[loops] 2 Mathematical background

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# 2 Mathematical background

### Sections

1. Quasigroups and loops
2. Translations
3. Homomorphisms and homotopisms
4. Extensions

We assume that you are familiar with the theory of quasigroups and loops, for instance with the textbook of Bruck Br or Pflugfelder Pf. Nevertheless, we did include definitions and results in this manual in order to unify the terminology and improve the intelligibility 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 · here) is called groupoid or magma, the latter name being used in GAP. Associative groupoid is a semigroup.

An element 1 of a groupoid G is a neutral element or an identity element if 1·x = x·1 = x for every x in G. Semigroup with a neutral element is a monoid.

Let G be a groupoid with neutral element 1. Then an element y is called a two-sided inverse of x in G if x·y = y·x = 1. A monoid in which every element has a two-sided inverse is called a group.

Groups can be reached in another way from groupoids, namely through quasigroups and loops.

A quasigroup Q is a groupoid such that the equation x·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., a 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, and one can show easily that an associative quasigroup is a group.

## 2.2 Translations

Given an element x of a quasigroup Q we can associative two permutations of Q with it: the left translation Lx:Q→ Q defined by y→ x·y, and the right translation Rx:Q→ Q defined by y→ y· x.

Although it is possible to compose two right (left) translations, the resulting permutation is not necessarily a right (left) translation. The set {Lx;x ∈ Q} is called the left section of Q, and {Rx;x ∈ Q} is the right section of Q.

Let SQ be the symmetric group on Q. Then the subgroup LMlt(Q)=〈Lx|x ∈ Q〉 of SQ generated by all left translations is the left multiplication group of Q. Similarly, RMlt(Q) = 〈Rx|x ∈ Q〉 is the right multiplication group of Q. The smallest group containing both LMlt(Q) and RMlt(Q) is called the multiplication group of Q and is denoted by Mlt(Q).

## 2.3 Homomorphisms and homotopisms

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

The ordered triple (α,β,γ) of maps α, β, γ:K→ H is a homotopism if α(x)·β(y) = γ(x·y) for every x, y ∈ K. If the three maps are bijections, (α,β,γ) 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, as we shall see.

Let (K,·), (K,°) be two quasigroups defined on the same set K. Then an isotopism (α,β,idK) is called a principal isotopism. An important class of principal isotopisms is obtained as follows:

Let (K,·) be a quasigroup, and let f, g be elements of K. Define a new operation ° on K by

x°y = Rg−1(x)·Lf−1(y),

where Rg, Lf are translations. Then (K,°) is a quasigroup isotopic to (K,·), in fact a loop with neutral element f·g. We call (K,°) a principal loop isotope of (K,·).

## 2.4 Extensions

Let K, F be loops. Then a loop Q is an extension of K by F if K is a normal subloop of Q such that Q/K is isomorphic to F. An extension Q of K by F is nuclear if K is an abelian group and K ≤ N(Q).

A map θ:F×F→ K is a cocycle if θ(1,x) = θ(x,1) = 1 for every x ∈ F.

The following theorem holds for loops Q, F and an abelian group K: Q is a nuclear extension of K by F if and only if there is a cocycle θ:F×F→ K and a homomorphism ϕ:F→AutQ such that K×F with multiplication (a,x)(b,y) = (aϕx(b)θ(x,y),xy) is isomorphic to Q.

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