Loops

A magma is a set with a binary operation.
An element $e$ in a magma $M$ is an identity element if $em=me=m$ for all $m\in M$. A groupoid is a magma with an identity.
A quasigroup is a magma $M$ with the property that for all $a,b\in M$, the equations $ax=b$ and $ya=b$ have unique solutions in $M$.

Proposition. An associative quasigroup is a groupoid.

A loop is a quasigroup with an identity element.

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 Bol-loop.

Consider two magmas $(L,\cdot)$ and $(L',\circ)$. If there are mappings $\alpha$, $\beta$, $\gamma$ from $L$ to $L'$ such that

$$\alpha(x)\circ\beta(y)=\gamma(x\cdot y)\quad \mbox{for all }x,y\in L,$$ 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.

A subset $T$ of a group $G$ is called a twisted subgroup of $G$ if (1) $1\in T$, (2) $T^{-1}=T$, (3) $xTx\subseteq T$ for all $x\in T$. We may call $T$ a gyrosubgroup of $G$ if (3) is replaced by (3') $xTx^{-1}\subseteq T$ for all $x\in T$ (cf. [FU00] and [FU01]).

Example. If $T$ is a transversal of a subgroup $H$ in $G$ and $T$ is a twisted subgroup, then $T$ is a Bol-loop.

References

[FU00] T. Foguel and A. A. Ungar. Involutory decomposition of groups into twisted subgroups and subgroups. J. Group Theory 3 (2000), 27--46.

[FU01] T. Foguel and A. A. Ungar. Cyrogroups and the decomposition of groups into twisted subgroups and subgroups. Pacific J. Math. 197 (2001), 1--11.