  
  [1X2 [33X[0;0YMathematical Background[133X[101X
  
  [33X[0;0YWe  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.[133X
  
  
  [1X2.1 [33X[0;0YQuasigroups and Loops[133X[101X
  
  [33X[0;0YA  set  with  one  binary  operation  (denoted [22X⋅[122X here) is called [13Xgroupoid[113X or
  [13Xmagma[113X, the latter name being used in [5XGAP[105X.[133X
  
  [33X[0;0YAn  element [22X1[122X of a groupoid [22XG[122X is a [13Xneutral element[113X or an [13Xidentity element[113X if
  [22X1⋅ x = x⋅ 1 = x[122X for every [22Xx[122X in [22XG[122X.[133X
  
  [33X[0;0YLet [22XG[122X be a groupoid with neutral element [22X1[122X. Then an element [22Xx^-1[122X is called a
  [13Xtwo-sided inverse[113X of [22Xx[122X in [22XG[122X if [22Xx⋅ x^-1 = x^-1⋅ x = 1[122X.[133X
  
  [33X[0;0YRecall  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.[133X
  
  [33X[0;0YA  [13Xquasigroup[113X  [22XQ[122X  is  a  groupoid such that the equation [22Xx⋅ y=z[122X has a unique
  solution in [22XQ[122X whenever two of the three elements [22Xx[122X, [22Xy[122X, [22Xz[122X of [22XQ[122X are specified.
  Note  that  multiplication  tables of finite quasigroups are precisely [13Xlatin
  squares[113X,  i.e.,  square  arrays  with  symbols  arranged so that each symbol
  occurs in each row and in each column exactly once. A [13Xloop[113X [22XL[122X is a quasigroup
  with a neutral element.[133X
  
  [33X[0;0YGroups  are  clearly  loops.  Conversely,  it  is  not  hard  to  show  that
  associative quasigroups are groups.[133X
  
  
  [1X2.2 [33X[0;0YTranslations[133X[101X
  
  [33X[0;0YGiven an element [22Xx[122X of a quasigroup [22XQ[122X, we can associative two permutations of
  [22XQ[122X  with it: the [13Xleft translation[113X [22XL_x:Q-> Q[122X defined by [22Xy↦ x⋅ y[122X, and the [13Xright
  translation[113X [22XR_x:Q-> Q[122X defined by [22Xy↦ y⋅ x[122X.[133X
  
  [33X[0;0YThe  binary  operation [22Xxbackslash y = L_x^-1(y)[122X is called the [13Xleft division[113X,
  and [22Xx/y = R_y^-1(x)[122X is called the [13Xright division[113X.[133X
  
  [33X[0;0YAlthough  it  is  possible  to  compose  two  left (right) translations of a
  quasigroup,  the  resulting  permutation  is  not necessarily a left (right)
  translation. The set [22X{L_x|x∈ Q}[122X is called the [13Xleft section[113X of [22XQ[122X, and [22X{R_x|x∈
  Q}[122X is the [13Xright section[113X of [22XQ[122X.[133X
  
  [33X[0;0YLet  [22XS_Q[122X be the symmetric group on [22XQ[122X. Then the subgroup [22XMlt_λ(Q)=⟨ L_x|x∈ Q⟩[122X
  of  [22XS_Q[122X  generated by all left translations is the [13Xleft multiplication group[113X
  of  [22XQ[122X. Similarly, [22XMlt_ρ(Q)= ⟨ R_x|x∈ Q⟩[122X is the [13Xright multiplication group[113X of
  [22XQ[122X.  The  smallest  group containing both [22XMlt_λ(Q)[122X and [22XMlt_ρ(Q)[122X is called the
  [13Xmultiplication group[113X of [22XQ[122X and is denoted by [22XMlt(Q)[122X.[133X
  
  [33X[0;0YFor  a  loop [22XQ[122X, the [13Xleft inner mapping group[113X [22XInn_λ(Q)[122X is the stabilizer of [22X1[122X
  in  [22XMlt_λ(Q)[122X.  The [13Xright inner mapping group[113X [22XInn_ρ(Q)[122X is defined dually. The
  [13Xinner mapping group[113X [22XInn(Q)[122X is the stabilizer of [22X1[122X in [22XQ[122X.[133X
  
  
  [1X2.3 [33X[0;0YSubquasigroups and Subloops[133X[101X
  
  [33X[0;0YA  nonempty  subset  [22XS[122X  of a quasigroup [22XQ[122X is a [13Xsubquasigroup[113X if it is closed
  under  multiplication  and the left and right divisions. In the finite case,
  it  suffices  for  [22XS[122X to be closed under multiplication. [13XSubloops[113X are defined
  analogously when [22XQ[122X is a loop.[133X
  
  [33X[0;0YThe  [13Xleft  nucleus[113X  [22XNuc_λ(Q)[122X  of [22XQ[122X consists of all elements [22Xx[122X of [22XQ[122X such that
  [22Xx(yz) = (xy)z[122X for every [22Xy[122X, [22Xz[122X in [22XQ[122X. The [13Xmiddle nucleus[113X [22XNuc_μ(Q)[122X and the [13Xright
  nucleus[113X  [22XNuc_ρ(Q)[122X  are  defined  analogously.  The  [13Xnucleus[113X  [22XNuc(Q)[122X  is  the
  intersection of the left, middle and right nuclei.[133X
  
  [33X[0;0YThe  [13Xcommutant[113X  [22XC(Q)[122X  of [22XQ[122X consists of all elements [22Xx[122X of [22XQ[122X that commute with
  all  elements  of [22XQ[122X. The [13Xcenter[113X [22XZ(Q)[122X of [22XQ[122X is the intersection of [22XNuc(Q)[122X with
  [22XC(Q)[122X.[133X
  
  [33X[0;0YA subloop [22XS[122X of [22XQ[122X is [13Xnormal[113X in [22XQ[122X if [22Xf(S)=S[122X for every inner mapping [22Xf[122X of [22XQ[122X.[133X
  
  
  [1X2.4 [33X[0;0YNilpotence and Solvability[133X[101X
  
  [33X[0;0YFor  a  loop  [22XQ[122X  define  [22XZ_0(Q)  = 1[122X and let [22XZ_i+1(Q)[122X be the preimage of the
  center  of [22XQ/Z_i(Q)[122X in [22XQ[122X. A loop [22XQ[122X is [13Xnilpotent of class[113X [22Xn[122X if [22Xn[122X is the least
  nonnegative  integer such that [22XZ_n(Q)=Q[122X. In such case [22XZ_0(Q)le Z_1(Q)le dots
  le Z_n(Q)[122X is the [13Xupper central series[113X.[133X
  
  [33X[0;0YThe  [13Xderived subloop[113X [22XQ'[122X of [22XQ[122X is the least normal subloop of [22XQ[122X such that [22XQ/Q'[122X
  is  a  commutative  group.  Define  [22XQ^(0)=Q[122X  and  let [22XQ^(i+1)[122X be the derived
  subloop  of  [22XQ^(i)[122X.  Then  [22XQ[122X  is  [13Xsolvable  of  class[113X  [22Xn[122X  if  [22Xn[122X is the least
  nonnegative integer such that [22XQ^(n) = 1[122X. In such a case [22XQ^(0)ge Q^(1)ge ⋯ ge
  Q^(n)[122X is the [13Xderived series[113X of [22XQ[122X.[133X
  
  
  [1X2.5 [33X[0;0YAssociators and Commutators[133X[101X
  
  [33X[0;0YLet  [22XQ[122X be a quasigroup and let [22Xx[122X, [22Xy[122X, [22Xz[122X be elements of [22XQ[122X. Then the [13Xcommutator[113X
  of  [22Xx[122X,  [22Xy[122X is the unique element [22X[x,y][122X of [22XQ[122X such that [22Xxy = [x,y](yx)[122X, and the
  [13Xassociator[113X  of  [22Xx[122X, [22Xy[122X, [22Xz[122X is the unique element [22X[x,y,z][122X of [22XQ[122X such that [22X(xy)z =
  [x,y,z](x(yz))[122X.[133X
  
  [33X[0;0YThe  [13Xassociator subloop[113X [22XA(Q)[122X of [22XQ[122X is the least normal subloop of [22XQ[122X such that
  [22XQ/A(Q)[122X is a group.[133X
  
  [33X[0;0YIt  is not hard to see that [22XA(Q)[122X is the least normal subloop of [22XQ[122X containing
  all  commutators,  and  [22XQ'[122X  is  the least normal subloop of [22XQ[122X containing all
  commutators and associators.[133X
  
  
  [1X2.6 [33X[0;0YHomomorphism and Homotopisms[133X[101X
  
  [33X[0;0YLet  [22XK[122X,  [22XH[122X be two quasigroups. Then a map [22Xf:K-> H[122X is a [13Xhomomorphism[113X if [22Xf(x)⋅
  f(y)=f(x⋅  y)[122X  for  every  [22Xx[122X, [22Xy∈ K[122X. If [22Xf[122X is also a bijection, we speak of an
  [13Xisomorphism[113X, and the two quasigroups are called isomorphic.[133X
  
  [33X[0;0YAn ordered triple [22X(α,β,γ)[122X of maps [22Xα[122X, [22Xβ[122X, [22Xγ:K-> H[122X is a [13Xhomotopism[113X if [22Xα(x)⋅β(y)
  =  γ(x⋅  y)[122X  for  every  [22Xx[122X,  [22Xy[122X  in [22XK[122X. If the three maps are bijections, then
  [22X(α,β,γ)[122X is an [13Xisotopism[113X, and the two quasigroups are isotopic.[133X
  
  [33X[0;0YIsotopic  groups  are necessarily isomorphic, but this is certainly not true
  for  nonassociative  quasigroups  or  loops.  In  fact,  every quasigroup is
  isotopic to a loop.[133X
  
  [33X[0;0YLet  [22X(K,⋅)[122X,  [22X(K,∘)[122X  be  two  quasigroups  defined on the same set [22XK[122X. Then an
  isotopism [22X(α,β, id_K)[122X is called a [13Xprincipal isotopism[113X. An important class of
  principal  isotopisms is obtained as follows: Let [22X(K,⋅)[122X be a quasigroup, and
  let  [22Xf[122X,  [22Xg[122X  be  elements  of  [22XK[122X.  Define  a  new  operation [22X∘[122X on [22XK[122X by [22Xx∘ y =
  R_g^-1(x)⋅  L_f^-1(y)[122X,  where  [22XR_g[122X,  [22XL_f[122X  are  translations. Then [22X(K,∘)[122X is a
  quasigroup  isotopic  to [22X(K,⋅)[122X, in fact a loop with neutral element [22Xf⋅ g[122X. We
  call [22X(K,∘)[122X a [13Xprincipal loop isotope[113X of [22X(K,⋅)[122X.[133X
  
