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A New Construction of Bol Loops of Order 8K

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A New Construction of Bol Loops of Order 8K View metadata, citation and similar papers at core.ac.uk brought to you by CORE Journal of Algebra 287 (2005) 103–122 provided by Elsevier - Publisher Connector www.elsevier.com/locate/jalgebra A new construction of Bol loops of order 8k Orin Chein a,∗, Edgar G. Goodaire b,1 a Temple University, Philadelphia, PA 19122, USA b Memorial University of Newfoundland, St. John’s, Newfoundland, Canada A1C 5S7 Received 9 December 2003 Communicated by Efim Zelmanov Abstract One of the most interesting constructions of nonassociative Moufang loops is the construction by the first author of the loops M(G,2) [Trans. Amer. Math. Soc. 188 (1974) 31–51], in which the new loop is constructed as a split extension of a nonabelian group G by a cyclic group of order 2. This same construction will not produce non-Moufang Bol loops, even if we start with a Moufang loop instead of a group. We generalize the construction to produce a large class of Bol loops as extensions of B by Cm × Cn,whereB can be any group or even any loop which satisfies the right Bol identity. We determine conditions under which the newly constructed loops will be Moufang and when they will be associative. We also find the nuclei, centrum, centre, and commutator/associator subloop of a loop constructed by this method, and we investigate which known Bol loops of small orders arise in this way. 2005 Elsevier Inc. All rights reserved. Keywords: Bol loop; Moufang loop; Centre; Construction * Corresponding author. E-mail addresses: [email protected] (O. Chein), [email protected] (E.G. Goodaire). 1 The author is grateful for the hospitality provided by the Department of Mathematics of Temple University where he was a visitor while this work was completed. His research was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada. 0021-8693/$ – see front matter 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jalgebra.2005.01.031 104 O. Chein, E.G. Goodaire / Journal of Algebra 287 (2005) 103–122 1. Introduction A(right) Bol loop is a loop which satisfies the (right) Bol identity (xy · z)y = x(yz· y), and a Moufang loop is a loop which satisfies any of three equivalent identities, one of which is the right Moufang identity (xy · z)y = x(y · zy). Every Moufang loop is a Bol loop, but not conversely. In fact, a Bol loop is Moufang if and only if it satisfies the left alternative law, x · xy = x2y, or, equivalently, if and only if it satisfies the flexible law, xy · x = x · yx [18, Theorem 2.7]. (Note: Both Bol and Moufang loops always satisfy the right alternative law, xy2 = xy · y.) Named after Garrit Bol, Bol loops came into prominence with the fundamental work of D.A. Robinson [18]. Today, they form an active area of research from many points of view. We refer the reader to [2] or [16] for more information about both Moufang and Bol loops. There are many papers in the literature, among them [1,3,7,8,11–13,15,17,19–25,27] that describe methods of constructing Bol loops. In some of these, the Bol loop is con- structed as a central extension of a group. In this paper, we introduce a new construction that produces a large number of Bol loops of order 8k as extensions of groups or loops of order 2k. Any group is a Moufang loop and, in some respects, Moufang loops in general are not far removed from groups. They are, for example, diassociative (the subloop generated by any two elements is always associative) and there are a number of constructions of Mo- ufang loops which consist of “doubling” a group in various ways. (See, for example, [4], [5], [9, §II.5] and [26].) In the latter paper, Vojtechovský˘ claims that this construction can- not result in a Bol loop which is not Moufang, but he corrects this claim in [27], where he not only shows that indeed this construction can result in a Bol loop, but he also determines exactly when a non-Moufang Bol loop will occur. 2. Main results We begin with a preliminary construction. Let m be a fixed positive integer, and, for every integer i,leti denote the least nonneg- ative residue of i, modulo m. Lemma 2.1. Let B be a loop, and let z be any fixed element in the centre of B, and let m be a positive integer. Define an operation on the set G = B × Cm by [a,i][b,j]= [abzq ,(i+ j)], where i + j − (i + j) 0 if i + j<m, q = = m 1 otherwise. Then G is a loop. Furthermore, G is Bol (respectively Moufang, respectively associative, respectively commutative) if and only if B is Bol (respectively Moufang, respectively asso- ciative, respectively commutative). Proof. For [a,i] and [b,j] in G,letk = (j − i) and let c = dz−q , where d is the unique element of B such that ad = b, and where q = (i + k − (i + k))/m. Then [a,i][c,k]= O. Chein, E.G. Goodaire / Journal of Algebra 287 (2005) 103–122 105 [adz−q zq ,(i + k)]=[b,j]. Similarly, if k and q are as above, e is the unique element of B such that ea = b, and f = ez−q , then [f,k][a,i]=[b,j]. Thus, G is a quasigroup. Clearly, (1, 0) is an identity element, so G is a loop. Since B =∼ B ={[b,0]|b ∈ B} is a subloop of G, B must be Bol (respectively Moufang, respectively associative) if G is. Conversely, suppose that B is a Bol loop. (In the calculations which follow, it is helpful to note that (r + s) = (r + s), for any integers r and s.) Let [a,i], [b,j] and [c,k] be three elements of G. Then [a,i][b,j]·[c,k] [b,j]= abzq1 ,(i+ j) [c,k] [b,j] + = ab · czq1 q2 ,(i+ j + k) [b,j] + + = (ab · c)b zq1 q2 q3 ,(i+ 2j + k) , where q1, q2 and q3 are defined by i + j = q1m + (i + j) , i + j + k = (q1 + q2)m + (i + j + k) , and i + 2j + k = (q1 + q2 + q3)m + (i + 2j + k) . Similarly, [a,i] [b,j][c,k]·[b,j] =[a,i] bczq4 ,(j + k) [b,j] + =[a,i] (bc · b)zq4 q5 ,(2j + k) + + = a(bc · b)zq4 q5 q6 ,(i+ 2j + k) , where q4, q5 and q6 are defined by j +k = q4m+(j +k) , i +2j = (q4 +q5)m+(i +2j) , and i +2j +k = (q4 +q5 +q6)m+(i +2j +k) . It follows that q1 +q2 +q3 = q4 +q5 +q6 and so the Bol identity holds in G. Furthermore, + [a,i][b,j] [a,i]= abzq1 ,(i+ j) [a,i]= (ab)azq1 q2 ,(2i + j) , and + [a,i] [b,j][a,i] =[a,i] bazq1 ,(j + i) = a(ba)zq1 q2 ,(2i + j) , where i +j = j +i = q1m+(i +j) and 2i +j = (q1 +q2)m+(2i +j) . Thus, G satisfies the flexible law {[a,i][b,j]}[a,i]=[a,i]{[b,j][a,i]} if and only if (ab)a = a(ba) holds for all a, b in B, i.e., if and only if B satisfies the flexible law. But a Bol loop is Moufang if and only if it satisfies the flexible law, so G is Moufang if and only if B is. Also, + [a,i][b,j] [c,k]= abzq1 ,(i+ j) [c,k]= (ab)czq1 q2 ,(i+ j + k) , where, in a manner similar to that above, i + j + k = (q1 + q2)m + (i + j + k) ; and + [a,i] [b,j][c,k] =[a,i] bczq3 ,(j + k) = a(bc)zq3 q4 ,(i+ j + k) , where i + j + k = (q3 + q4)m + (i + j + k) . Thus q3 + q4 = q1 + q2, and G is associative if and only if B is. 106 O. Chein, E.G. Goodaire / Journal of Algebra 287 (2005) 103–122 Finally, since [a,i][b,j]=[abzq ,(i+j)] and [b,j][a,i]=[bazq ,(j+i)], it is clear that [a,i] and [b,j] commute if and only if a and b commute. Thus, G is commutative if and only if B is. 2 Remark 2.2. In a similar manner, it is not difficult to see that if B is any loop with central element z, and u is any cyclic group whose order is either infinite or divisible by m, then any associativity-like identity (such as the Bol identity, the Moufang identities or the extra law, (xy · z)x = x(y · zx)) which holds in B also holds in the direct product B ×u, and hence also in G = (B ×u)/N, where N is the cyclic subgroup generated by zu−m. (Clearly N is a normal subloop, since it is central.) We are now ready for the main construction. Theorem 2.3. Let m and n be even positive integers, let B be a loop satisfying the right Bol identity, and let r, s, t, z and w be (not necessarily distinct) elements in Z(B),the m n 2 2 centre of B, such that r = r = s = t = 1. Let L = B × Cm × Cn with multiplication defined by [a,i,π][b,j,ρ]= abrjπsijρ tjπρzpwq ,(i+ j),(π + ρ) , where, for any integer i, i and i denote the least nonnegative residues of i modulo m and n, respectively; i + j − (i + j) 0 if i + j<m, p = = m 1 otherwise, π + ρ − (π + ρ) 0 if π + ρ<n, q = = n 1 otherwise.
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