Quasi-Groups

A permutation x can be regarded as a one-to-one mapping of the integers
{1,2,..,n} to themselves, and we can write j = x(i) to signify that the
permutation x maps i to j. Any group can be represented by a set of
permutations together with the operation of "composition".  

To indicate that z is the composition of two permutations x and y
we write z = xy, and this operation is defined by

                       z(i) = y(x(i))                           (1)

Of course, this operation yields a result z whose effect on a set of n
objects is the same as the effect of first applying x and then y to 
that set objects.  While this is certainly the most natural and useful 
way of defining "composition", it's not the only way in which two
permutations could be "composed" to give a third permutation.

Suppose we define z=xy by the relation

                       z(x(i)) = y(i)                        (2)

It appears that a given set of permutations together with this 
operation constitute a quasi-group if and only if the same set of 
permutations together with ordinary composition (1) constitute a 
group.  In some cases the quasi-group is actually a group, and 
isomorphic to the corresponding group using ordinary composition.  
However, in many cases the quasi-group is not associative and 
does not possess a unique identity element.

For example, consider the set of all possible permutations of 3 items.
The elements of this set are

  a = 123     b = 132     c = 213     d = 231     e = 312     f = 321

The "multiplication tables" for these six elements based on operations
(1) and (2) are shown below:

             a b c d e f               a b c d e f

        a    a b c d e f          a    a b c e d f
        b    b a d c f e          b    b a e c f d
        c    c e a f b d          c    c d a f b e
        d    d f b e a c          d    d c f a e b
        e    e c f a d b          e    e f b d a c
        f    f d e b c a          f    f e d b c a

The right hand table is a quasi-group, because there is no single 
"unit" (note that ad=e) and it is not associative (note that [eb]c=d  
and e[bc]=a).  Each of these has a sub-group (or quasi) of 3 elements

                 a d e                a d e

            a    a d e           a    a e d
            d    d e a           d    d a e
            e    e a d           e    e d a

as well as three isomorphic subgroups of 2 elements, {a,b}, {a,c}, and 
{a,f}.  As these examples confirm, every quasi-group generated by (2)
has the property that xx=a for every x in the set, and of course any
two of x,y,z in the equation xy=z uniquely determine the third (meaning
that each element appears exactly once in each row and each column).

If we consider the twenty-four possible permutations four items

  a = 1234    b = 1243    c = 1324    d = 1342    e = 1423    f = 1432
  g = 2134    h = 2143    i = 2314    j = 2341    k = 2413    l = 2431
  m = 3124    n = 3142    o = 3214    p = 3241    q = 3412    r = 3421
  s = 4123    t = 4132    u = 4213    v = 4231    w = 4312    x = 4321

we find that the following sets of four permutations constitute groups 
with either (1) or (2) as the group operation:

      {a,b,g,h}    {a,c,v,x}    {a,f,o,q}    {a,h,q,x}

and these groups are all isomorphic to

                         a b g h

                    a    a b g h
                    b    b a h g
                    g    g h a b
                    h    h g b a

However, the three sets

               {a,h,r,w}     {a,j,q,s}     {a,k,n,x}

constitute either a group or a quasi-group, depending on whether (1) 
or (2) is taken as the group operation.  In each case, all three are
isomorphic to one of the following:

               a h r w                    a h r w

          a    a h r w                a   a h r w
          h    h a w r                h   h a w r
          r    r w h a                r   w r a h
          w    w r a h                w   r w h a

Axiomatically a quasi-group can be defined as a set S together with
some "multiplication" such that 

   (i)   If x and y are in S then xy and yx are in S.
   (ii)  Any two of x,y,z in the equation xy=z uniquely determine 
         the third (meaning that each element appears exactly once 
         in each row and each column of the group table).

By the way, here's an example of a quasi-group with four elements 
where the condition x*x=u doesn't hold:
 
                  a b c d

              a   b a d c
              b   c b a d
              c   a d c b
              d   d c b a

If this quasi-group is represented by a set of permutations, I'd be 
interested to know the rule of composition that generates this table.  

Questions: 
   -What fraction of all the quasi-groups of a given order
    contain an element u such that xx=u for all x?
   -Can every quasi-group be represented by a set of
    permutations with some "composition" rule?

I suppose in a sense the answer to the second question is trivially 
"yes", because we can dream up an "ad hoc" composition rule that 
generates any given quasi-group when applied to any given set of 
objects.  What I'm really wondering is whether there is a "cannonical
form" of composition rule that will generate all quasi-groups.  For 
example, if I define the operation z=x*y using the formula z(x(y(i)))=i
then the set of permutations {a=1234, b=2143, c=3421, d=4312} yields 
the quasi-group
                      a b d c
                      b a c d
                      d c b a
                      c d a b

whereas the set {a=1342,b=2431,c=3124,d=4213} gives the quasi-group

                      a d b c
                      c b d a
                      d a c b
                      b c a d

This illustrates how a single rule of composition can yield distinct
quasi-groups depending on the choice of elements of S_4 to which it is
applied.  The latter quasi-group can also be generated using the rule
z(i) = y(x(y(x(i)))) on the same four elements of S_4, so for that
particular quasi-group we have a choice of 

      z(x(y(i))) = i        or          z(i) = y(x(y(x(i))))

I imagine we could "index" all the nested formulas of this type and
choose the "lowest" one as the cannonical operation for a given
quasi-group.  My question is whether EVERY quasi-group can be
generated by an operation of this type applied to an appropriate 
set of permutations.