The Structure of Certain Triple Systems

For each prime power q = 1 (mod 12), there is a triple system of order q whose automorphism group is transitive on unordered pairs. The object of this paper is to study these systems. This is done by analyzing how pairs of elements are linked. The linkage of a and b consists of a triple (a, b, c) and of some cycles in which adjacent pairs of elements form triples alternately with a and with b. Because of the transitivity, the lengths of the cycles will be independent of the choice of a and b. Using a computer, the linkage between two elements was determined for each q < 1000. Some curious facts concerning the lengths of the cycles were uncovered; for example, the number of cycles of length greater than 4 is even. The systems of prime order p < 1000 were found to have no proper subsystems of order greater than 3. In the remaining case, q = 343, there are subsystems of orders 7 and 49, and all subsystems of the same order are isomorphic. For no q with 1 < q < 1000 is the automorphism group doubly transitive. Finally, some general results are proved. The cycles of lengths 4 and 6 are determined. Using this result, it is shown that there can be no subsystem of order 7 or 9, except for the subsystems of order 7 when q is a power of 7. Hence, by a theorem of Marshall Hall, the automorphism group cannot be doubly transitive, except possibly when ? isa power of 7. (Added August 1974. In a postscript, it is shown that the automorphism group is not doubly transitive in this case either.) 0. Doubly Transitive Triple Systems. A (Steiner) triple system on a given set of elements is a set of triples of these elements such that each pair of elements is included in one and only one triple. Triple systems are discussed, for example, in Hall [5]. If « is a positive integer, then there is a triple system on « elements if and only if « = 1,3 (mod 6). The system is unique for « < 9, but for larger «, this is no longer the case. Of especial interest are triple systems with a high degree of symmetry. The only systems which are known to have a doubly transitive automorphism group are the projective spaces over the 2-element field and the affine spaces over the 3-element field, where the triples in each case are the lines of the geometry. These yield triple systems with 2r — 1 and 3r elements. Another description of these systems is obtained by starting with the field of 2r elements with 0 deleted, or with the field of 3r elements, and letting {x, y, z) form a triple whenever x + y + z = 0. A paper by Hall Received July 8, 1974. AMS (MOS) subject classifications (1970). Primary 05B05. Copyright © 1975. American Mathematical Society 223 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 224 RAPHAEL M. ROBINSON [4] raises the question whether there are any other triple systems with doubly transitive automorphism groups, and makes a contribution to the solution by showing that these are the only systems satisfying a stronger condition. The automorphisms of the system with 3r elements clearly include the linear transformations xa = ax + b with a ¥= 0. These transformations are sufficient to guarantee the double transitivity. In this section, we shall show that there is no other triple system based on a field and having all these automorphisms. Indeed, we find that there is no other system having even the automorphisms x° = ± x + b. It will be convenient to discuss first cyclic triple systems, that is, triple systems having an automorphism which permutes the elements cyclically. We may use as elements the residue classes mod «, and assume that x° = x + 1 is an automorphism. Associated with a triple {x, y, z) is the difference triangle {u, u, w) with u = y — x, v = z y, w = x z. Here u + v + w = 0, but u =£ 0, v ¥= 0, w ¥= 0. Only the cyclic order of u, v, w is important. We may represent the system geometrically by placing the element h at e2nih^". A difference triangle can be visualized as a triangle inscribed in the unit circle, its vertices being elements of the triple. However, the lengths of the sides are measured by the arcs cut off. Rotating the triangle through the angle 27r/« corresponds to the automorphism x° = x + 1. If the triangle is scalene, it will produce « different triples by rotation. It is not permissible for the triangle to be isosceles, since it would produce two different triples with a pair in common. However, if « is a multiple of 3, we could use an equilateral triangle with side «/3. It will produce only «/3 triples by rotation. The various difference triangles used must be such that the sides and their negatives exhaust the nonzero residue classes mod «. If « = 6k + 1, there must be k scalene triangles, whereas if « = 6k + 3, there must be k scalene triangles and an equilateral triangle. The difference triangle ( — w, v, u) is equivalent to («, d, w). But the reversed triangle {w, v, u) or (— u, — v, — w) is not equivalent to this, and is indeed inconsistent with it, unless the triangle is equilateral. More generally, consider triple systems on an Abelian group of order « which are invariant under addition of constants. These reduce to cyclic triple systems if the group is cyclic. Similar considerations hold in general, except that there may be more than one equilateral triangle. The side of such a triangle will satisfy the equation 3u = 0, that is, must be a group element of order 3. If a triple system associated with an Abelian group also has the automorphism xa = x, then with every difference triangle we also have the reversed triangle, hence all difference triangles are equilateral. We now apply this conclusion to triple systems on a field having the automorphisms xa = ± x + b. The argument will be based solely on the fact that all difference triangles are equilateral. If u is the side of such a triangle, then 3u = 0 but u =£ 0. It follows that 3 = 0. Thus p = 3 and so q = 3r. Also, in any triple {x, y, z), we License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use STRUCTURE OF CERTAIN TRIPLE SYSTEMS 225 would have x+y+z=x + {x+u) + ix + 2u) = 0, which leads back to the triple system of order 3r already discussed. 1. Systems Transitive on Unordered Pairs. The group of automorphisms x° = ax+b with ai=0 discussed in Section 0 has a subgroup of index 2, where a is restricted to be a square. The subgroup takes 0 and 1 into any two elements differing by a nonzero square. Now if — 1 is not a square, then these squares and their negatives exhaust the field, except for 0. Hence the subgroup will be transitive on unordered pairs. In this section, we shall determine the triple systems on a field which admit this group. Since — 1 is not a square, we have p = 3 (mod 4) and r odd, or, what is equivalent, q = 3 (mod 4). For a triple system, we must have q = 1, 3 (mod 6). The case q = 3 (mod 6) arises only when q = 3r. In this case, there must be at least one equilateral difference triangle. Multiplication by all nonzero squares then yields {q l)/2 equilateral difference triangles, so all difference triangles are equilateral. As shown at the end of Section 0, this leads to the doubly transitive system of order 3r discussed there. Thus we need consider only the case q = 1 (mod 6). Then we must have q = l (mod 12), which is equivalent to p = 7 (mod 12) and r odd. Let q = 6k + 1. There must be k scalene difference triangles. We may assume that at least two sides of each triangle are squares; otherwise, we reverse the cyclic order of the sides and change their signs. Multiplying the sides of any triangle by the 3k nonzero squares, we obtain each of the triangles three times, with their sides permuted cyclically. Hence all of the sides must be squares. Let the triangle containing 1 be il,u,v). Multiplying by w_1 gives the triangle {u~1,l,vu~ ) = (1, vu~l, u~l). Hence vu~l=u and u~l = v, and so v = u2 and u = v2. It follows that u3 v3 = 1. Since p = 1 (mod 6), the equation x3 = 1 has three integer solutions, that is, three solutions in the p-element subfield. If co is either of the solutions other than 1, then we may take u = oj and v = co2. All difference triangles are obtained from the basic triangle (l,co, co2) by multiplying by nonzero squares. This does yield a triple system with the prescribed automorphisms. The two choices for co lead to two isomorphic copies of the triple system. One differs from the other by reversing the cyclic order of the differences, which corresponds to changing the signs of all the elements. Except for a few remarks about triple systems in general and about the systems considered in Section 0, the rest of this paper will be devoted to these triple systems. It will be convenient to introduce the character x(«)> which has the value 1, 0, or — 1 according as a is a nonzero square, 0, or a nonsquare. If q = p, then xfaO is the Legendre symbol {a/p). More generally, if q = pr with r odd, and a is an integer, then x{a) — {<*/p)Since p = 1 (mod 6) in the cases considered, we have X(— 3) = 1. Furthermore, we assumed that x(— 1) = — 1, hence also xQ) — — 1. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 226 RAPHAEL M. ROBINSON The third element of a triple (x, y, z) is determined from x and y by the formula z = y + {y x)co if \iy -x)=l. In case x{y x) = — I, then we have x(* y) = U hence the same formula may be used with x and y interchanged. When r > 1, we may also consider the field of q = pr elements as an /--dimensional affine space over the p-element field. Since to is an integer, the preceding paragraph shows that the three elements of any triple are collinear. Every linear subspace will therefore define a subsystem of the given triple system. In the computer programs discussed in Section 2, I chose the numerically smaller value of co. For theoretical purposes, a different choice may be better. Since to = (to2)2 and to + 1 = — to2, we have x(w) = 1 and x(w + 1) = — 1These do not furnish any distinction between the two choices for. to. But the latter shows that x{cs>2 1) = x(w 1). In other words, for one choice of to we have X(to 1) = 1, for the other we have x(w-l) = -l. We may suppose that co is chosen so that x(w — 1) = 1. This choice of to may also be characterized by assuming that (1, to, to2) is a triple as well as a difference triangle. Indeed, the triple (1, to, to2) corresponds to the difference triangle (to — 1, to2 — to, 1 — to2), which is obtained from the basic difference triangle (1, to, to2) by multiplying by to 1. Furthermore, since (to — 1)(to + 2) = to2 + to — 2 = — 3, it follows that x(w + 2) = 1. The prescribed linear automorphisms x° = ax + b with x{a) = 1 form a group of order q{q l)/2. The r automorphisms of the field of q = pr elements are also automorphisms of the triple system which we constructed. We see that a field automorphism commutes with the group of linear automorphisms. Hence, together, the prescribed linear automorphisms and the field automorphisms generate a group of order rq{q — l)/2. For q = 7, the automorphism group is in fact larger, and is doubly transitive. Using a theorem of Marshall Hall, we show in Sections 7—9 that the automorphism group is not doubly transitive in any other case, except possibly when q = 7''. (These sections are independent of Sections 3-6.) Using the results of a computer calculation, we show in Section 5 that it is also not doubly transitive when q = I3. (This section is independent of Sections 3—4.) Thus only the cases q = lr (r = 5, 7, 9, • • • ) remain open. The triple systems of this section have been characterized by Lüneburg [8] and Kantor [7]. Their work is discussed by Dembowski [3, pp. 96-99]. The description which was given above is on a more elementary level, starting from somewhat special assumptions. I have not been able to find who first studied the triple system of order p = 7 (mod 12) considered here. Dembowski [3, p. 98], ascribes it to Netto [9], but this is License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use STRUCTURE OF CERTAIN TRIPLE SYSTEMS 227 incorrect, since the system discussed by Netto is different. Netto's system also appears in his book [10, pp. 220-221] ; in a note to the second edition, pp. 329-331, Th. Skolem introduces a class of systems including both Netto's and the one discussed here. However, Skolem makes no mention of the special properties of the present system, so it hardly can be ascribed to him. The first appearance of the system in print may be in a problem in Carmichael [1, p. 436]. 2. Linkages. A useful tool in studying isomorphisms and automorphisms of triple systems is the type of linkage between two elements. Let S be any triple system, and let a and b be two elements of S. They determine a triple {a, b, c), which we call the key triple of the linkage. All of the rest of the elements fall into cycles of the form {cl, c2, • • • , c2l), where / ~> 2 and {a, cv c2), {b, c2, c3), {a, c3, c4), • • • ,{a, c2l_v c2l), {b, c2l, c¡) are all triples. Thus any cycle has an even length not less than 4. Such linkages were used by Reiss [11] in the special case where all the elements not in the key triple form a single cycle. He constructed triple systems of all possible orders in which a suitable pair of elements are linked in this way. Linkages as a tool in studying isomorphisms seem to have been introduced by Cole, Cummings, and White [2]. Details are given in [12, Parts 3-5]. This method was used again by Hall and Swift [6]. If we want more information than is furnished by the cycles described above, we can compute the cross-links joining pairs of elements in the same or different cycles which form a triple with the third element c of the key triple. If S is mapped isomorphically onto another system S', then a and b must be mapped onto elements a' and b' such that the linkage between a' and b' has the same structure as the linkage between a and b. In the linkage between a and b, the pairs of consecutive elements of a cycle form triples alternately with a and with b. The linkage between b and a is the same, except for the interchange of the first and second pairings. This difference may, however, be enough to make it possible to show that there can be no automorphism interchanging a and b. In general, the number and lengths of the cycles for the linkage between a and b in S will depend on a and b. However, this will not be the case if the automorphism group is doubly transitive, or even transitive on unordered pairs. Now in the two known kinds of triple systems with doubly transitive automorphism groups, the systems of orders 2r — 1 and 3'', the cycles are all of the same length. Consider the systems as based on the field of 2r elements with 0 deleted, or on the field of 3r elements. For the system of order 2r — 1, the key triple is {a, b, a + b), and every cycle has the form {x, x + a, x + a + b, x + b). For the system of order 3r, the key triple is {a, b, — a b), and every cycle has the form {x, x a, x + a b, x + a + b, x — a + b, — x — b). Furthermore, in both cases, every element is cross-linked to the License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 228 RAPHAEL M. ROBINSON opposite element of the same cycle. The key triple with each cycle forms a subsystem of order 7 or 9, respectively. For the triple systems of order q = 7 (mod 12) discussed in Section 1, we have a new situation. The linkage structure for each q is unique, except for the possible interchange of the two pairings of consecutive elements of a cycle, but it varies with q in a manner which is not easily predictable. Thus the number and lengths of the cycles for each q seems an interesting object of study. They may be computed taking a = 0 and è=l. This was done for q < 1000 during March and April 1974 using the CDC 6400 at the Computer Center of the University of California, Berkeley. The main program covered primes p < 1000. A special program was written for the only other case, q = 3A3. In the computer programs, the numerically smaller value of to was chosen. The key triple is (0, 1, to + 1). The remaining elements fall into cycles, and these were computed. In addition, the computer printed out the number to which each element of a cycle was cross-linked, and the name of the cycle to which this number belonged. It is impossible to reproduce all this information here, although it is necessary to refer to it at times. The numbers of cycles of various lengths are given in Table 1. The arrangement is as follows: For each value of q, the total number of cycles is given, then the number of cycles of lengths 6, 12, 18, 24, 30, and then the lengths of all other cycles. Here are some interesting facts about the lengths of the cycles for q < 1000 which may be read from Table 1. (1) There is a cycle of length 4 if and only if q = 7 (mod 24). (2) The number of other cycles is even. (3) The average length of the cycles is less than 12, but seems to be approaching 12. (4) In the prime cases, the cycles all have lengths divisible by 6 except for one congruent to 4 mod 6 (40 cases) or two congruent to 2 mod 6 (4 cases). For q = 343, there is a cycle of length 4 and three of length 14 besides the cycles whose lengths are divisible by 6. A general proof of (1) is given in Sections 7-8. The statements (2) and (3) lead to obvious conjectures. Finally, with regard to (4), it is clear that there is a strong preference for cycles whose lengths are divisible by 6, but it is not clear whether we should expect the number of exceptions to remain bounded. We shall also be interested in determining the subsystems of the triple systems considered. In general, if a triple system has « elements and a proper subsystem has s elements, then n>2s + 1. Otherwise, there would not be enough elements to complete the triples determined by elements of the subsystem and a fixed element not in the subsystem. A subsystem of the given system of order q which contains 0 and 1 will consist of the key triple and a certain number of cycles. Furthermore, all cycles may be divided into equivalence classes of cycles which are connected by cross-links. There may be a number of minor classes, with fewer than q/2 elements each, and perhaps one major class, with more than q¡2 elements. A proper subsystem containing 0 and 1 must consist of the key triple and some minor classes. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use STRUCTURE OF CERTAIN TRIPLE SYSTEMS 229