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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