60 V. BeloUSOV AEQ. MATH.

or

The converse statement is obvious.

Definition . A permutation ф is said to be special for the loop (•) if the mapping Ф' defined by х-^фх = хфх~^ (x~^ the right inverse of x) is also a permutation.

Lemma 2. Let ф be a special permutation for the loop ß(-) and let ф satisfy (17).

Then {•) is a quasigroup operation. Ф Proof The loop ( • ) is distributive with respect to ( • ) :

Ф x{y.z) = xy.xz. Ф Ф

If z = 0 - the unit element for the loop Q{'), then

x ( y , 0 ) = xy.x. (21)

Ф Ф

On the other hand у.О=^уф(у\0)=уфу~'^=фу, since y\0=y~^ {УУ~^=0). From

Ф (21) we get

хф' у = xy.x. (22)

Ф

Let xy = z i.e. y = x\z, therefore we find from (22):

z . x = хф' {x\z).

Ф

The equations a.x = b, y.a = b sltq equivalent to аф{а\х) = Ь and аф'{а\у) = Ь respec-

Ф Ф

tively . It follows that (•) is a quasigroup if and only if ф and ф' are permutations. Ф Corollary.

^х , уФ' = Ф' ^x,y^ (23)

хф'х'^ = фх. (24)

For , we have

^1уФ' { у\^У ] = 2[.хФ{АуУ] = zx.ф{zx\zy) = zy.ф'{zy\zx), i.e.

Ь^Ьуф'Uy x = Ь^уф'L~y L^x.

From the last equality we obtain (23). The second equality (24) is obvious :

хф'х~^ =хф'{х\0) = Оф{0\х) = фх.

Lemma 3. IfA = (*)for some special permutation ф satisfying (17) then A~^={-), Ф A

~^A= ( ' ) , where к and[i are also special satisfying (17).