SYLOW THEORY FOR QUASIGROUPS II: SECTIONAL ACTION MICHAEL K. KINYON1, JONATHAN D. H. SMITH2, AND PETR VOJTECHOVSKˇ Y´ 3 Abstract. The first paper in this series initiated a study of Sy- low theory for quasigroups and Latin squares, based on orbits of the left multiplication group. The current paper is based on so- called pseudo-orbits, which are formed by the images of a subset under the set of left translations. The two approaches agree for groups, but differ in the general case. Subsets are described as sectional if the pseudo-orbit that they generate actually partitions the quasigroup. Sectional subsets are especially well-behaved in the newly identified class of conflatable quasigroups, which pro- vides a unified treatment of Moufang, Bol, and conjugacy-closure properties. Relationships between sectional and Lagrangean prop- erties of subquasigroups are established. Structural implications of sectional properties in loops are investigated, and divisors of the order of a finite quasigroup are classified according to the behavior of sectional subsets and pseudo-orbits. An upper bound is given on the size of a pseudo-orbit. Various interactions of the Sylow theory with design theory are discussed. In particular, it is shown how Sylow theory yields readily computable isomorphism invari- ants with the resolving power to distinguish each of the 80 Steiner triple systems of order 15. 1. Introduction This paper is part of a research program to extend Sylow theory from finite groups to finite quasigroups and Latin squares. General approaches to the extension have been both top-down and bottom- up. The top-down approach works with the Burnside order, a labeled 2010 Mathematics Subject Classification. 20N05, 05B07. Key words and phrases. Latin square, quasigroup, Sylow theorem, Moufang loop, loop coset, Steiner triple system. Work on this paper was begun while the second author enjoyed the hospital- ity of the Department of Mathematics, University of Denver, partially supported by the Simons Foundation under the auspices of Collaboration Grant 210176 to P. Vojtˇechovsk´y. 1 2 M.K. KINYON, J.D.H. SMITH, AND P. VOJTECHOVSKˇ Y´ order structure on the set of isomorphism classes of irreducible permu- tation representations of a quasigroup [19, xx10{13]. The bottom-up approach, which is modeled on Wielandt's treatment of the Sylow the- ory for groups [22], studies the combinatorial or geometric structure of systems of subsets of a quasigroup. In a predecessor paper, the systems studied were the orbits of a given subset under the full left multiplica- tion group of the quasigroup [19, xx5{9]. In the current paper, which again works solely with the bottom-up approach, the systems studied are the images of a subset under the set of left multiplications by ele- ments of the quasigroup. For groups, these two versions of the bottom- up approach coincide, but they separate for general quasigroups. If S is a subset of a quasigroup Q, then the pseudo-orbit generated by S is the system fSL(q) j q 2 Qg of images of S under the left multiplications L(q) by elements q of Q (see Example 2.2). The left multiplications are the permutations represented by the rows of the Latin square forming the body of the bordered multiplication table. The images under the left multiplications are the cosets that appeared in [13], but here they are described as left translates. A subset S is said to be sectional if the pseudo-orbit it generates forms a partition of Q. Note that in the finite case, a nonempty subset S is sectional if and only if the columns indexed by S in the multiplica- tion table of Q can be partitioned into jQj=jSj pairwise disjoint Latin subsquares of order jSj. In Wielandt's approach to the Sylow theory for groups, p-subgroups of a finite group Q (for a prime p) appear in the pseudo-orbits of sectional subsets of p-power order. The basic properties of pseudo- orbits and sectional subsets are studied in Section 2. One of the primary motivations for the extension of Sylow theory to quasigroups is the insight that the extension provides into various aspects of quasigroup theory, and the way that it leads to a unification of these diverse aspects. This effect is seen clearly in Section 3, which investigates when each element of the orbit of a subset under the full left multiplication group is actually sectional. Theorem 3.7 shows that this happens within the class of (left) conflatable quasigroups Q, satisfying the property 8 x 2 Q; 9 αx 2 Q! : 8 y 2 Q;L(x)L(y)αx 2 L(Q) | compare (3.1). Conflatable quasigroups form the framework for a unified treatment of various classes of quasigroups and loops that have appeared in the literature, such as Moufang loops, Bol quasigroups, conjugacy-closed loops, and distributive quasigroups. SECTIONAL ACTION 3 Section 4 investigates the relationships between the sectional and Lagrangean properties for subquasigroups. The concepts coincide for loops, but differ for general quasigroups. Sufficient conditions for sec- tionality are also given. Section 5 provides a classification of the divi- sors d of the order of a finite quasigroup, based on the behavior of the pseudo-orbits of subsets of size d. This classification is correlated with the classification from [19], which was based on the behavior of orbits under the full left multiplication group. Theorem 6.5 provides an upper bound for the size of the pseudo- orbit of a (non-sectional) subgroup of a quasigroup, in terms of the semilattice of subgroups containing the given subgroup. This bound is sharp, for example, when applied to Klein 4-subgroups of the smallest simple, non-associative Moufang loop, the Paige loop PSL1;3(2) of order 120 [17]. The concluding Sections 7 and 8 offer applications and illustrations of the current Sylow theory in design theory and loop theory respectively. In particular, x7.1 shows how Sylow theory produces readily computed invariants that are powerful enough to distinguish each of the 80 Steiner triple systems of order 15 (compare Table 4 and [6, Table II.1.28]). Readers are referred to [18] and [20] for quasigroup-theoretic and general algebraic concepts and conventions that are not otherwise ex- plicitly clarified here. 2. Pseudo-orbits and sectional subsets A quasigroup Q is a set with a binary operation · such that the equation x·y = z has a unique solution in Q whenever two of the three elements x, y, z of Q are specified. This is equivalent to saying that the left multiplications L(q): Q ! Q; x 7! qx and the right multiplications R(q): Q ! Q; x 7! xq by each element q of Q are bijections of Q. In particular, unlabeled multiplication tables of finite quasigroups (which we will employ throughout the paper and whose rows and columns we tacitly label 1, ::: , jQj) are precisely Latin squares. We also set ynx = xL(y)−1, and x=y = xR(y)−1 for elements x; y of a quasigroup Q. Finally, a loop is defined as a quasigroup Q with an identity element 1 2 Q satisfying 1 · x = x · 1 = x for all x 2 Q. 2.1. Pseudo-orbits. Definition 2.1. Let S be a subset of a quasigroup Q. Then fSL(q) j q 2 Qg is called the (left) pseudo-orbit generated by S in Q. Its elements are described as (left) translates of S in Q. 4 M.K. KINYON, J.D.H. SMITH, AND P. VOJTECHOVSKˇ Y´ Example 2.2. Consider the quasigroup with multiplication table 1 2 3 4 5 6 2 1 4 5 6 3 3 4 5 6 1 2 4 6 2 1 3 5 5 3 6 2 4 1 6 5 1 3 2 4 { } The set f1; 3g, f1; 6g, f2; 4g, f3; 5g, f5; 6g is the pseudo-orbit of f1; 3g. We begin with an elementary observation. Lemma 2.3. Let S be a non-empty subset of a quasigroup Q. Then the pseudo-orbit generated by S is a cover of Q. Proof. Let s be an element of S, and let x be an element of Q. Then x = (x=s)s lies in the translate SL(x=s) of S. Note that common membership in a pseudo-orbit is not a transitive relation. More specifically, define a relation ∼ on subsets of a quasi- group Q by S1 ∼ S2 , 9 S ⊆ Q: 9 q1; q2 2 Q:S1 = q1S and S2 = q2S: Certainly, ∼ is symmetric. Lemma 2.4. Let Q be a quasigroup. Then the relation ∼ is reflexive. Proof. If Q is empty, the result is immediate. Otherwise, let S1 be a −1 subset of Q. For an element q of Q, consider S = S1L(q) . Then S1 = SL(q), so S1 ∼ S1. In general, ∼ is not a transitive relation. For instance, in the quasi- group of Example 2.2, we have f1; 2; 3g ∼ f1; 2; 4g ∼ f1; 2; 5g, but f1; 2; 3g f1; 2; 5g. Recall that for a quasigroup Q, the left multiplication group LMlt Q of Q is the permutation group hL(q) j q 2 QiQ! of the underlying set Q generated by the full set fL(q) j q 2 Qg of left multiplications by elements of Q. Proposition 2.5. If two subsets of a quasigroup Q are related by the transitive closure of ∼, then they lie in the same orbit under the left multiplication group LMlt Q. SECTIONAL ACTION 5 Proof. Suppose that S and S0 lie in the transitive closure of ∼, and are thus related by a chain of the form 0 (2.1) S = S0 ∼ S1 ∼ · · · ∼ Sr = S : It will be shown by induction on r that S0 lies in the LMlt Q-orbit of S.