An Extension Theory for a Certain Class of Loops

Introduction. If £ is a group with a normal subgroup K one may form the quotient group E/K^M. Conversely, for preassigned groups K, M, there is the extension problem: to determine (in some sense) all groups E with K as normal subgroup such that E/K^M. Much progress has been made on this problem, particularly through the work of Baer [l, 2, 3] and the cohomology theory of Eilenberg and MacLane [l, 2, 3]. The latter authors make it clear that insight is gained by relinquishing part of the associative law; specifically, by requiring that E be merely a loop such that the associative law (̂ 1̂ 2)̂ 3 = 1̂(̂ 2̂ 3) holds if at least one of the e% belongs to a distinguished subgroup of K. We take this to be K itself. It then becomes evident that the subclass of loops E consisting of the groups is not the only one of interest; one may consider, for example, the Moufang loops, in which case it seems natural to allow M also to be Moufang. Thus we approach the extension problem actually studied in the paper: M is a given loop, K is a group (not given, but with given centre G) and E is to be any loop with l a s a normal subloop contained in the "associator" of E, such that E/K=M. This problem is more typical of group theory than of loop theory but is, nevertheless, a natural and significant special topic in the theory of loops. For the sake of brevity no examples or applications are given and references to the bibliography are kept to a minimum. The EilenbergMacLane kernels, important for constructions, have been ignored. I may signal out as new: the inverse of a (noncentral) extension (§1), the specific results on central Moufang extensions (§6) and the allpervading functions F which generalize (even for M a group) the Eilenberg-MacLane cocycles. As indicated by Theorem 8 (§4), additional information about the functions F would probably increase our knowledge of cohomology groups.