Loops and the Lagrange Property

Let F be a family of finite loops closed under subloops and factor loops. Then every loop in F has the strong Lagrange property if and only if every simple loop in F has the weak Lagrange property. We exhibit several such families, and indicate how the Lagrange property enters into the problem of existence of finite simple loops. The two most important open problems in loop theory, namely the existence of a finite simple Bol loop and the Lagrange property for Moufang loops, have been around for more than 40 years. While we did not solve these problems, we show that they are closely related. Some of the ideas developed here have been present in the loop-theoretical community, however, in a rather vague form. We thus felt the need to express them more precisely and in a more definite way. We assume only basic familiarity with loops, not reaching beyond the introductory chapters of [Pflugfelder(1990)]. All loops mentioned below are finite. Let us get started with the crucial notion: the Lagrange property. A loop L is said to have the weak Lagrange property if |K| divides |L|, for each subloop K of L. It has the strong Lagrange property if every subloop K of L has the weak Lagrange property. A loop may have the weak Lagrange property but not the strong Lagrange property [Pflugfelder(1990), p. 13]. Our main result depends on the following lemma, which is a restatement of [Bruck (1958), Lemma 2.1]. Lemma 1. Let L be a loop with a normal subloop N such that (i) N has the weak (resp. strong) Lagrange property, and (ii) L/N has the weak (resp. strong) Lagrange property. Then L has the weak (resp. strong) Lagrange property. There are some classes of loops studied in the literature to which the lemma applies directly. For each of these, the normal subloop in question is associative. Date: November 20, 2001. 1991 Mathematics Subject Classification. 20N05.