups and subgroups. Pacific J. Math. 197 (2001), 111. 
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