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Non-Isomorphic Reverse Steiner Triple Systems of Order 19
Annals of Discrete vathematics 7 (1980) 255-264 @ North-Holland Publishing Company
NON-ISOMORPHIC REVERSE STEINER TRIPLE SYSTEMS OF ORDER 19 R.H.F. DENNISTON Deparrmenf of Mafhematics, Uniuersiry of Leicesfer, Leicesfer LEI 7RH, England
1. Introduction It has been known since 1917 that, up to isomorphism, there are just 80 Steiner triple systems of order 15. The next admissible order for such systems is 19; but the corresponding problem is too big to be feasible, and it may not be without interest if a restricted form of it can be solved. One severe restriction is to require a system to have what we may call a “reversal”, namely an involutory automorphism with only one fixed point. Triple systems with reversals have in fact been given the name of reverse Steiner triple systems, and have recently been the subject of various papers, beginning with [ 13 and [4].The present paper establishes that the maximum number of non-isomorphic reverse systems of order 19 is 184.
2. Uniqueness of reversal
Theorem 1. If a Steiner triple system of order u has two different reversals, then u is divisible by 3 . Proof. Let r and s be the fixed points of different reversals p and a. Then points x and y correspond in p if and only if rxy is a line; so s must be distinct from r, since otherwise a would be the same as p. Let R be the group of automorphisms generated by p and a: one orbit under R consists of r, s, and the point r collinear with r and s. Let x be any other point. Then a p x and papx are in line with r ; using a, we find that px and a p p x are in line with t ; using p, we find that x is in line with s and papapx, and therefore coincides with a p p p x . So we see that, in the orbit to which x belongs, there are just six points. That is, the group R partitions the set of u points into orbits, each of cardinality three or six, and t h e Theorem is proved. We may observe that apa is a reversal with t as fixed point, and that R is naturally isomorphic with the group of permutations of { r , s, t}. An affine space of order three and any number of dimensions has, of course, as many reversals as it has points. I have found it easy to construct, on 33 points, a system with at least 255
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three reversals; and I conjecture that such a system exists whenever u is congruent to 3 or 9 modulo 24.
3. Projections and coverings Let S be a reverse Steiner triple system of order 19. We suppose that S has its points so named that its reversal is the permutation
Then we may construct a design D on nine varieties 1 , 2 , . . . , 9 , by taking in turn each pair of corresponding lines (not through 0) of S, and simply dropping the primes. If, for instance, 1’2‘3’, 1”2”3“, 1’2”4‘, l”2’4” are lines of S, then 123 and 124 are blocks of D. We naturally call D “the projection” of S ; we see at once that, given any two of the nine varieties, we can find just two distinct blocks of D to which they belong. So the projection D is a balanced incomplete block design without repeated blocks, and its parameters are given by v=9,
b=24,
k=3, r=8,
h=2.
The situation would be complicated if a system could have more than one reversal, and so more than one projection: but Theorem 1 reassures us that this will never happen with a system of order 19. We can therefore see that the projections of two isomorphic reverse systems will be isomorphic ( 9 , 2 4 , 3 , 8 , 2 ) designs. So,if we are to solve the problem of non-isomorphic reverse systems, we should begin with the problem of non-isomorphic (9,24,3,8,2)-designs without repeated blocks. After that, we should consider the problem of setting up “a covering” for a given design D - namely, a reverse system S which has D as its projection. We shall see that there is a design which has no covering, but that other designs can be covered in 213 or 214 ways. The appropriate point of view will be that the varieties of D are numbered 1 , 2 , . . . , 9 in a fixed way, and that the labelling 0, l‘, . . . .9“ of the points of the covering S is required to be consistent with this; also that different arrangements into triads of the symbols 0. l‘, . . . , 9 ” are to be regarded as “distinct” coverings. Suppose, then, that S is a covering of 0,and that we interchange the two symbols 1’ and 1” wherever they occur in S. This will give a covering of D, distinct from S according to the convention just made, but isomorphic with S as a Steiner system. We might say that we have “re-primed” the variety 1 ; and we could go o n to re-prime other varieties of D. However, the result of re-priming all nine varieties would be a reverse system which, by definition, coincided exactly with S, On the other hand, we shall certainly get a covering distinct from S if we re-prime any proper subset of ( 1 , . . . ,9}. To prove this, we have only to consider
Non-isomorphic rewrse Steiner triple systems of order 19
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the effect on the four lines 1’2’3’, 1”2”3”, 1’24‘, l”2‘4“ of any re-priming that includes 1 but not 2-and to remember that any two varieties of D will be covered by four lines in essentially this way. It follows that, by taking S as equivalent to any covering obtained from S by re-priming, we partition the set of all coverings of D into equivalence classes (“prime classes”, shall we say), each of cardinality 2’ exactly. Two coverings of D, if they are in the same prime class, are isomorphic as reverse Steiner systems. Now let p be an automorphism of the given design D. Then we can transform any covering S into an isomorphic reverse Steiner system, p*S say, by making the obvious permutation of the points: if, for instance, 1 4 2 in p, then 1 ’ 4 2 ‘ and 1”+2” in p * , while 0 is fixed. Moreover, p* preceded by the re-priming of 1, and p* followed by the re-priming of 2, will in this instance have the same effect: we can assert that p will permute the prime classes in a well-defined way. If p happens to fix the prime class of a given covering S, we can get an automorphism of S if we follow p* with a suitable re-priming: and conversely. We shall see that, when a design D has coverings, a non-trivial automorphism of D never fixes all t h e prime classes, though it may fix a quarter of them. The conclusion is that the automorphism group of a design D will act as a group of permutations on the set of prime classes of coverings of D. If the prime class of a covering S is in the same orbit as that of a covering S*, then the reverse Steiner systems S and S* are isomorphic. Conversely, we remarked above that isomorphic systems have isomorphic projections: if we think concretely of two isomorphic coverings of the same design, we see that their prime classes are in the same orbit. Or, if we are given two isomorphic reverse systems that are specified in some other way, we can label their points so that they will have the same design as projection - and then, over that design, they will be in the same prime class (or, at least, in two prime classes in the same orbit). We can now see how the investigation should go. First, we exhibit a maximal set of non-isomorphic (9,24,3,8,2)-designs without repeated blocks, each design having its automorphism group determined. Then, for each of these designs, we find the number of prime classes, and sort these into orbits under the automorphism group (by looking for cases where an automorphism can fix a class). The total number of orbits will be the required number of non-isomorphic reverse Steiner systems.
4. A set of designs A solution to the problem of non-isomorphic designs, obtained by means of a computer, has been published by Gibbons [2]. I had independently solved the problem by hand, and have now checked that my solution agrees with Gibbons’. I will write out the designs as I found them, with Gibbons’ Roman numbers in brackets.
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D,: 123 124 134 156 159 167 178 189 234 256 259 267 (I) 278 289 357 358 368 369 379 457 458 468 469 479 We do not need the automorphism group of D , (which is of order 80). D,: 123 129 134 145 156 167 178 189 234 246 257 258 (VII) 267 289 357 358 368 369 379 459 468 478 479 569 Group of order 2 generated by (12) (39) (48) (5) (67).
D,:
123 129 134 145 156 167 178 189 234 246 258 259 (IX) 267 278 357 358 368 369 379 457 468 479 489 569 Group is trivial. 123 129 134 145 156 167 178 189 235 246 247 257 (HI) 268 289 348 359 367 369 378 458 469 479 568 579 Group of order 2 generated by (18) (2) (36)(45) (7) (9). D,:
D,: 123 129 134 145 156 167 178 189 235 246 247 257 (IV) 268 289 348 358 367 369 379 459 469 478 568 579 Group is trivial. Dh: 123 129 134 145 156 167 178 189 234 246 257 259 (V) 268 278 357 358 368 369 379 458 467 479 489 569 Group is trivial. D,: 123 129 134 145 156 167 178 189 235 246 247 257 (VI) 268 289 348 358 367 369 379 459 468 479 569 578 Group of order 2 generated by (14) (28) (3) (5) (69) (7).
D,: 124 127 136 139 146 157 158 189 235 238 245 268 (X) 269 279 347 349 356 378 458 467 489 569 579 678 Group of order 6 generated by (123) (456) (789) and (14) (26) (35) (7) (89). D,: 126 127 134 139 145 158 168 179 237 (XI) 256 289 348 356 359 367 467 469 478 Group of order 8 generated by (1 2345678) (9).
238 245 249 578 579 689
Ill,,: 124 127 136 138 145 159 169 178 235 238 249 256 (XII) 267 289 346 349 357 379 457 468 478 568 589 679 Group of order 6 generated by (123456) (789). D l l : 123 126 139 147 148 156 157 189 234 247 258 259 (VIII) 268 279 345 358 367 369 378 456 469 489 579 678 Group of order 6 generated by ( I 23456) (789).
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D12: 123 125 134 148 157 168 169 179 236 247 249 259 (XIII) 268 278 349 357 358 367 389 456 458 467 569 789 Group of order 18 generated by (123) (456) (789) and (174) (235689). D13: 126 129 137 139 147 148 156 158 237 238 246 248 (11) 257 259 346 349 356 358 457 459 678 679 689 789 Group of order 8 generated by (1234) (5) (67) (89) and (1) (24) (3) (5) (68) (79). 5. Construction of coverings
The problem of finding coverings, for a given design D, will be easier to handle if we set up some kind of arithmetic for it. Let us think again of the example at the beginning of Section 3, where 123 and 124 are blocks of D. We may express the assertion that each of the pairs { l’,2’) and { l”,2”) is collinear with a point that projects onto 3, while {l’,2”) and 11”. 2’) are collinear with points that project onto 4, by writing [l, 2; 31 = + 1,
[ 1,2; 41 = -1.
The symbol [p, q ; r ] is defined only when pqr is a block of the given design D. The following equations are satisfied when the symbols in them are defined:
[P, 4 ; rl = [q, p; rl. [ p , q ; r l . [p, q ; s I = -1
-
(1) ( r f s),
[P, 4 ; r l . [r, p; ql Eq, 7; PI= + I .
(2) (3)
In fact, we can see the truth of (3) by imagining that we have tossed three coins, and want to take one away so as to leave a head and a tail; there will be either two ways of doing this, or none. Equations (l),(2), (3) are not only necessary, but sufficient: if we have written down symbols [p, q; r ] , three for each block of D, that satisfy these equations, we have effectively set up a covering of D. A topologist might visualise the blocks of D as two-dimensional simplexes of a pseudomanifold, and might arrange a covering by setting up a one-dimensional cochain. However, a dual interpretation can be made in terms of graph theory. Let us consider the blocks of D as the vertices of a graph, two vertices being adjacent when the blocks have a pair of points in common [3]. Eleven of our designs will then have connected graphs; one exception is DI3,where the four vertices 789, 689, 679, 678 make up one component, and the other twenty make up a second. The graph of D, has three components: one consists of ten vertices 156, 167, 178, 189, 195, 256, 267, 278, 289, 295; another has ten vertices, and the third has four. The choice of a covering €or D will now correspond to the standard process of converting the graph into a directed graph (will “orientation” serve as a name for
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this process?). Taking once more the example symbolized by the equations [ 1.2; 31 = + I and [ 1.2; 41 = - 1, we may say that the edge which represents {1,2} has been oriented, the head end being at the vertex which represents the block 123, and the tail end at 124. Equation (3) now tells us that, of the three edges through any given vertex, either all have their head ends there, or one has a head end and two have tail ends. A simple counting argument then shows that, in any component of such a directed graph, exactly a quarter of the vertices have the former property (of being at the head ends of all their edges). But this means that a graph. if ten of its vertices make up a component, cannot possibly be oriented according to t h e rule. We conclude that there is no reverse system that has D , as its projection. As it turns out. each of the other twelve designs has some coverings, and accordingly has a graph for which there is, shall we say, at least one “admissible orientation”. This fact, which we shall establish in the next section, does not seem easy to infer from the graph theory. However, it does seem obvious what happens when G. the graph of a design D, has been provided with two admissible orientations A , and AZ. Let C be the set of edges of G which we have to reverse in changing from A , to A,. Then at any vertex, since the number of head ends is odd for A , , and likewise for A,. there must be an even number of edges that belong to C.That is, C will be a cycle of G. Conversely, given any cycle C of G, and one admissible orientation A , , we can change from A , to another admissible orientation by reversing all the edges of C. It follows that a given graph with first Betri number v, i f it has any admissible orientations at all, will have exactly 2” of them To find v, we subtract 24, the number of vertices of any one of our graphs, from 36, the number of edges, and add the number of components: so v is 13 for each of the designs from D, to D,, inclusive, and 14 for DI3.According to section 3. the respective numbers of prime classes are 2’ and 2‘, for the coverings of these designs.
6. Construction of coverings: an effective procedure It remains to be seen how the symbols [ p , q ; r ] can be used to handle these large sets of coverings. Let us take the design D, by way of example. We may choose to say that a covering of D, is “standardised” when the lines through 7’ are 7’7”0, 7’1’2”. 7’2’9”, 7‘9’5”. 7’5’1’’, 7’4’6, 7‘6’8”, 7’8‘3’’, 7’3‘4“. It is in fact easy to see that, in any prime class of coverings of D,, there is one and only one standardised covering. For such a covering of D,, we see that [ 1 , 2 ; 71 = - 1; and it follows, by equation (2) of section 5 , that [ l , 2; 41 = + I . Likewise, [2,9;61 = + 1,
[ 5 , 9 ;61 = + 1,
[ 1 , 5 ; 81 = + 1,
Non-isomorphic reverse Steiner triple systems of order 19
26 1
Let us now give names to five (suitably chosen) of the other symbols; it turns out that a convenient set is a = [2,8; 61,
fi = [8, 9; 41,
6 = [3,9; 41,
= [l,9; 81.
y = [2,5; 31,
Then we should be able to find all the other symbols in terms of these. First, we use Eq. (2) again:
[2,8;3]=-a,
[8,9; 1]=-/3,
[3,9; 1]=-6,
[1,9;5]=-~.
[2,5;4]=-y,
Now, using Eq. (3) as well, we deduce from the equations [3,8; 21 = +1 and [2,8; 3]= -a that [2,3; 81 = -a, whence again [2,3; 51 = a. Likewise, we find successively that
[2,6; 81 = a, [ 5 , 6; 91 = a, P - 6 ; 51 = Y, [4,8;9]=-p6, [ 5 , 8 ; 11= B E , [2,4; 5]=-y&, [ 1,6; 41 = - Y ~ E ,
[2,6; 9]= - a ; [5,6;3]=-a, [3,6; 1]=-7, [4,8;5]=@; [5,8;4]=--P~; [2,4; 1 ] = ~ 6 ~ [ 1,6; 31 = y& ;
[6,9; 2]= -a, [3,5;2]=ay, [4,9;3]=6, [1,8;9]=-/3~, [4,5; 8]=-&, ;[1,4;2]=~6~, [ 1,3; 61 = - 6 ~ ,
[6,9; 5]= a ; [3,5;6]=-ay; [4,9; 8]= -6; [1,8;5]=@, [4,5; 2 ] = 6 ~ ; [1,4;6]=-yb; [ 1,3; 91 = SE.
In the course of this arithmetic, we have ensured that condition (3) is satisfied for all but one of the 24 blocks of D,. We now observe that
[3,9; 1]=-6,
[1,9;3]=-~,
[1,3;9]=6~;
so condition (3) is satisfied for the last block, and there is no doubt that the system exists. Each of the symbols a,p, y, S, E can take the values +1 and -1 independently. So we have effectively constructed 2’ coverings of D8,all of them standardised as above. No two of these are in the same prime class, and we can obtain from them, by re-priming, the complete set of 213 “distinct” coverings of D,. It turns out that the same algorithm works for each of the designs D,, . . . , DI3,except that D,, with its higher Betti number needs six Greek letters.
7. Effect of an automorphism
Our next task, according to the programme at the end of Section 3, is to determine how the automorphism group of a design acts on the set of prime classes of its coverings. In particular, we should consider how often a non-trivial automorphism will leave a prime class fixed.
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By way of example. suppose that C , a prime class of coverings of D8,is fixed by the automorphism (123) (456) (789). In C there is one system, S say, which is "standardised" as in Section 5. According to Section 3, we are concerned with the "obvious" transformation 1'+2', 1"+2", 2'+3', . . . , determined by our automorphism. This will take S to a system, 7-say, which by hypothesis is also in the class C. Let us choose one of the two re-primings that take T back to S ; and let us write nq= -1 if the re-priming interchanges q' and q", but nq= +1 if it does not. Now the standardised system S will certainly have 7'9'5" as one of its lines; this will correspond to a line 8'7'6 in T, and the re-priming will take this to one or the other of two lines 7'6'8" and 7"6"8' that belong to S. So definitely n,r6 = - 1. In terms of the symbols a. 0, y , 6, E that belong to S, and of its numbers [p, q ; r], we can work out necessary and sufficient conditions for the re-primed system to be a standardised one: [ 9 . 3 ; 11 = -6, [9, 1: 81 = E , [9,2: 6]= + I , [9.6; 51 = a, [9.4; 31 = 6, [ 9 , 5 ;71-1, [9,7:2]=-1, [9.8; 41 = p,
[7,1: 2]= + l ; [7,2;9]=+1; [7,3; 41 = + 1; [7 , 4 ;6] = + 1; [7,5; 1 ] = + 1 ; [7,6; 81 = + 1; [7,8; 3]= + I ; [7,9; 51 = + 1;
These equations determine the set of numbers 7rq (to within multiplication of the whole set through by -1). But we are assuming, not merely that the re-primed system is standardised, but that it actually coincides with S. So we need five more conditions:
This is as far as we need to go, because the standardised covering of D, is uniquely determined by its symbols a, p, y, 6, €.'The only restrictions on S arc thc last five, which in fact reduce to two independent conditions a = ps = E .
The conclusion is that (123) (456) (789) fixes just a quarter of t h e prime classes of coverings of D, (namely those whose standardised representatives satisfy these two conditions). The same arithmetic, applied to the involution (14) (26) (35) (7) (89) of D,, leads to an impossible condition a = -a. So this automorphism does not fix any
Non-isomorphic reverse Steiner triple systems of order 29
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prime class. It turns out, likewise, that no prime class is fixed by any involution of any standard design (nor, consequently, by any automorphism of period 4 or 8).
8. Counting the systems Accordingly, we see in the case of D9that the automorphism (12345678) (9) must permute the 32 prime classes in four orbits of length 8; and this means that, up to automorphism, D9is the projection of only four reverse systems. Likewise, each of the designs D,, D4, D,, having an automorphism group of order two, is the projection of 16 systems. For D13, there are 64 prime classes, permuted by the eight automorphisms in eight orbits of length 8; so D,, is the projection of eight reverse systems. Going back to D,, we see that its automorphism group has two generators, one of order three fixing eight of the 32 prime classes, and one of order two fixing none. It follows that there are four orbits of length 2 and four of length 6. And therefore a maximal set of non-isomorphic coverings of D, will consist of eight reverse systems, four having automorphism groups of order 6 (which turn out to be cyclic), and four of order 2. The designs D,, and D,,, each of which has an automorphism group of order 6, give results analogous to those for D,. Each of these has eight nonisomorphic coverings, of which four have cyclic automorphism groups of order 6 . The design D,,, with a group of order 18, has two prime classes which are fixed by all its automorphisms of period 3 (but interchanged by all its involutions). Then there are six classes that are fixed by (123) (456) (789), but the automorphism group is transitive on this set of six; and likewise for (147) (258) (369). Finally, there is a set of 18 prime classes on which the group is sharply transitive. The conclusion is that D,, has four non-isomorphic coverings, with automorphism groups of orders 18, 6, 6, 2. These groups turn out to be Abelian, even though the group for D,, is not. To summarise, we have 32 non-isomorphic coverings for each of the designs that have trivial automorphism groups, namely D,, D,, and D,. We have 16 for each of D,, D4, D,; eight for each of D,, Dlo,D I 1DI3; , four for each of D, and D,,: but none for D,.
Theorem 3. Up to isomorphism, there are just 184 reuerse Steiner triple systems of order 19. One of these has an automorphism group of order 18 (Abelian, not cyclic), and fourteen have (cyclic) groups of order 6. For each of rhe others, the unique reversal and the identity are the only automorphisms. We may observe that the system with the group of order 18 is the one that was discovered by Rosa [4].
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References [ I ] J . Doyen, A note on reverse Steiner triple systems, Discrete Math. 1 (1972) 315-319. [2] P.B. Gibbons, Computing techniques for the construction and analysis of block designs, Ph.D. thesis, University of Toronto 1976 (=Technical Report no. 92, Department of Computer Science, University of Toronto, May 1976). [3] E.S. Kramer, Indecomposable triple systems, Discrete Math. 8 (1974) 173-180. [ I ]A. Rosa, On reverse Steiner triple systems, Discrete Math. 2 (1972) 61-71.
NON-ISOMORPHIC REVERSE STEINER TRIPLE SYSTEMS OF ORDER 19 R.H.F. DENNISTON Deparrmenf of Mafhematics, Uniuersiry of Leicesfer, Leicesfer LEI 7RH, England
1. Introduction It has been known since 1917 that, up to isomorphism, there are just 80 Steiner triple systems of order 15. The next admissible order for such systems is 19; but the corresponding problem is too big to be feasible, and it may not be without interest if a restricted form of it can be solved. One severe restriction is to require a system to have what we may call a “reversal”, namely an involutory automorphism with only one fixed point. Triple systems with reversals have in fact been given the name of reverse Steiner triple systems, and have recently been the subject of various papers, beginning with [ 13 and [4].The present paper establishes that the maximum number of non-isomorphic reverse systems of order 19 is 184.
2. Uniqueness of reversal
Theorem 1. If a Steiner triple system of order u has two different reversals, then u is divisible by 3 . Proof. Let r and s be the fixed points of different reversals p and a. Then points x and y correspond in p if and only if rxy is a line; so s must be distinct from r, since otherwise a would be the same as p. Let R be the group of automorphisms generated by p and a: one orbit under R consists of r, s, and the point r collinear with r and s. Let x be any other point. Then a p x and papx are in line with r ; using a, we find that px and a p p x are in line with t ; using p, we find that x is in line with s and papapx, and therefore coincides with a p p p x . So we see that, in the orbit to which x belongs, there are just six points. That is, the group R partitions the set of u points into orbits, each of cardinality three or six, and t h e Theorem is proved. We may observe that apa is a reversal with t as fixed point, and that R is naturally isomorphic with the group of permutations of { r , s, t}. An affine space of order three and any number of dimensions has, of course, as many reversals as it has points. I have found it easy to construct, on 33 points, a system with at least 255
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three reversals; and I conjecture that such a system exists whenever u is congruent to 3 or 9 modulo 24.
3. Projections and coverings Let S be a reverse Steiner triple system of order 19. We suppose that S has its points so named that its reversal is the permutation
Then we may construct a design D on nine varieties 1 , 2 , . . . , 9 , by taking in turn each pair of corresponding lines (not through 0) of S, and simply dropping the primes. If, for instance, 1’2‘3’, 1”2”3“, 1’2”4‘, l”2’4” are lines of S, then 123 and 124 are blocks of D. We naturally call D “the projection” of S ; we see at once that, given any two of the nine varieties, we can find just two distinct blocks of D to which they belong. So the projection D is a balanced incomplete block design without repeated blocks, and its parameters are given by v=9,
b=24,
k=3, r=8,
h=2.
The situation would be complicated if a system could have more than one reversal, and so more than one projection: but Theorem 1 reassures us that this will never happen with a system of order 19. We can therefore see that the projections of two isomorphic reverse systems will be isomorphic ( 9 , 2 4 , 3 , 8 , 2 ) designs. So,if we are to solve the problem of non-isomorphic reverse systems, we should begin with the problem of non-isomorphic (9,24,3,8,2)-designs without repeated blocks. After that, we should consider the problem of setting up “a covering” for a given design D - namely, a reverse system S which has D as its projection. We shall see that there is a design which has no covering, but that other designs can be covered in 213 or 214 ways. The appropriate point of view will be that the varieties of D are numbered 1 , 2 , . . . , 9 in a fixed way, and that the labelling 0, l‘, . . . .9“ of the points of the covering S is required to be consistent with this; also that different arrangements into triads of the symbols 0. l‘, . . . , 9 ” are to be regarded as “distinct” coverings. Suppose, then, that S is a covering of 0,and that we interchange the two symbols 1’ and 1” wherever they occur in S. This will give a covering of D, distinct from S according to the convention just made, but isomorphic with S as a Steiner system. We might say that we have “re-primed” the variety 1 ; and we could go o n to re-prime other varieties of D. However, the result of re-priming all nine varieties would be a reverse system which, by definition, coincided exactly with S, On the other hand, we shall certainly get a covering distinct from S if we re-prime any proper subset of ( 1 , . . . ,9}. To prove this, we have only to consider
Non-isomorphic rewrse Steiner triple systems of order 19
251
the effect on the four lines 1’2’3’, 1”2”3”, 1’24‘, l”2‘4“ of any re-priming that includes 1 but not 2-and to remember that any two varieties of D will be covered by four lines in essentially this way. It follows that, by taking S as equivalent to any covering obtained from S by re-priming, we partition the set of all coverings of D into equivalence classes (“prime classes”, shall we say), each of cardinality 2’ exactly. Two coverings of D, if they are in the same prime class, are isomorphic as reverse Steiner systems. Now let p be an automorphism of the given design D. Then we can transform any covering S into an isomorphic reverse Steiner system, p*S say, by making the obvious permutation of the points: if, for instance, 1 4 2 in p, then 1 ’ 4 2 ‘ and 1”+2” in p * , while 0 is fixed. Moreover, p* preceded by the re-priming of 1, and p* followed by the re-priming of 2, will in this instance have the same effect: we can assert that p will permute the prime classes in a well-defined way. If p happens to fix the prime class of a given covering S, we can get an automorphism of S if we follow p* with a suitable re-priming: and conversely. We shall see that, when a design D has coverings, a non-trivial automorphism of D never fixes all t h e prime classes, though it may fix a quarter of them. The conclusion is that the automorphism group of a design D will act as a group of permutations on the set of prime classes of coverings of D. If the prime class of a covering S is in the same orbit as that of a covering S*, then the reverse Steiner systems S and S* are isomorphic. Conversely, we remarked above that isomorphic systems have isomorphic projections: if we think concretely of two isomorphic coverings of the same design, we see that their prime classes are in the same orbit. Or, if we are given two isomorphic reverse systems that are specified in some other way, we can label their points so that they will have the same design as projection - and then, over that design, they will be in the same prime class (or, at least, in two prime classes in the same orbit). We can now see how the investigation should go. First, we exhibit a maximal set of non-isomorphic (9,24,3,8,2)-designs without repeated blocks, each design having its automorphism group determined. Then, for each of these designs, we find the number of prime classes, and sort these into orbits under the automorphism group (by looking for cases where an automorphism can fix a class). The total number of orbits will be the required number of non-isomorphic reverse Steiner systems.
4. A set of designs A solution to the problem of non-isomorphic designs, obtained by means of a computer, has been published by Gibbons [2]. I had independently solved the problem by hand, and have now checked that my solution agrees with Gibbons’. I will write out the designs as I found them, with Gibbons’ Roman numbers in brackets.
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D,: 123 124 134 156 159 167 178 189 234 256 259 267 (I) 278 289 357 358 368 369 379 457 458 468 469 479 We do not need the automorphism group of D , (which is of order 80). D,: 123 129 134 145 156 167 178 189 234 246 257 258 (VII) 267 289 357 358 368 369 379 459 468 478 479 569 Group of order 2 generated by (12) (39) (48) (5) (67).
D,:
123 129 134 145 156 167 178 189 234 246 258 259 (IX) 267 278 357 358 368 369 379 457 468 479 489 569 Group is trivial. 123 129 134 145 156 167 178 189 235 246 247 257 (HI) 268 289 348 359 367 369 378 458 469 479 568 579 Group of order 2 generated by (18) (2) (36)(45) (7) (9). D,:
D,: 123 129 134 145 156 167 178 189 235 246 247 257 (IV) 268 289 348 358 367 369 379 459 469 478 568 579 Group is trivial. Dh: 123 129 134 145 156 167 178 189 234 246 257 259 (V) 268 278 357 358 368 369 379 458 467 479 489 569 Group is trivial. D,: 123 129 134 145 156 167 178 189 235 246 247 257 (VI) 268 289 348 358 367 369 379 459 468 479 569 578 Group of order 2 generated by (14) (28) (3) (5) (69) (7).
D,: 124 127 136 139 146 157 158 189 235 238 245 268 (X) 269 279 347 349 356 378 458 467 489 569 579 678 Group of order 6 generated by (123) (456) (789) and (14) (26) (35) (7) (89). D,: 126 127 134 139 145 158 168 179 237 (XI) 256 289 348 356 359 367 467 469 478 Group of order 8 generated by (1 2345678) (9).
238 245 249 578 579 689
Ill,,: 124 127 136 138 145 159 169 178 235 238 249 256 (XII) 267 289 346 349 357 379 457 468 478 568 589 679 Group of order 6 generated by (123456) (789). D l l : 123 126 139 147 148 156 157 189 234 247 258 259 (VIII) 268 279 345 358 367 369 378 456 469 489 579 678 Group of order 6 generated by ( I 23456) (789).
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D12: 123 125 134 148 157 168 169 179 236 247 249 259 (XIII) 268 278 349 357 358 367 389 456 458 467 569 789 Group of order 18 generated by (123) (456) (789) and (174) (235689). D13: 126 129 137 139 147 148 156 158 237 238 246 248 (11) 257 259 346 349 356 358 457 459 678 679 689 789 Group of order 8 generated by (1234) (5) (67) (89) and (1) (24) (3) (5) (68) (79). 5. Construction of coverings
The problem of finding coverings, for a given design D, will be easier to handle if we set up some kind of arithmetic for it. Let us think again of the example at the beginning of Section 3, where 123 and 124 are blocks of D. We may express the assertion that each of the pairs { l’,2’) and { l”,2”) is collinear with a point that projects onto 3, while {l’,2”) and 11”. 2’) are collinear with points that project onto 4, by writing [l, 2; 31 = + 1,
[ 1,2; 41 = -1.
The symbol [p, q ; r ] is defined only when pqr is a block of the given design D. The following equations are satisfied when the symbols in them are defined:
[P, 4 ; rl = [q, p; rl. [ p , q ; r l . [p, q ; s I = -1
-
(1) ( r f s),
[P, 4 ; r l . [r, p; ql Eq, 7; PI= + I .
(2) (3)
In fact, we can see the truth of (3) by imagining that we have tossed three coins, and want to take one away so as to leave a head and a tail; there will be either two ways of doing this, or none. Equations (l),(2), (3) are not only necessary, but sufficient: if we have written down symbols [p, q; r ] , three for each block of D, that satisfy these equations, we have effectively set up a covering of D. A topologist might visualise the blocks of D as two-dimensional simplexes of a pseudomanifold, and might arrange a covering by setting up a one-dimensional cochain. However, a dual interpretation can be made in terms of graph theory. Let us consider the blocks of D as the vertices of a graph, two vertices being adjacent when the blocks have a pair of points in common [3]. Eleven of our designs will then have connected graphs; one exception is DI3,where the four vertices 789, 689, 679, 678 make up one component, and the other twenty make up a second. The graph of D, has three components: one consists of ten vertices 156, 167, 178, 189, 195, 256, 267, 278, 289, 295; another has ten vertices, and the third has four. The choice of a covering €or D will now correspond to the standard process of converting the graph into a directed graph (will “orientation” serve as a name for
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this process?). Taking once more the example symbolized by the equations [ 1.2; 31 = + I and [ 1.2; 41 = - 1, we may say that the edge which represents {1,2} has been oriented, the head end being at the vertex which represents the block 123, and the tail end at 124. Equation (3) now tells us that, of the three edges through any given vertex, either all have their head ends there, or one has a head end and two have tail ends. A simple counting argument then shows that, in any component of such a directed graph, exactly a quarter of the vertices have the former property (of being at the head ends of all their edges). But this means that a graph. if ten of its vertices make up a component, cannot possibly be oriented according to t h e rule. We conclude that there is no reverse system that has D , as its projection. As it turns out. each of the other twelve designs has some coverings, and accordingly has a graph for which there is, shall we say, at least one “admissible orientation”. This fact, which we shall establish in the next section, does not seem easy to infer from the graph theory. However, it does seem obvious what happens when G. the graph of a design D, has been provided with two admissible orientations A , and AZ. Let C be the set of edges of G which we have to reverse in changing from A , to A,. Then at any vertex, since the number of head ends is odd for A , , and likewise for A,. there must be an even number of edges that belong to C.That is, C will be a cycle of G. Conversely, given any cycle C of G, and one admissible orientation A , , we can change from A , to another admissible orientation by reversing all the edges of C. It follows that a given graph with first Betri number v, i f it has any admissible orientations at all, will have exactly 2” of them To find v, we subtract 24, the number of vertices of any one of our graphs, from 36, the number of edges, and add the number of components: so v is 13 for each of the designs from D, to D,, inclusive, and 14 for DI3.According to section 3. the respective numbers of prime classes are 2’ and 2‘, for the coverings of these designs.
6. Construction of coverings: an effective procedure It remains to be seen how the symbols [ p , q ; r ] can be used to handle these large sets of coverings. Let us take the design D, by way of example. We may choose to say that a covering of D, is “standardised” when the lines through 7’ are 7’7”0, 7’1’2”. 7’2’9”, 7‘9’5”. 7’5’1’’, 7’4’6, 7‘6’8”, 7’8‘3’’, 7’3‘4“. It is in fact easy to see that, in any prime class of coverings of D,, there is one and only one standardised covering. For such a covering of D,, we see that [ 1 , 2 ; 71 = - 1; and it follows, by equation (2) of section 5 , that [ l , 2; 41 = + I . Likewise, [2,9;61 = + 1,
[ 5 , 9 ;61 = + 1,
[ 1 , 5 ; 81 = + 1,
Non-isomorphic reverse Steiner triple systems of order 19
26 1
Let us now give names to five (suitably chosen) of the other symbols; it turns out that a convenient set is a = [2,8; 61,
fi = [8, 9; 41,
6 = [3,9; 41,
= [l,9; 81.
y = [2,5; 31,
Then we should be able to find all the other symbols in terms of these. First, we use Eq. (2) again:
[2,8;3]=-a,
[8,9; 1]=-/3,
[3,9; 1]=-6,
[1,9;5]=-~.
[2,5;4]=-y,
Now, using Eq. (3) as well, we deduce from the equations [3,8; 21 = +1 and [2,8; 3]= -a that [2,3; 81 = -a, whence again [2,3; 51 = a. Likewise, we find successively that
[2,6; 81 = a, [ 5 , 6; 91 = a, P - 6 ; 51 = Y, [4,8;9]=-p6, [ 5 , 8 ; 11= B E , [2,4; 5]=-y&, [ 1,6; 41 = - Y ~ E ,
[2,6; 9]= - a ; [5,6;3]=-a, [3,6; 1]=-7, [4,8;5]=@; [5,8;4]=--P~; [2,4; 1 ] = ~ 6 ~ [ 1,6; 31 = y& ;
[6,9; 2]= -a, [3,5;2]=ay, [4,9;3]=6, [1,8;9]=-/3~, [4,5; 8]=-&, ;[1,4;2]=~6~, [ 1,3; 61 = - 6 ~ ,
[6,9; 5]= a ; [3,5;6]=-ay; [4,9; 8]= -6; [1,8;5]=@, [4,5; 2 ] = 6 ~ ; [1,4;6]=-yb; [ 1,3; 91 = SE.
In the course of this arithmetic, we have ensured that condition (3) is satisfied for all but one of the 24 blocks of D,. We now observe that
[3,9; 1]=-6,
[1,9;3]=-~,
[1,3;9]=6~;
so condition (3) is satisfied for the last block, and there is no doubt that the system exists. Each of the symbols a,p, y, S, E can take the values +1 and -1 independently. So we have effectively constructed 2’ coverings of D8,all of them standardised as above. No two of these are in the same prime class, and we can obtain from them, by re-priming, the complete set of 213 “distinct” coverings of D,. It turns out that the same algorithm works for each of the designs D,, . . . , DI3,except that D,, with its higher Betti number needs six Greek letters.
7. Effect of an automorphism
Our next task, according to the programme at the end of Section 3, is to determine how the automorphism group of a design acts on the set of prime classes of its coverings. In particular, we should consider how often a non-trivial automorphism will leave a prime class fixed.
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By way of example. suppose that C , a prime class of coverings of D8,is fixed by the automorphism (123) (456) (789). In C there is one system, S say, which is "standardised" as in Section 5. According to Section 3, we are concerned with the "obvious" transformation 1'+2', 1"+2", 2'+3', . . . , determined by our automorphism. This will take S to a system, 7-say, which by hypothesis is also in the class C. Let us choose one of the two re-primings that take T back to S ; and let us write nq= -1 if the re-priming interchanges q' and q", but nq= +1 if it does not. Now the standardised system S will certainly have 7'9'5" as one of its lines; this will correspond to a line 8'7'6 in T, and the re-priming will take this to one or the other of two lines 7'6'8" and 7"6"8' that belong to S. So definitely n,r6 = - 1. In terms of the symbols a. 0, y , 6, E that belong to S, and of its numbers [p, q ; r], we can work out necessary and sufficient conditions for the re-primed system to be a standardised one: [ 9 . 3 ; 11 = -6, [9, 1: 81 = E , [9,2: 6]= + I , [9.6; 51 = a, [9.4; 31 = 6, [ 9 , 5 ;71-1, [9,7:2]=-1, [9.8; 41 = p,
[7,1: 2]= + l ; [7,2;9]=+1; [7,3; 41 = + 1; [7 , 4 ;6] = + 1; [7,5; 1 ] = + 1 ; [7,6; 81 = + 1; [7,8; 3]= + I ; [7,9; 51 = + 1;
These equations determine the set of numbers 7rq (to within multiplication of the whole set through by -1). But we are assuming, not merely that the re-primed system is standardised, but that it actually coincides with S. So we need five more conditions:
This is as far as we need to go, because the standardised covering of D, is uniquely determined by its symbols a, p, y, 6, €.'The only restrictions on S arc thc last five, which in fact reduce to two independent conditions a = ps = E .
The conclusion is that (123) (456) (789) fixes just a quarter of t h e prime classes of coverings of D, (namely those whose standardised representatives satisfy these two conditions). The same arithmetic, applied to the involution (14) (26) (35) (7) (89) of D,, leads to an impossible condition a = -a. So this automorphism does not fix any
Non-isomorphic reverse Steiner triple systems of order 29
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prime class. It turns out, likewise, that no prime class is fixed by any involution of any standard design (nor, consequently, by any automorphism of period 4 or 8).
8. Counting the systems Accordingly, we see in the case of D9that the automorphism (12345678) (9) must permute the 32 prime classes in four orbits of length 8; and this means that, up to automorphism, D9is the projection of only four reverse systems. Likewise, each of the designs D,, D4, D,, having an automorphism group of order two, is the projection of 16 systems. For D13, there are 64 prime classes, permuted by the eight automorphisms in eight orbits of length 8; so D,, is the projection of eight reverse systems. Going back to D,, we see that its automorphism group has two generators, one of order three fixing eight of the 32 prime classes, and one of order two fixing none. It follows that there are four orbits of length 2 and four of length 6. And therefore a maximal set of non-isomorphic coverings of D, will consist of eight reverse systems, four having automorphism groups of order 6 (which turn out to be cyclic), and four of order 2. The designs D,, and D,,, each of which has an automorphism group of order 6, give results analogous to those for D,. Each of these has eight nonisomorphic coverings, of which four have cyclic automorphism groups of order 6 . The design D,,, with a group of order 18, has two prime classes which are fixed by all its automorphisms of period 3 (but interchanged by all its involutions). Then there are six classes that are fixed by (123) (456) (789), but the automorphism group is transitive on this set of six; and likewise for (147) (258) (369). Finally, there is a set of 18 prime classes on which the group is sharply transitive. The conclusion is that D,, has four non-isomorphic coverings, with automorphism groups of orders 18, 6, 6, 2. These groups turn out to be Abelian, even though the group for D,, is not. To summarise, we have 32 non-isomorphic coverings for each of the designs that have trivial automorphism groups, namely D,, D,, and D,. We have 16 for each of D,, D4, D,; eight for each of D,, Dlo,D I 1DI3; , four for each of D, and D,,: but none for D,.
Theorem 3. Up to isomorphism, there are just 184 reuerse Steiner triple systems of order 19. One of these has an automorphism group of order 18 (Abelian, not cyclic), and fourteen have (cyclic) groups of order 6. For each of rhe others, the unique reversal and the identity are the only automorphisms. We may observe that the system with the group of order 18 is the one that was discovered by Rosa [4].
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References [ I ] J . Doyen, A note on reverse Steiner triple systems, Discrete Math. 1 (1972) 315-319. [2] P.B. Gibbons, Computing techniques for the construction and analysis of block designs, Ph.D. thesis, University of Toronto 1976 (=Technical Report no. 92, Department of Computer Science, University of Toronto, May 1976). [3] E.S. Kramer, Indecomposable triple systems, Discrete Math. 8 (1974) 173-180. [ I ]A. Rosa, On reverse Steiner triple systems, Discrete Math. 2 (1972) 61-71.