Aequationes Mathematicae 22 (1981) 302-306 University of Waterloo

0001-9054/81/002302-05501.50+0.2010 © 1981 Birkhiiuser Verlag, Basel

A note on Bol loops of order 2"k

KARL RoBrNSON

Abstract. It is shown that, for each n >I 2 and k t> 3, there exist at least 2"-3 non-isomorphic loops of order 2"k which are Bol but not Moufang, In most cases this bound can be improved.

1. Introduction. For the definition of a loop, the reader should consult [1]. A loop (L, .) is said to be a Bol loop if

[ ( a . b ) - c ] . b = a - [ ( b . c ) . b ] for all a ,b , c e L .

(L, .) is said to be a Moufang loop if

[ ( a . b ) - c ] . b = a . [ b . ( c - b ) ] for all a , b , c ~ L .

Moufang loops are Bol loops, the converse is not in general true. In [2] it has been shown that the smallest non-associative Bol loop is of order

8 and that there are six non- isomorphic Bol loops of ~that order. From results in [2] the next possible order is 12. The smallest Moufang loop [4], provides us with at least one non-associative Bol loop of order 12. Here we exhibit non-Moufang (and hence non-associative) Bol loops of order 4k for all integers k~>3. In addition, if n is an integer i>2 we show that there are at least 2"-3 non- isomorphic Bol loops of order 2"k.

It should be noted that, in [3], [4] and [5], it has been shown that non- associative Moufang loops of order 4k exist for all integers k >t 3.

2. L E M M A 1. Let G and H be groups with identity elements 1 and e

AMS (1980) subject classification: Primary 20N05.

Manuscript received March 9, 1979, and, in final form, August 20, 1980.

302

Vol. 22, 1981 A note on Bol loops of order 2"k 303

respectively, and let f : G --~ Aut (H) be a mapping. I f B = G × H and multiplica- tion in B is defined according to

(y, b)(x, a) = (yx, br~)a) for all x, y ~ G, a, b ~ H (1)

(where f (x) : b ~ H ~ b f(~) ~ H), then B is a Bol loop with identity element (1, e) if[

(i) f(xyx) = f ( x ) f ( y ) f ( x ) for all x, y ~ G, (ii) f (1 )= 1H, the identity map on H.

Furthermore, B is Moufang if[ B is associative if[ f is a homomorphism of G into Aut (H).

Proof. B is Bol iff

[[(z, c)(x, a)](y, b)](x, a) = (z, c)[[(x, a)(y, b)](x, a)],

for all (x, a), (y, b), (z, c ) ~ B . But

[[(z, c)(x, a)](y, b)](x, a) = [(zx, cf~*~a)(y, b)](x, a)

= (zxy, c¢(~f(~af(~b)(x, a) = (zxyx, c f (~a f (~ f (~b f (~a) .

and

(z, c)[[(x, a)(y, b)](x, a)] = (z, c)[(xy, af(Y~b)(x, a)]

= (z, c)(xyx, af(Y~f~bf(~a) = (zxyx, c r ( ~ a f ~ f ( ~ b ~ a ) .

Thus B is Bol iff f (xyx) = f ( x ) f ( y ) f ( x ) for all x, y ~ G. That (1, e)(x, a) = (x, a) = (x, a)(1, e) is easily verified. Furthermore, if B satisfies the Moufang identity then

[(x, a)(y, b)](x, a) = (x, a)[(y, b)(x, a)].

But

[(x, a)(y, b)](x, a) = (xy, af(~b)(x, a) = (xyx, af(~>f(~bf(~a)

and

(x, a)[(y, b)(x, a)] = (x, a)(yx, bf('~a) = (xyx, af~Y'°b~("~a).

Thus f ( y x ) = f ( y ) f ( x ) V x , y ~ G is a necessary condition for B to be Moufang.

304 KARL ROBINSON AEO, MATH.

Tha t B is associative iff f (yx) = f (y)f(x) for all x, y ~ G is easily verified, and since groups are also Moufang loops this condition is sufficient for B to be Moufang.

T H E O R E M 1. Let k>-3 and n>12 be integers. There are at least 2"-3 non-isomorphic, non-Moufang Bol loops of order 2nk.

Proof. Le t k >/3, n ~> 2, G be the e lementa ry abelian group of order 2" i.e. G = C 2 x C 2 x . - - x C 2 (n times), H = C k , the cyclic group of order k, and ~-e Aut (H) be ~- : h --~ h -1. Le t us list the elements of G as

G = { x 1 , x 2 , x 3 . . . . . x2,~ }

where we require that xl = 1 and x 4 = X.2X 3. For each integer j such that 4 <~ j <~ 2" define f~ : G --~ Aut (H) by

fi(x~) = {1", if l < i ~ < j , 1H, otherwise.

For each x, y e G , fj(xyx)=f~(y)=f~(x)fi(y)=f~(x)fi(y)fi(x), fi is not a h o m o m o r p h i s m since fi(x2x3)=fi(x4) = ~'~ lr~ = r2 =fi(x2)f~(x3). By L e m m a 1 we are able to construct a Bol loop, having underlying set G × H, which we shall denote by B(2 ", j, k). B(2", ], k) has order 2"k.

We now show that, if 4<~s, t~<2" and s ~ t, then B(2 ", s, k) and B(2 ~, t, k) are not isomorphic. Le t y be a genera tor of H = Ck. In B(2 ~, s, k)

(:q,y~)2 = ~(1, e), if a = 0 or l<i<~s, [(1, y2,,), otherwise.

Thus the number of e lements of o rder 2 in B(2", s, k) (including the identity e lement) is 2" + (k - 1)(s - 1) + ~k(2" - s + 1) where

1, if k is even, 8 k = O, i f k i s o d d .

Similarly, the number of e lements of order 2 in B(2 ", t, k) is 2" + ( k - 1 ) ( t - I ) + 8k(2" - t + 1). If s ~ t these numbers are distinct and thus B(2", s, k) and B(2 ", t, k) are non-isomorphic .

C O R O L L A R Y . For each integer k >13, there exists at least one non-Moufang Bol loop of order 4k.

Proof. This follows f rom T h e o r e m 1 with n = 2.

Vol. 22, 1981 A note on Bol loops of order 2"k 305

R e m a r k 1. In most cases the lower bound provided in Theorem 1 can be considerably improved. In fact, if k >t 3 is odd and n I>2, there are at least 3.2 " - 1 - 3n + 2 non-isomorphic Bol Loops of order 2~k. Also for n > 3 there are at least 2 " - 1 - 3n + 5 non-isomorphic Bol loops of order 2". These results can be proven as follows: if l >~ 3 is even, r >t 2 and 4 ~< s <~ 2 r, there is an element of order I in B(2 r, s , / ) , viz. (1, y) where y is a generator of H = Ci, also (xi, y,)Z =(1 , e) for all (x~, y~)~B(2r , s, l). If l is odd and s # 2 r, there is an element of order 21 in B(2r, s , l ) , viz. (xi, y) where s < i ~ 2 ~ and y is a generator of H = C I , also (xi, y~)Zl = (1, e) for all (x~, y~) ~ B(2 r, s, l). If s = 2 ~, then B(2 r, s, l) has exponent l.

Thus, if k is even, the 2 "+~- 3 n - 1 loops B(2 "-r, s, 2"k) 0 ~< r ~ n - 2 , 4 ~ < s <~ 2 "-~, each of order 2"k, are non-isomorphic. This, however, provides an improve- ment on the bound provided in Theorem 1 only when k = 4. Since, if k # 4, then k = 21, l >t 3, and thus, by Theorem 1, there are at least 2 "+t - 3 Bol loops of order 2"k = 2~+1/. If k is odd then the 3.2 "-1 - 3n + 2 loops B(2 ~-~, s, 2~k), 0 ~ r <~ n - 2, r ~ 1, 4<~s~<2 " ~ are all non-isomorphic. To determine whether the loops B ( 2 " - ' , s , 2~k) for r = 0 , 1 , 4~<s<~2 "-~ and odd k ~ 3 are all non-isomorphic seems to require deeper analysis. It would also be of interest to determine which of the above loops, ff any, are isotopic to each other.

R e m a r k 2. Mappings satisfying conditions (i) and (ii) of Lemma 1 have occurred in another context in the study of Bol loops, namely the special embeddings of Bol loops in groups of [6].

Acknowledgement

The author wishes to suggestions.

thank the referees for their helpful comments and

REFERENCES

[1] BRUCK, R. H., A survey of binary systems. Springer-Verlag, Berlin, 1958. [2] BURN, R. P., Finite BoI loops. Math. Proc. Cambridge Philos. Soc. 84 (1978), 377-385. [3] CI-mlN, O., Moufang loops of small order. Mem. Amer. Math. Soc., #197, (1978). [4] CREIN, O. and PFLUGFELDER, H. 0., The smallest Moufang loop. Arch. Math. (Basel) 22 (1971),

573-576. [5] CHEIN, O. and PFLUGFELDER, H. O., On maps x ~ x" and the isotopy-isomorphy properly of

Moufang loops. Aequationes Math. 6 (1971), 157-161.

306 KARL ROBINSON AEQ. MATH.

[6] ROBINSON, D. A., A special embedding of Bol loops in groups. Acta Math. Acad. Sci. Hungar. 30 (1977), 95-103.

Department of Mathematics, University of the West Indies,

Mona, Kingston 7, Jamaica.