loops/doc/chap2.txt

  
  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 ⋅ 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⋅ x = x⋅ 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⋅ x^-1 = x^-1⋅ 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⋅ 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-> Q defined by y↦ x⋅ y, and the right
  translation R_x:Q-> Q defined by y↦ y⋅ x.
  
  The  binary  operation xbackslash 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∈ Q} is called the left section of Q, and {R_x|x∈
  Q} is the right section of Q.
  
  Let  S_Q be the symmetric group on Q. Then the subgroup Mlt_λ(Q)=⟨ L_x|x∈ Q⟩
  of  S_Q  generated by all left translations is the left multiplication group
  of  Q. Similarly, Mlt_ρ(Q)= ⟨ R_x|x∈ Q⟩ is the right multiplication group of
  Q.  The  smallest  group containing both Mlt_λ(Q) and Mlt_ρ(Q) is called the
  multiplication group of Q and is denoted by Mlt(Q).
  
  For  a  loop Q, the left inner mapping group Inn_λ(Q) is the stabilizer of 1
  in  Mlt_λ(Q).  The right inner mapping group Inn_ρ(Q) is defined dually. The
  inner mapping group 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  Nuc_λ(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 Nuc_μ(Q) and the right
  nucleus  Nuc_ρ(Q)  are  defined  analogously.  The  nucleus  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 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 ⋯ 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-> 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.
  
  An 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  in K. If the three maps are bijections, then
  (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.
  
  Let  (K,⋅),  (K,∘)  be  two  quasigroups  defined on the same set K. Then an
  isotopism (α,β, id_K) 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 =
  R_g^-1(x)⋅  L_f^-1(y),  where  R_g,  L_f  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,⋅).