ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 17 (2019) 617-626 https://doi.org/10.26493/1855-3974.1891.3b7 (Also available at http://amc-journal.eu) On flag-transitive automorphism groups of symmetric designs* Seyed Hassan AlaviAshraf Daneshkhah, Narges Okhovat Department of Mathematics, Faculty of Science, Bu-Ali Sina University, Hamedan, Iran Received 27 December 2018, accepted 4 November 2019, published online 12 December 2019 In this article, we study flag-transitive automorphism groups of non-trivial symmetric (v, k, A) designs, where A divides k and k > A2. We show that such an automorphism group is either point-primitive of affine or almost simple type, or point-imprimitive with parameters v = A2 (A + 2) and k = A(A +1), for some positive integer A. We also provide some examples in both possibilities. Keywords: Symmetric design, flag-transitive, point-primitive, point-imprimitive, automorphism group. Math. Subj. Class.: 05B05, 05B25, 20B25 1 Introduction A t-design D = (P, B) with parameters (v, k, A) is an incidence structure consisting of a set P of v points, and a set B of k-element subsets of P, called blocks, such that every t-element subset of points lies in exactly A blocks. The design D is non-trivial if t < k < v - t, and is symmetric if |B| = v. By [7, Theorem 1.1], if D is symmetric and non-trivial, then t < 2, see also [12, Theorem 1.27]. Thus we study non-trivial symmetric 2-designs with parameters (v, k, A) which we simply call non-trivial symmetric (v, k, A) designs. A flag of D is an incident pair (a, B), where a and B are a point and a block of D, respectively. An automorphism of a symmetric design D is a permutation of the points permuting the blocks and preserving the incidence relation. An automorphism group G of D is called flag-transitive if it is transitive on the set of flags of D. If G leaves invariant a non-trivial partition of P, then G is said to be point-imprimitive; otherwise G is called * The authors would like to thank anonymous referees for providing us helpful and constructive comments and suggestions. t Corresponding author. E-mail addresses: alavi.s.hassan@basu.ac.ir, alavi.s.hassan@gmail.com (Seyed Hassan Alavi), adanesh@basu.ac.ir, daneshkhah.ashraf@gmail.com (Ashraf Daneshkhah), okhovat.nargeshh@gmail.com (Narges Okhovat) Abstract ©® This work is licensed under https://creativecommons.org/licenses/by/4.0/ 618 Ars Math. Contemp. 17 (2019) 493-514 point-primitive. We here adopt the standard notation as in [8, 23] for finite simple groups of Lie type. For example, we use PSLn(q), PSpn(q), PSUn(q), PQ2n+1 (q) and PQ±„(q) to denote the finite classical simple groups. Symmetric and alternating groups on n letters are denoted by Sn and An, respectively. Further notation and definitions in both design theory and group theory are standard and can be found, for example in [10, 12, 14]. We also use the software GAP [21] for computational arguments. Flag-transitive incidence structures have been of most interest. In 1961, Higman and McLaughlin [11] proved that a flag-transitive automorphism group of a linear space must act primitively on its points set, and then Buekenhout, Delandtsheer and Doyen [5] studied this action in details and proved that a linear space admitting a flag-transitive automorphism group (which is in fact point-primitive) is either of affine, or almost simple type. Thereafter, a deep result [6], namely the classification of flag-transitive finite linear spaces relying on the Classification of Finite Simple Groups (CFSG) was announced. Although, flag-transitive symmetric designs are not necessarily point-primitive, Regueiro [18] proved that a flag-transitive and point-primitive automorphism group of such designs for A < 4 is of affine or almost simple type, and so using CFSG, she determined all flag-transitive and point-primitive biplanes (A = 2). In conclusion, she gave a classification of flag-transitive biplanes except for the 1-dimensional affine case [17]. Tian and Zhou [22] proved that a flag-transitive and point-primitive automorphism group of a symmetric design with A < 100 must be of affine or almost simple type. Generally, Zieschang [25] proved in 1988 that a flag-transitive automorphism group of a 2-design with gcd(r, A) = 1 is (point-primitive) of affine or almost simple type, and this result has been generalised by Zhuo and Zhan [24] for A > gcd(r, A)2. 1.1 Main result In this paper, we study flag-transitive automorphism groups of symmetric (v, k, A) designs, where A divides k and k > A2, and we show that such an automorphism group is not necessarily point-primitive: Theorem 1.1. Let D = (P, B) be a non-trivial symmetric (v, k, A) design with A > 1, and let G be a flag-transitive automorphism group of D. If A divides k and k > A2, then one of the following holds: (a) G is point-primitive of affine or almost simple type; (b) G is point-imprimitive and v = A2 (A + 2) and k = A(A + 1), for some positive integer A. In particular, if G has d classes of imprimitivity of size c, then there is a constant l such that, for each block B and each class A, the size \B n A| is either 0, or l, and (c, d, l) = (A2, A + 2, A) or (A + 2, A2,2). We highlight here that if A divides k, then gcd(k, A)2 = A2 > A which does not satisfy the conditions which have been studied in [24, 25]. Moreover, in Section 1.2, we provide some examples to show that both possibilities in Theorem 1.1 can actually occur. In order to prove Theorem 1.1(a), we apply O'Nan-Scott Theorem [15] and discuss possible types of primitive groups in Section 3. We further note that our proof for part (a) relies on CFSG. To prove part (b), we use an important result by Praeger and Zhou [20, Theorem 1.1] on characterisation of imprimitive flag-transitive symmetric designs. S. H. Alavi et al.: Onflag-transitive automorphism groups of symmetric designs 619 1.2 Examples and comments on Theorem 1.1 Here, we give some examples of symmetric (v, k, A) designs admitting flag-transitive automorphism groups, where A divides k and k > A2. In Table 1, we list some small examples of such designs with A < 3. To our knowledge the design in Line 2 is the only point-primitive example of symmetric designs with v < 2500 satisfying the conditions of Theorem 1.1 and this motivates the authors to investigate symmetric designs admitting symplectic automorphism groups [3]. More examples of symmetric designs admitting flag-transitive and point-imprimitive automorphism groups can be found in [20] and references therein. Line 1. Hussain [13] showed that there are exactly three symmetric (16,6,2) designs, and Regueiro proved that exactly two of such designs are flag-transitive and point-imprimitive [18, p. 139]. Line 2. The symmetric design in this line arises from the study of primitive permutation groups with small degrees. This design belongs to a class of symmetric designs with parameters (3m(3m +1)/2, 3m-1(3m -1)/2, 3m-1(3m-1 -1)/2), for some positive integer m > 1, see [4, 9]. If m = 2, then we obtain the symmetric (45,12, 3) design admitting PSp4(3) or PSp4(3) : 2 as flag-transitive automorphism group of rank 3, see [4]. Lines 3-4. Mathon and Spence [16] constructed 2616 pairwise non-isomorphic symmetric (45,12, 3) designs with non-trivial automorphism groups. Praeger [19] proved that there are exactly two flag-transitive symmetric (45,12, 3) designs, exactly one of which admits a point-imprimitive group, and this example satisfies Line 4, but not Line 3. Table 1: Some symmetric designs satisfying the conditions in Theorem 1.1. Line v k A c d l Case Examples Reference Comments 1 16 6 2 4 4 2 (b) 2 [13], [18] imprimitive 2 45 12 3 - - - (a) 1 [4] primitive 3 45 12 3 5 9 2 (b) None [19] imprimitive 4 45 12 3 9 5 3 (b) 1 [19] imprimitive 2 Preliminaries In this section, we state some useful facts in both design theory and group theory. Lemma 2.1 ([1, Lemma 2.1]). Let D be a symmetric (v, k, A) design, and let G be a flag-transitive automorphism group of D. If a is a point in P and H := Ga, then (a) k(k - 1) = A(v - 1); (b) k divides |H | and Av < k2. Lemma 2.2 ([2, Corollary 4.3]). Let T be a finite simple classical group of dimension n over a finite field Fq of size q. Then 620 Ars Math. Contemp. 17 (2019) 493-514 (a) If T = PSLn(q) with n > 2, then |T| > q"2-2; (b) If T = PSUn(q) with n > 3, then |T| > (1 - q-1)q"2-2; (c) If T = PSpn(q) with n > 4, then |T| > q1 "("+1)/(2a), where a = gcd(2, q - 1); (d) If T = PQ"(q) with n > 7, then |T| > q2"("-1)/(4,0), where £ = gcd(2,n). Lemma 2.3. Let T be a non-abelian finite simple group satisfying |T| < 8 Out(T)|3. (2.1) Then T is isomorphic to A5 or A6. Proof. If T is a sporadic simple group or an alternating group An with n > 7, then | Out(T)| G {1, 2}, and so by (2.1), we must have |T| < 64, which is a contradiction. Note that the alternating groups A5 and A6 satisfy (2.1) as claimed. Therefore, we only need to consider the case where T is a finite simple group of Lie type. In what follows, we discuss each case separately. Let T = PSLn(q) with q = p° and n > 2. If n = 2, then q > 4 and | Out(T)| = a • gcd(2, q - 1), and so by Lemma 2.2(a) and (2.1), we have that q2 < | PSL2(q)| < 8a3 • gcd(2, q - 1)3 ^ 64a3. Thus, q2 < 64a3. This inequality holds only for (p, a) G {(2,1), (2, 2), (2, 3), (2,4), (2, 5), (2, 6), (2, 7), (3,1), (3, 2), (3, 3), (5,1), (7,1)}. Note in this case that q > 4, and hence by (2.1), we conclude that T is either PSL2(4) = PSL2(5) = As, or PSL2(9) = ¿6, as claimed. If n = 3, then by Lemma 2.2(a), we have that q7 < 64a3 • gcd(3, q - 1)3 < 64a3q3, and so q4 < 64a3. If q would be odd, then we would have 34a < 64a3, which is impossible. If q = 2°, then 2° < 64a3 would hold only for a = 1, 2. Therefore, T is isomorphic to PSL3(2) or PSL3(4). These simple groups do not satisfy (2.1). If n > 4, then (2.1) implies that q11 < 64a3, but this inequality has no possible solution. Let T = PSUn(q) with q = p° and n > 3. By Lemma 2.2(b), we have that |T| > (1 -q-1)q"2-2, and so (2.1) implies that (1 - q-1)q"2-2 < 64a3 • gcd(n, q +1)3. If n = 3, then (1 - q-1)q7 < 64a3 • gcd(n, q + 1)3, and so q6 < 27 • 64a3. This inequality holds only for (p, a) G {(2,1), (2, 2), (3,1)}. Note that PSU3(2) is not simple. Therefore, T is isomorphic to PSU3(3) or PSU3(4). These simple groups do not satisfy (2.1). If n > 4, thensince (q +1)3 < 4 • q3(q -1), we would have q" -3 < 64a3 • gcd(n, q +1)3/(q - 1) < 4 • 64a3 (q +1)3/4(q - 1) < 4 • 64a3q3,andso q"2-6 < 4 • 64a3, and hence q10 < 4 • 64a3, which is impossible. Let T = PSp"(q) with q = p° and n > 4. By Lemma 2.2(c), we observe that |T| > q1 "("+1)/2 gcd(2, q-1) > q1 "("+1)/4. By (2.1), we have that q10 < q2"("+1) < 4-64a3, and so q10 < 4 • 64a3, which is impossible. Let T = P^"(q) with q = p° odd and n > 7. Then we conclude by Lemma 2.2(d) that |T| > q2"("-1)/8. Since | Out(T)| = 2a and n > 7, it follows from (2.1) that q21 < 83a3, which is impossible. Let T = P^"(q) with q = p° and n > 8 and e = ±. It follows from Lemma 2.2(d) that |T| > q2"(n-1)/8. Note that | Out(T)| < 6a • gcd(4,qn - e) < 24a. Then (2.1) implies that q28 < 82 • 243a3, which is impossible. S. H. Alavi et al.: Onflag-transitive automorphism groups of symmetric designs 621 Let T be one of the finite exceptional groups F4(q), Ee(q), Eg(q), ^(q) (q = 22m+1), 3D4(q) and 2Ee(q). Then |T| > q20, and so (2.1) implies that q20 < 8 • 23 • 33a3, which is impossible. If T = G2(q) with q = pa = 2. Then by (2.1), we have that q12 < q6(q2 - 1)(q6 - 1) < 8 • 23a3, and so q12 < 8 • 23a3, which is impossible. Similarly, if T is one of the groups 2B2 (q) with q = 22m+1 and 2G2(q) with q = 32m+1, then |T| > q4, and so (2.1) implies that q4 < 8a3, which is impossible. □ 3 Point-primitive designs In what follows, we assume that D = (P, B) is a non-trivial symmetric (v, k, A) design admitting a flag-transitive and point-primitive automorphism group G. Let also A divide k and k > A2 and set t := k/A. Notice that A < k, and so t > 2. We moreover observe by Lemma 2.1(a) that k = ; (3.1) A = ^. (3.2) Since also G is a primitive permutation group on P, by O'Nan-Scott Theorem [15], G is of one of the following types: (a) Affine; (b) Almost simple; (c) Simple diagonal; (d) Product; (e) Twisted wreath product. 3.1 Product and twisted wreath product type In this section, we assume that G is a primitive group of product type on P, that is to say, G < HI Si, where H is of almost simple or diagonal type on the set r of size m := | r | > 5 and I > 2. In this case, P = Lemma 3.1. Let G be a flag-transitive point-primitive automorphism group of product type. Then k divides A^(m — 1). Proof. See the proof of Lemma 4 in [18]. □ Proposition 3.2. If D = (P, B) is a non-trivial symmetric (v,k, A) design admitting a flag-transitive and point-primitive automorphism group G, where A divides k and k > A2, then G is not of product type. Proof. Assume the contrary. Suppose that G is of product type. Then v = m£. Note by Lemma 3.1 that k divides A^(m — 1), and so t = k/A divides ¿(m — 1). We also note by 622 Ars Math. Contemp. 17 (2019) 493-514 Lemma 2.1(b) that Av < k2. Then v < At2, and since A < t, we have that v < t3. Recall that t divides i(m - 1). Hence m£ < i3(m - 1)3. (3.3) Then m£ < i3m3, or equivalently, m£-3 < i3. Since m > 5, it follows that 5£-3 < i3, and this is true for 2 < i < 6. If i = 6, then since m6-3 < 63, we conclude that m = 5, but (m, i) = (5, 6) does not satisfy (3.3). Therefore, 2 < i < 5. Suppose first that i =5. Then by (3.3), we have that m5 < 53(m - 1)3, and so 5 < m < 9. It follows from (3.1) that t divides m5 — 1. For each 5 < m < 9, we can obtain divisors t of m5 - 1. Note by (3.2) that t2 must divide m5 -1 +1. This is true only for m = 7 when t = 2 or 6 for which (v, k, A) = (16807, 8404,4202) or (16807, 2802, 467), respectively. Since A2 < k, these parameters can be ruled out. Suppose that i = 4. Then by (3.3), we have that m5 < 43(m - 1)3, and so 5 < m < 9. By the same argument as in the case where i = 5, by (3.1) and (3.2), we obtain possible parameters (m, t, v, k, A) as in Table 2. Note by Lemma 3.1 that k must divide 4A(m - 1), and this is not true, for all parameters in Table 2. Table 2: Possible values for (m, t, v, k, A) when i = 4. m t v k A 13 51 28561 561 11 31 555 923521 1665 3 47 345 4879681 14145 41 57 416 10556001 25376 61 Suppose now that i = 3. We again apply Lemma 3.1 and conclude that t divides 3(m - 1). Then there exists a positive integer x such that 3(m - 1) = tx, and so m = (tx + 3)/3. By (3.2), we have that m2 +1 - 1 t2x3 + 9tx2 + 27x + 27 A — -- — -. t2 27t Then 27At = t2x3 + 9tx2 + 27x + 27. Therefore, t must divide 27x + 27, and so ty = 27x + 27, for some positive integer y. Thus, _ t(ty - 27)3 + 9 ■ 27(ty - 27)2 + 273y A = 27^ , (3.4) for some positive integers t and y. Since A2 < k, we have that A < t, and so t(ty - 27)3 + 9 ■ 27(ty - 27)2 + 273y < 274t. (3.5) If y > 32, then t(ty - 27)3 + 9 ■ 27(ty - 27)2 + 273y > t(32t - 27)3 + 9 ■ 27(32t - 27)2 + 32 ■ 273 > 274t, S. H. Alavi et al.: Onflag-transitive automorphism groups of symmetric designs 623 for t > 2. Thus 1 < y < 31, and so by (3.5), we conclude that 2 < t < 107. For each such y and t, by straightforward calculation, we observe that A as in (3.4) is not a positive integer. Suppose finally that I = 2. Recall by Lemma 3.1 that t divides 2(m - 1). Then 2(m — 1) = tx for some positive integer x, and so m = (tx + 2)/2. It follows from (3.2) that A = (tx2 + 4x + 4)/4t, or equivalently, 4tA = tx2 + 4x + 4. This shows that t divides 4x + 4, and so ty = 4x + 4, for some positive integer y. Therefore, 43 A = (ty — 4)2 + 16y. Since A2 < k, we have that A < t, and so (ty — 4)2 + 16y < 43t. If y > 6, then (6t — 4)2 + 6 • 16 < 43t, which has no possible solution for t. Thus 1 < y < 5. Since also (t — 4)2 + 16 < 43t, we conclude that 2 < t < 71, and so (3.1) and (3.2) imply that k = t(t2y2 — 8ty + 16y + 16) d A = (ty — 4)2 + 16y = 64 an = 64 ' where 2 < t < 71 and 1 < y < 5. For these values of t and y, considering the fact that m > 5, k > A2 and A divides k, we obtain (v, k, A) = (121,25,5) or (441,56, 7) respectively when (t, y) = (5,4) or (8, 3). These possibilities can be ruled out by [4] or [22, Theorem 1.1]. □ Proposition 3.3. If D = (P, B) is a non-trivial symmetric (v,k, A) design admitting a flag-transitive and point-primitive automorphism group G, where A divides k and k > A2, then G is not of twisted wreath product type. Proof. If G would be of twisted wreath product type, then by [15, Remark 2(ii)], it would be contained in the wreath product H I Sm with H = T x T of simple diagonal type, and so G would act on P by product action, and this contradicts Proposition 3.2. □ 3.2 Simple diagonal type In this section, we suppose that G is a primitive group of diagonal type. Let M = Soc(G) = Ti x • • • x Tm, where T = T is a non-abelian finite simple group, for i = 1,..., m. Then G may be viewed as a subgroup of M • (Out(T) x Sm). Here, Ga is isomorphic to a subgroup of Aut(T) x Sm and Ma = T is a diagonal subgroup of M, and so |P| = |T|m-i. Lemma 3.4. Let G be a flag-transitive point-primitive automorphism group of simple diagonal type with socle Tm. Then k divides Am1h, where m1 < m and h divides |T|. Proof. See the proof of Proposition 3.1 in [22]. □ Proposition 3.5. If D = (P, B) is a non-trivial symmetric (v,k, A) design admitting a flag-transitive and point-primitive automorphism group G, where A divides k and k > A2, then G is not ofsimple diagonal type. Proof. Suppose by contradiction that G is a primitive group of simple diagonal type. Then v = |T|m-1, and so by Lemma 2.1(b), Av < k2. This implies that A|T|m-1 < k2 = A2t2. Since A2 < k, we must have A < t, and hence |T |m-1 60, we must have m < 6. If m = 5, then |T| < 53, and it follows that T = A5. Note that k divides A(v - 1) = A(|T|m-1 - 1). Then t divides |T|m-1 -1 = 604 - 1 = 13 • 59 • 61 • 277. Since t < m|T| = 300 and t > 2, it follows that t G {13, 59, 61,277}. For each such t, we have that A < t and k = tA, and so we easily observe that these parameters does not satisfy Lemma 2.1(a). Therefore m g {2, 3,4}. Note that Ga is isomorphic to a subgroup of Aut(T) x Sm. Then by Lemma 2.1(b), the parameter k divides |Ga|, and so k divides (m!) • |T| • | Out(T)|. On the other hand, Lemma 2.1(a) implies that k divides A(|T|m-1 - 1), and so t divides |T|m-1 - 1 implying that gcd(t, |T|) = 1. Since k divides (m!) • |T| • | Out(T)| and t is a divisor of k, we conclude that t divides (m!) • | Out(T)|. Recall by (3.6) that |T|m-1 < t3. Therefore, |T|m-1 < (m!)3 ft Out(T)|3, (3.7) where m G {2, 3,4}. If m = 2, then |T| < 8 • | Out(T)|3. If m = 3, then |T|2 < 63| Out(T)|3, and so |T| < 621 Out(T)|. If m = 4, then |T|3 < 243| Out(T)|3, and |T| < 241 Out(T)|. Thus for m < 4, we always have |T| < 8 • | Out(T)|3, where T is a non-abelian finite simple group. We now apply Lemma 2.3 and conclude that T is isomorphic to A5 or A6. If m = 2, then since t divides |T|m-1 - 1 = |T| - 1, we have that t divides 59 or 359 when T is isomorphic to A5 or A6, respectively. Thus (v, k, A) = (60,59A, A) or (v, k, A) = (360,359A,A). Since A > 1, in each case, we conclude that k > v, which is a contradiction. For m = 3,4, since | Out(A5)| = 2 and | Out(A6)| = 4, it follows from (3.7) that |T| < 48 or |T| < 96 when T is isomorphic to A5 or A6, respectively, which is a contradiction. □ 4 Proof of the main result In this section, we prove Theorem 1.1. Suppose that D = (P, B) is a non-trivial symmetric (v, k, A) design with A divides k and k > A2. Suppose also that G is a flag-transitive automorphism group of D. Proof of Theorem 1.1. If G is point-primitive, then by O'Nan-Scott Theorem [15] and Propositions 3.2, 3.3 and 3.5, we conclude that G is of affine or almost simple type. Suppose now that G is point-imprimitive. Then G leaves invariant a non-trivial partition C of P with d classes of size c. By [20, Theorem 1.1], there is a constant l such that, for each B g B and A g C, |B n A| g {0, l} and one of the following holds: (a) k < A(A - 3)/2; (b) (v, k, A) = (A2(A + 2), A(A +1),A) with (c,d,l) = (A2, A + 2, A) or (A + 2, A2, 2); (c) '(A + 2)(A2 - 2A + 2) A2 a + 2 A2 - 2A + 2 (v, k, A, c, d, l) = -, —, A,-,-, 2 k ,,,,, j y 4 '22 2 and either A = 0 (mod 4), or A = 2u2, where u is odd, u > 3, and 2(u2 — 1) is a square; S. H. Alavi et al.: Onflag-transitive automorphism groups of symmetric designs 625 / , ^ , / (A + 6)(A2 +4 A - 1) A(A + 5) _ A2 +4 A - 1 (v,k,A,c,d,l) = ( + /-), ( ■+ ) ,A,A + 6,- + (d) 4 where A = 1 or 3 (mod 6). We easily observe that the cases (a) and (c) can be ruled out as k > A2. If case (d) occurs, then A(A + 5)/2 = k > A2 implying that A < 5. Since A = 1 or 3 (mod 6), it follows that A = 3 for which (v, k, A, c, d, l) = (45,12, 3, 9,5,3) which satisfies the condition in Theorem 1.1(b). Therefore, the case (b) can occur as claimed. □ References [1] S. H. 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