We assume that you are familiar with the theory of quasigroups and loops, for instance with the textbook of Bruck [Bru58] or Pflugfelder [Pfl90]. Nevertheless, we did include definitions and results in this manual in order to unify terminology and improve legibility of the text. Some general concepts of quasigroups and loops can be found in this chapter. More special concepts are defined throughout the text as needed.

A set with one binary operation (denoted \(\cdot\) here) is called *groupoid* or *magma*, the latter name being used in **GAP**.

An element \(1\) of a groupoid \(G\) is a *neutral element* or an *identity element* if \(1\cdot x = x\cdot 1 = x\) for every \(x\) in \(G\).

Let \(G\) be a groupoid with neutral element \(1\). Then an element \(x^{-1}\) is called a *two-sided inverse* of \(x\) in \(G\) if \( x\cdot x^{-1} = x^{-1}\cdot x = 1\).

Recall that groups are associative groupoids with an identity element and two-sided inverses. Groups can be reached in another way from groupoids, namely via quasigroups and loops.

A *quasigroup* \(Q\) is a groupoid such that the equation \(x\cdot y=z\) has a unique solution in \(Q\) whenever two of the three elements \(x\), \(y\), \(z\) of \(Q\) are specified. Note that multiplication tables of finite quasigroups are precisely *latin squares*, i.e., square arrays with symbols arranged so that each symbol occurs in each row and in each column exactly once. A *loop* \(L\) is a quasigroup with a neutral element.

Groups are clearly loops. Conversely, it is not hard to show that associative quasigroups are groups.

Given an element \(x\) of a quasigroup \(Q\), we can associative two permutations of \(Q\) with it: the *left translation* \(L_x:Q\to Q\) defined by \(y\mapsto x\cdot y\), and the *right translation* \(R_x:Q\to Q\) defined by \(y\mapsto y\cdot x\).

The binary operation \(x\backslash y = L_x^{-1}(y)\) is called the *left division*, and \(x/y = R_y^{-1}(x)\) is called the *right division*.

Although it is possible to compose two left (right) translations of a quasigroup, the resulting permutation is not necessarily a left (right) translation. The set \(\{L_x|x\in Q\}\) is called the *left section* of \(Q\), and \(\{R_x|x\in Q\}\) is the *right section* of \(Q\).

Let \(S_Q\) be the symmetric group on \(Q\). Then the subgroup \({\rm Mlt}_{\lambda}(Q)=\langle L_x|x\in Q\rangle\) of \(S_Q\) generated by all left translations is the *left multiplication group* of \(Q\). Similarly, \({\rm Mlt}_{\rho}(Q)= \langle R_x|x\in Q\rangle\) is the *right multiplication group* of \(Q\). The smallest group containing both \({\rm Mlt}_{\lambda}(Q)\) and \({\rm Mlt}_{\rho}(Q)\) is called the *multiplication group* of \(Q\) and is denoted by \({\rm Mlt}(Q)\).

For a loop \(Q\), the *left inner mapping group* \({\rm Inn}_{\lambda}(Q)\) is the stabilizer of \(1\) in \({\rm Mlt}_{\lambda}(Q)\). The *right inner mapping group* \({\rm Inn}_{\rho}(Q)\) is defined dually. The *inner mapping group* \({\rm Inn}(Q)\) is the stabilizer of \(1\) in \(Q\).

A nonempty subset \(S\) of a quasigroup \(Q\) is a *subquasigroup* if it is closed under multiplication and the left and right divisions. In the finite case, it suffices for \(S\) to be closed under multiplication. *Subloops* are defined analogously when \(Q\) is a loop.

The *left nucleus* \({\rm Nuc}_{\lambda}(Q)\) of \(Q\) consists of all elements \(x\) of \(Q\) such that \(x(yz) = (xy)z\) for every \(y\), \(z\) in \(Q\). The *middle nucleus* \({\rm Nuc}_{\mu}(Q)\) and the *right nucleus* \({\rm Nuc}_{\rho}(Q)\) are defined analogously. The *nucleus* \({\rm Nuc}(Q)\) is the intersection of the left, middle and right nuclei.

The *commutant* \(C(Q)\) of \(Q\) consists of all elements \(x\) of \(Q\) that commute with all elements of \(Q\). The *center* \(Z(Q)\) of \(Q\) is the intersection of \({\rm Nuc}(Q)\) with \(C(Q)\).

A subloop \(S\) of \(Q\) is *normal* in \(Q\) if \(f(S)=S\) for every inner mapping \(f\) of \(Q\).

For a loop \(Q\) define \(Z_0(Q) = 1\) and let \(Z_{i+1}(Q)\) be the preimage of the center of \(Q/Z_i(Q)\) in \(Q\). A loop \(Q\) is *nilpotent of class* \(n\) if \(n\) is the least nonnegative integer such that \(Z_n(Q)=Q\). In such case \(Z_0(Q)\le Z_1(Q)\le \dots \le Z_n(Q)\) is the *upper central series*.

The *derived subloop* \(Q'\) of \(Q\) is the least normal subloop of \(Q\) such that \(Q/Q'\) is a commutative group. Define \(Q^{(0)}=Q\) and let \(Q^{(i+1)}\) be the derived subloop of \(Q^{(i)}\). Then \(Q\) is *solvable of class* \(n\) if \(n\) is the least nonnegative integer such that \(Q^{(n)} = 1\). In such a case \(Q^{(0)}\ge Q^{(1)}\ge \cdots \ge Q^{(n)}\) is the *derived series* of \(Q\).

Let \(Q\) be a quasigroup and let \(x\), \(y\), \(z\) be elements of \(Q\). Then the *commutator* of \(x\), \(y\) is the unique element \([x,y]\) of \(Q\) such that \(xy = [x,y](yx)\), and the *associator* of \(x\), \(y\), \(z\) is the unique element \([x,y,z]\) of \(Q\) such that \((xy)z = [x,y,z](x(yz))\).

The *associator subloop* \(A(Q)\) of \(Q\) is the least normal subloop of \(Q\) such that \(Q/A(Q)\) is a group.

It is not hard to see that \(A(Q)\) is the least normal subloop of \(Q\) containing all commutators, and \(Q'\) is the least normal subloop of \(Q\) containing all commutators and associators.

Let \(K\), \(H\) be two quasigroups. Then a map \(f:K\to H\) is a *homomorphism* if \(f(x)\cdot f(y)=f(x\cdot y)\) for every \(x\), \(y\in K\). If \(f\) is also a bijection, we speak of an *isomorphism*, and the two quasigroups are called isomorphic.

An ordered triple \((\alpha,\beta,\gamma)\) of maps \(\alpha\), \(\beta\), \(\gamma:K\to H\) is a *homotopism* if \(\alpha(x)\cdot\beta(y) = \gamma(x\cdot y)\) for every \(x\), \(y\) in \(K\). If the three maps are bijections, then \((\alpha,\beta,\gamma)\) is an *isotopism*, and the two quasigroups are isotopic.

Isotopic groups are necessarily isomorphic, but this is certainly not true for nonassociative quasigroups or loops. In fact, every quasigroup is isotopic to a loop.

Let \((K,\cdot)\), \((K,\circ)\) be two quasigroups defined on the same set \(K\). Then an isotopism \((\alpha,\beta,{\rm id}_K)\) is called a *principal isotopism*. An important class of principal isotopisms is obtained as follows: Let \((K,\cdot)\) be a quasigroup, and let \(f\), \(g\) be elements of \(K\). Define a new operation \(\circ\) on \(K\) by \(x\circ y = R_g^{-1}(x)\cdot L_f^{-1}(y)\), where \(R_g\), \(L_f\) are translations. Then \((K,\circ)\) is a quasigroup isotopic to \((K,\cdot)\), in fact a loop with neutral element \(f\cdot g\). We call \((K,\circ)\) a *principal loop isotope* of \((K,\cdot)\).

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