Automorphism groups of Steiner triple systems

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1 Automorphism groups of Steiner triple systems Alexander Rosa Department of Mathematics and Statistics, McMaster University Rosa (McMaster) Automorphism groups of Steiner triple systems 1 / 20

2 Steiner triple systems A Steiner triple system is a pair (V, B) where V is a set of elements (points, vertices), where V = v (the order), and B is a collection of 3-subsets of V called triples or blocks such that every 2-subset of V is contained in exactly one triple. (Kirkman 1847) A Steiner triple system of order v [STS(v), S(2, 3, v)] exists if and only if v 1 or 3 (mod 6). Two classes of STS(v): projective triple systems of order 2 n 1; PG(d, 2) (n = d + 1) affine triple systems of order 3 n ; AG(d, 3) (d = n) Rosa (McMaster) Automorphism groups of Steiner triple systems 2 / 20

3 Enumeration N(v) = number of nonisomorphic STS(v) A(v) = number of nonisomorphic automorphism-free STS(v) v N(v) ,084,874,829 A(v) ,084,710,071 R.M. Wilson (MZ 1974): log N(v) 1 6 v 2 log v. Rosa (McMaster) Automorphism groups of Steiner triple systems 3 / 20

4 Enumeration N(v) = number of nonisomorphic STS(v) A(v) = number of nonisomorphic automorphism-free STS(v) v N(v) ,084,874,829 A(v) ,084,710,071 R.M. Wilson (MZ 1974): log N(v) 1 6 v 2 log v. Rosa (McMaster) Automorphism groups of Steiner triple systems 3 / 20

5 Automorphism groups of Steiner triple systems Eric Mendelsohn (JCT A 1985): Every group is an automorphism group of a finite Steiner triple system. Proof (sketch). Basically, three ingredients are needed. 1. Amicable directed 4-path system: a set X, together with a collection of directed 4-paths (i.e. sets of directed edges {(w, x), (x, y), (y, z)} with w, x, y, z X, w, x, y, z = 4) such that any two 4-paths of the system have at most one vertex in common, and if they do have one vertex in common then it is the end-vertex in both 4-paths; moreover, every x X is contained in at least one 4-path. 2. An automorphism-free and subsystem-free STS(15). 3. A sufficiently large projective STS + the replacement property for STSs. Rosa (McMaster) Automorphism groups of Steiner triple systems 4 / 20

6 1. Every group G is the automorphism group of a graph. Fix G, and let Γ = (V, E) be a graph such that Aut Γ = G. For every edge e = {x, y} E, replace e with two directed edges (x, ē), (y, ē) where ē is a new vertex. Add a new vertex and a directed edge (, x) for each x V. The automorphism group of the resulting directed graph (Y, A) is isomorphic to G. Now form a directed 4-path system: let Z = Y (A {0, 1}) be its set of elements, and for each a = (x, y) A, form a directed 4-path (x, (a, 0), (a, 1), y). The resulting directed 4-path system has again G as its automorphism group, and is amicable if Γ has no isolated vertices. 2. Let (W, D) be an STS(15) which is automorphism-free and has no subsystem of order 7. Rosa (McMaster) Automorphism groups of Steiner triple systems 5 / 20

7 3. Let (X, P) be an amicable directed 4-path system where X = {x 1,..., x n } and G is its automorphism group. Take PG(n 1, 2), i.e. projective triple system of order 2 n 1, with elements non-zero binary vectors of length n. Fix 4 coordinate positions, then the 15 vectors in which all other n 4 coordinate positions are zero induce a subsystem of order 15. For every 4-path (x a, x b, x c, x d ), replace the subsystem of order 15 induced by coordinate positions a, b, c, d with our STS(15) (W, D) from 2. Do this replacement for each directed 4-path; each of these replacements may be done independently. The result is another STS(2 n 1), say (V, B ), whose automorphism group is isomorphic to G. Rosa (McMaster) Automorphism groups of Steiner triple systems 6 / 20

8 Automorphism-free Steiner triple systems Lindner - Rosa (J. Alg. 1975): For every order v 1, 3 (mod 6), v 15 there exists an automorphism-free STS(v), and the number of nonisomorphic automorphism-free STS(v) tends to infinity with v. L. Babai (ADM 1980): Almost all Steiner triple systems are automorphism-free. In his proof, Babai uses Wilson s result and (a weaker form) of van der Waerden s permanent conjecture to obtain A(v) N(v). Rosa (McMaster) Automorphism groups of Steiner triple systems 7 / 20

9 Highly symmetric Steiner triple systems 2-transitive 2-homogeneous flag-transitive block-transitive transitive Rosa (McMaster) Automorphism groups of Steiner triple systems 8 / 20

10 2-transitive Steiner triple systems M. Hall Jr. (IBM J. Res. Dev. 1960): Conjecture. The 2-transitive Steiner triple systems are precisely the projective triple systems and the affine triple systems. Sample of results in that paper: 1. A 2-transitive STS must contain a subsystem of order 7 or If in an STS any triangle generates STS(7) then the STS is projective. 3. If for every element of an STS there is an involutory automorphism fixing just that element then any triangle generates an STS(9). Such STSs are called Hall triple systems. However, not every Hall triple system is an affine space. But if the automorphism group is 2-transitive then it is an affine space. Rosa (McMaster) Automorphism groups of Steiner triple systems 9 / 20

11 Proving Hall s conjecture The proof of Hall s conjecture had to await CFSG. J.Key - E. Shult (JCT A 1984) W.M. Kantor (JCT A 1985) M. Hall Jr. (JCT A 1985) Burnside s Theorem. If G is 2-transitive of degree v then either (i) G contains a transitive normal subgroup A which is elementary abelian of order A = p r = v, or (ii) G contains a unique minimal normal subgroup N, where N is simple nonabelian, and N G Aut N. For an STS(v) with a 2-transitive automorphism group, case (i) leads to affine spaces AG(d, 3) with v = 3 d, and case (ii) to projective spaces PG(d, 2) with v = 2 d+1 1. Rosa (McMaster) Automorphism groups of Steiner triple systems 10 / 20

12 2-homogeneous Steiner triple systems Netto triple systems Let q = p n 7 (mod 12), let ω be a primitive root in GF(p n ). Then {{ω 2i + j, ω 2t+2i + j, ω 4t+2i + j} : 0 i < t, ; j GF(q)} are triples of an STS(p n ) called Netto triple system. Another construction: Let ɛ be a primitive sixth root of unity in F p n. Then the orbit of {0, 1, ɛ} under the action of Γ is a Netto triple system where Γ = {a 2 xσ + b : a, b F p n, σ Aut(F p n)}. Example. Netto triple system of order 19: {0, 1, 8}, {0, 2, 5}, {0, 4, 13} mod 19 Rosa (McMaster) Automorphism groups of Steiner triple systems 11 / 20

13 Classification of 2-homogeneous STSs A. Delandtsheer, J.Doyen, J. Siemons, C. Tamburini (JCT A 1985) The only Steiner triple systems which are 2-homogeneous but not 2-transitive are the Netto systems. In fact, Netto triple systems are the only Steiner systems S(2, k, v) whose full automorphism group is 2-homogeneous but not 2-transitive. (Their proof does not use CFSG!) Note: R. M. Robinson (Math. Comput. 1975) investigated Netto triple systems in great detail, especially from the point of view of uniformity properties. Rosa (McMaster) Automorphism groups of Steiner triple systems 12 / 20

14 Flag-transitive and block-transitive Steiner triple systems A classical work here is that of H. Lüneburg (MA 1963). Further references are P.C. Clapham (Discrete Math. 1976) and F. Buekenhout, A. Delandtsheer, J. Doyen (JCT A 1988). For Steiner triple systems, flag-transitivity is equivalent to 2-homogeneity. That is, the only flag-transitive STSs other than projective and affine triple systems are the Netto triple systems. Flag-transitivity clearly implies block-transitivity. Clapham characterized (without CFSG) the block-transitive Steiner triple systems: they are exactly those which are flag-transitive (and thus also 2-homogeneous). Rosa (McMaster) Automorphism groups of Steiner triple systems 13 / 20

15 Transitive Steiner triple systems A transitive Steiner triple system exists for all admissible orders v 1, 3 (mod 6). Indeed, a cyclic STS(v) exists if and only if v 1, 3 (mod 6), v 9. This was first proved by Rose Peltesohn in And STS(9) AG(2, 3) is 2-transitive. Question. Which permutations are automorphisms of Steiner triple systems? In an STS, if α is an automorphism, the elements fixed by α induce a subsystem. If STS(v) has an automorphism α with exactly v 1 2 fixed points then α is an involution and v 7 or 15 (mod 24). Rosa (McMaster) Automorphism groups of Steiner triple systems 14 / 20

16 Transitive Steiner triple systems A transitive Steiner triple system exists for all admissible orders v 1, 3 (mod 6). Indeed, a cyclic STS(v) exists if and only if v 1, 3 (mod 6), v 9. This was first proved by Rose Peltesohn in And STS(9) AG(2, 3) is 2-transitive. Question. Which permutations are automorphisms of Steiner triple systems? In an STS, if α is an automorphism, the elements fixed by α induce a subsystem. If STS(v) has an automorphism α with exactly v 1 2 fixed points then α is an involution and v 7 or 15 (mod 24). Rosa (McMaster) Automorphism groups of Steiner triple systems 14 / 20

17 Transitive Steiner triple systems A transitive Steiner triple system exists for all admissible orders v 1, 3 (mod 6). Indeed, a cyclic STS(v) exists if and only if v 1, 3 (mod 6), v 9. This was first proved by Rose Peltesohn in And STS(9) AG(2, 3) is 2-transitive. Question. Which permutations are automorphisms of Steiner triple systems? In an STS, if α is an automorphism, the elements fixed by α induce a subsystem. If STS(v) has an automorphism α with exactly v 1 2 fixed points then α is an involution and v 7 or 15 (mod 24). Rosa (McMaster) Automorphism groups of Steiner triple systems 14 / 20

18 Transitive Steiner triple systems A transitive Steiner triple system exists for all admissible orders v 1, 3 (mod 6). Indeed, a cyclic STS(v) exists if and only if v 1, 3 (mod 6), v 9. This was first proved by Rose Peltesohn in And STS(9) AG(2, 3) is 2-transitive. Question. Which permutations are automorphisms of Steiner triple systems? In an STS, if α is an automorphism, the elements fixed by α induce a subsystem. If STS(v) has an automorphism α with exactly v 1 2 fixed points then α is an involution and v 7 or 15 (mod 24). Rosa (McMaster) Automorphism groups of Steiner triple systems 14 / 20

19 Some automorphism types 1-rotational (i.e., one fixed point and one cycle of length v 1) A 1-rotational STS(v) exists if and only v 3 or 9 (mod 24). 2-rotational (i.e. one fixed point and two cycles of length v 1 2 ) A 2-rotational STS(v) exists if and only if v 1, 3, 7, 9, 15, 19 (mod 24). reverse (i.e. an involution fixing exactly one point) A reverse STS(v) exists if and only if v 1, 3, 9, 19 (mod 24). Many other types are described in C.J. Colbourn, A. Rosa, Triple Systems, Oxford Rosa (McMaster) Automorphism groups of Steiner triple systems 15 / 20

20 Steiner triple systems with multiplier automorphisms Suppose (Z v, B) is a cyclic triple system of order v. Define B α = {{αx, αy, αz} : {x, y, z} B} where α Z v. If (Z v, B α ) is also an STS(v) then α is a multiplier, and if B α = B then α is a multiplier automorphism. Question. For which orders v does there exist a cyclic STS(v) having a nontrivial multiplier automorphism? Indication of difficulty: No such cyclic STS(v) exists for v {9, 25, 27, 45, 55, 81}. A cyclic STS(v) cannot have 1 as a multiplier automorphism when v > 3. A cyclic STS(p β ) cannot have a nontrivial automorphism if p = 2 α + 1 is prime. Rosa (McMaster) Automorphism groups of Steiner triple systems 16 / 20

21 Steiner triple systems with multiplier automorphisms Suppose (Z v, B) is a cyclic triple system of order v. Define B α = {{αx, αy, αz} : {x, y, z} B} where α Z v. If (Z v, B α ) is also an STS(v) then α is a multiplier, and if B α = B then α is a multiplier automorphism. Question. For which orders v does there exist a cyclic STS(v) having a nontrivial multiplier automorphism? Indication of difficulty: No such cyclic STS(v) exists for v {9, 25, 27, 45, 55, 81}. A cyclic STS(v) cannot have 1 as a multiplier automorphism when v > 3. A cyclic STS(p β ) cannot have a nontrivial automorphism if p = 2 α + 1 is prime. Rosa (McMaster) Automorphism groups of Steiner triple systems 16 / 20

22 On the positive side: If v = p β, p 1 (mod 6), p prime then there exist a cyclic STS(v) for which cubic roots of unity are multiplier automorphisms. Let p 5 (mod 6), p prime. Let ω be an element of order p(p 1) satisfying certain additional conditions, then there exists a cyclic STS(p 2 ) with a multiplier automorphism (whose order is the largest odd divisor of p 1). A direct product type construction: If there exists a cyclic STS(v) with a nontrivial multiplier automorphism, and a cyclic STS(w) where v and w are coprime then there exists a cyclic STS(v.w) with a nontrivial multiplier automorphism. If v 1, 3 (mod 6) and v has a prime divisor p 1 (mod 6) then there exists a cyclic STS(v) with a nontrivial multiplier automorphism except possibly when v = 9p. Rosa (McMaster) Automorphism groups of Steiner triple systems 17 / 20

23 On the positive side: If v = p β, p 1 (mod 6), p prime then there exist a cyclic STS(v) for which cubic roots of unity are multiplier automorphisms. Let p 5 (mod 6), p prime. Let ω be an element of order p(p 1) satisfying certain additional conditions, then there exists a cyclic STS(p 2 ) with a multiplier automorphism (whose order is the largest odd divisor of p 1). A direct product type construction: If there exists a cyclic STS(v) with a nontrivial multiplier automorphism, and a cyclic STS(w) where v and w are coprime then there exists a cyclic STS(v.w) with a nontrivial multiplier automorphism. If v 1, 3 (mod 6) and v has a prime divisor p 1 (mod 6) then there exists a cyclic STS(v) with a nontrivial multiplier automorphism except possibly when v = 9p. Rosa (McMaster) Automorphism groups of Steiner triple systems 17 / 20

24 Uniform and perfect Steiner triple systems In an STS(v), say, (V, B), the interlacing graph Γ xy where x, y V, is defined as follows: the vertex set of Γ xy is the set V \ {x, y, z} where z is the third element of the triple containing x, y; the edges are all pairs {a, b} such that {x, a, b} B or {y, a, b} B. A Steiner triple system is uniform if the interlacing graphs Γ xy are mutually isomorphic for all x, y V. It is perfect if all interlacing graphs are Hamiltonian cycles. Rosa (McMaster) Automorphism groups of Steiner triple systems 18 / 20

25 Perfect STSs All projective and affine triple systems, Hall triple systems and Netto triple system are uniform but none is perfect, apart from the STS(7) and STS(9) which are "trivially" perfect. There exist no perfect Steiner triple systems of orders 13, 15, 19 and 21. Perfect STSs are known to exist only for 12 further orders: 25, 33, 79, 139, 367, 811, 1531, 25771, 50923, 61339, 69991, Rosa (McMaster) Automorphism groups of Steiner triple systems 19 / 20

26 THANK YOU FOR YOUR ATTENTION! Rosa (McMaster) Automorphism groups of Steiner triple systems 20 / 20

arxiv: v1 [math.co] 31 Mar 2016

arxiv: v1 [math.co] 31 Mar 2016 3-pyramidal Steiner Triple Systems Marco Buratti Gloria Rinaldi Tommaso Traetta arxiv:1603.09645v1 [math.co] 31 Mar 2016 October 1, 2018 Abstract A design is said to be f-pyramidalwhen it has an automorphism