## 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 of 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. For
example, suppose we define z=xy by the relation

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

A given set of permutations together with this operation does not
necessarily constitute a group, but it appears that if the same set
of permutations together with ordinary composition (1) constitute a
group, then those permutation with composition (2) constitute a structure
that we may call a "quasi-group", which we define 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).

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
three 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

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.
```

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