A class of simple proper Bol loops ∗ Gábor

For a loop Q, we call the maps La(x) = ax,Ra(x) = xa left and right translations, respectively. These are permutations of Q, generating the left and right multiplication groups LMlt(Q),RMlt(Q) of Q, respectively. The group closure Mlt(Q) of LMlt(Q) and RMlt(Q) is the full multiplication group of Q. Just like for groups, normal subloops are kernels of homomorphisms of loops. The loop Q is simple if it has no proper normal subloop. The commutator-associator subloop Q is the smallest normal subloop of Q such that Q/Q is an Abelian group. For basic introductory reference on loops see [11]. The loop Q is a left (right) Bol loop if the identity x(y(xz)) = (x(yx))z (((xy)z)y = x((yz)y)) holds in Q. Loops which satisfy both identities are called Moufang loops. For any field F , L. J. Paige [10] constructed a simple nonassociative Moufang loop M(F ). Using the classification of finite simple groups, M. Liebeck [5] showed that the only finite simple nonassociative Moufang loops are M(Fq). The existence of finite simple non-Moufang Bol loops was considered as the one of the main open problems in the theory of loops and quasigroups, cf. [12] and [1, Question 4]. In this paper, Bol loops are left Bol loops and with proper Bol loops we mean left Bol loops which are not Moufang.