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2 Mathematical Background

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.

2.1 Quasigroups and Loops

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.

2.2 Translations

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 $$Q$$.

2.3 Subquasigroups and Subloops

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 $$Q$$.

2.4 Nilpotence and Solvability

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 $$Q$$.

2.5 Associators and Commutators

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.

2.6 Homomorphism and Homotopisms

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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