/^creative ^commor
ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 14 (2018) 187-195
https://doi.org/10.26493/1855-3974.1165.105
(Also available at http://amc-journal.eu)
Flag-transitive automorphism groups of 2-designs with A > (r, A)2 and an application to symmetric designs*
Shenglin Zhou Xiaoqin Zhan
School of Mathematics, South China University of Technology, Guangzhou 510641, P.R. China
Received 23 July 2016, accepted 12 April 2017, published online 17 August 2017 Abstract
Let D be a 2-(v, k, A) design with A > (r, A)2. If G < Aut(D) is flag-transitive, then G cannot be of simple diagonal or twisted wreath product type, and if G is product type then the socle of G has exactly two components and G has rank 3. Furthermore, we prove that if D is symmetric, then G must be an affine or almost simple group.
Keywords: 2-design, automorphism group, primitivity, flag-transitivity. Math. Subj. Class.: 05B05, 05B25, 20B25
1 Introduction
A 2-(v, k, A) design is an incidence structure D = (P, B) where P is a set of v points and B is a set of b blocks with incidence relation such that every block is incident with exactly k points, and every 2-element subset of P is incident with exactly A blocks. Let r be the number of blocks incident with a given point. The numbers v, b, r, k, and A are the parameters of D. A design D is called simple if it has no repeated blocks, and is called symmetric if v = b, and nontrivial if 2 <k<v - 1. Here we always assume that D is simple and nontrivial. An automorphism of D is a permutation of the points which also permutes the blocks and preserves the incidence relation. The set of all automorphisms of D with the composition of maps is a group, denoted by Aut(D). Let G < Aut(D). If G is a primitive permutation group on the point set P, then G is called point-primitive,
* The authors sincerely thank the referees for their helpful suggestions and comments which led to the improvement of the paper. This work is supported by the National Natural Science Foundation of China (Grant No. 11471123).
t Corresponding author.
E-mail addresses: slzhou@scut.edu.cn (Shenglin Zhou), zhanxiaoqinshuai@126.com (Xiaoqin Zhan) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/
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otherwise point-imprimitive. A flag in a design is an incident point-block pair, and G is called flag-transitive if G is transitive on the set of flags.
There are many research works on flag-transitive 2-designs. It was shown in [1] that if a linear space admits a flag-transitive automorphism group G, then G is either of affine or almost simple type. Then the classification of flag-transitive linear spaces was announced in [2], and the complete proof was given by Liebeck [11] for affine type and Saxl [15] for almost simple type. In 1988, Zieschang [19] proved that if G is a flag-transitive automorphism group of a 2-design with (r, A) = 1, then G is an affine or almost simple group. This paper study flag-transitive 2-(v, k, A) designs under the condition that A > (r, A)2. This condition has significance in design theory. On the one hand, the condition A > (r, A)2 and the flag-transitivity of G implies that G is primitive [5, (2.3.7)] (also see Lemma 2.3 below), so we can use the O'Nan-Scott Theorem to analyze this type of designs. On the other hand, there exists many flag-transitive 2-designs satisfying the conditions A > (r, A)2 and (r, A) > 1. Before stating our main results, we give an example in the following.
Example 1.1. Let P = {1,2,3,4, 5, 6}, G = ((3546), (162)(345)) = S5 be a primitive group of degree 6 acting on P. Let B = {1, 2,4}. It is easily known that
BG = {{1, 2, 4}, {1, 3, 5}, {4, 5, 6}, {1, 3,4}, {1, 2, 6}, {2, 3, 6}, {1, 2, 5}, {2,4, 6}, {1,4, 6}, {3, 5, 6}, {2, 3, 4}, {1, 2, 3}, {1, 5, 6}, {3,4, 5}, {1,4, 5}, {1, 3, 6}, {2, 5, 6}, {2, 4, 5}, {2, 3, 5}, {3, 4, 6}},
and Gb = ((124)(356), (12)(56)) ^ D6 is transitive on B. Let B = BG. Then D = (P, B) is a 2-(6, 3,4) design, and G acts flag-transitively on it.
More examples of flag-transitive 2-designs with A > (r, A)2 and (r, A) > 1 can be found in [18]. Our main theorem is the following partial improvement of Zieschang's result.
Theorem 1.2. Let D be a 2-(v, k, A) design with A > (r, A)2. If G is a flag-transitive automorphism group of D, then G is of affine, almost simple type, or product type with Soc(G) = T x T, where T is a nonabelian simple group and G has rank 3.
Flag-transitive symmetric designs with A small have been investigated by many researchers, including Kantor [10] for finite projective planes, Regueiro [13] for A < 3, Fang et al. [8] and Regueiro [14] for A = 4, Tian and Zhou [16] for A < 100. In all these cases, it was proved that if a 2-(v, k, A) symmetric design D admits a flag-transitive, point-primitive automorphism group G, then G must be of affine or almost simple type. As an application of Theorem 1.2, we get the following theorem on symmetric designs.
Theorem 1.3. Let D be a 2-(v, k, A) symmetric design with A > (r, A)2, which admits a flag-transitive automorphism group G. Then G is an affine or almost simple group.
The structure of the paper is organized as follows. Section 2 gives some preliminary lemmas on flag-transitive designs and permutation groups that will apply directly to our situation. In Section 3, we prove Theorem 1.2. Our strategy is based on the O'Nan-Scott Theorem [12] on finite primitive permutation groups, so we deal with the simple diagonal type, the twisted wreath product type, and the product type in Subsections 3.1, 3.2 and 3.3, respectively. In Section 4, we give a proof of Theorem 1.3.
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2 Preliminaries
The following lemma is well known.
Lemma 2.1. The parameters v, b, k, r, A of a 2-(v, k, A) design satisfy the following conditions:
(i)	vr = bk.
(ii)	A(v - 1) = r(k - 1).
(iii)	b > v and k < r.
Lemma 2.2. Let D be a 2-(v, k, A) design, and G be a flag-transitive automorphism group of D. Then
(i)	v < Av < r2.
(ii)	r | A(v — 1, |Ga|), where Ga is the stabilizer of a point a.
(iii)	r | Ad for all nontrivial subdegrees d of G, i.e., the lengths of the Ga-orbits.
Proof. (i) By Lemma 2.1(ii), we have Av = r(k — 1) + A = rk — (r — A), and the result follows by combining it with k < r and 1 < A < r.
(ii) Since G is flag-transitive and A(v — 1) = r(k —1), we have r | |Ga| and r | A(v — 1). It follows that r divides (A(v — 1), |Ga|), and hence r | A(v — 1, |Ga|).
For (iii), r | Ad was proved in [4] and [3, p. 91].	□
The following lemma first appears in [5, (2.3.7)].
Lemma 2.3. Let D be a 2-(v, k, A) design with A > (r, A)2. If G < Aut(D) acts flag-transitively on D, then G is point-primitive.
Proof. Suppose that G < Aut(D) is flag-transitive and {Ci, C2,..., Ct} is a system of t sets of imprimitivity each of size s. Then v = st. The set of imprimitivity containing a point a is a union of Ga-orbits, one of which is {a}, hence by Lemma 2.2(iii) we have
s = 1 (mod (Tx)). Then v = st = t (mod (Txy), which implies t = r(fc-1) +1 = 1 (mod (T^y). Now let s = aj^x) +1 and t = t(T^y + 1. Then
r(k — 1)	r	r
v=rVi+1=st= (a ^+1)(T M)+1)
and thus
rA	A
W + (a + T= k " 1 (2.1)
Since G is flag-transitive and imprimitive, we must have a solution of (2.1) with ctt = 0. Hence if A > (r, A)2, then (2.1) implies r < arr < k — 1 < k, a contradiction.	□
Lemma 2.4 ([6, Lemma 2.5]). Let D be a symmetric design and assume that G < Aut(D) is a primitive rank 3 permutation group on points and blocks. If N = Soc(G) is non-abelian, then N is simple.
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3 Proof of Theorem 1.2
In this section, we will assume that D is a 2-(v, k, A) design with A > (r, A)2 and G < Aut(D) is flag-transitive. By Lemma 2.3, G is point-primitive. The O'Nan-Scott Theorem classifies primitive groups into five types: (i) Affine type; (ii) Almost simple type; (iii) Simple diagonal type; (iv) Product type; (v) Twisted wreath product type, see [12] for details. We will rely on the O'Nan-Scott Theorem to prove Theorem 1.2 by dealing with the cases of simple diagonal action, twisted wreath product action and product action separately.
3.1 Simple diagonal action
Proposition 3.1. Let D be a 2-(v, k, A) design with A > (r, A)2. If G is a flag-transitive automorphism group of D, then G is not of simple diagonal type.
Proof. Suppose that G is of simple diagonal type. Then
G < W = j(ai,..., am)n | ai G Aut(T), n G Sm, ai = aj mod Inn(T) for all i, j}, and there is a G P such that
Ga < {(a,...,a)n | a G Aut(T), n G Sm} = Aut(T) x Sm,
and
Ma = D = {(a,..., a) | a G Inn(T)}
is a diagonal subgroup of M = Ti x • • • x Tm = Tm. Put £ = {Ti,..., Tm}, where T is identified with the group {(1,1,..., t,..., 1) 11 G T} where t is in the i-th position. Then G acts on £ [12]. Moreover the set P of points can be identified with the set M/D of right cosets of D in M so that a = D(1,..., 1), v = |P| = |T|m-1, and for p = D(t1,..., tm), s = (s1,..., sm) G M, a G Aut(T), n G Sm, we have the actions
ps = D(tisi,...,tmsm), pa = D(tJ ,...,tm), = D(tin-1 ,...,tmw-i ).
Since M < G and G is primitive on P, M is transitive on P. Since T\ < M it follows that Ti acts 2-transitively on P ([17, Theorem 10.3]), and so all its orbits have equal length c > 1. Let ri be the orbit of T\ containing the point a. For any ti = (t, 1,..., 1) G T\, we have a*1 = D(t, 1,..., 1), so that
ri = aTl = {D(t, 1,..., 1) | t G T}
and |ri| = |T| = c. Similarly, define ri = aTi for 1 < i < m. Clearly ri n rj = {a} for i = j provided that m > 2.
Choose an orbit A of Ga in P — {a} such that |A n ri| = d = 0. Let mi = |Ga : NGa(Ti)|. Since Ga < Aut(T) x Sm and Gs is transitive on £, it follows that mi < m, and thus
|A| = mid < m|T|.
Lemma 2.2(iii) implies r | Amid, so r < (r, A)mid < (r, A)m|T|. From Av < r2 and A > (r, A)2 we have
A|T|m-i < r2 < ((r,A)m|T|)2 < Am2|T|2.
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As T is a nonabelian simple group, we have
60m-3 < |T|m-3 < m2,
from which it follows that m < 3. Since T = Ma < Ga < Aut(T) x Sm and r | |Ga|, r also divides |T||Out(T)|m!. Let a = (r, A), so that a(k - 1) = £(|T|m-1 - 1). It follows that a divides |T|m-1 - 1, and so (a, |T|) = 1, which implies a ] |Out(T)|m!. Therefore,
|T|m-1 = v < AV < r22 < (|Out(T)|m!)2. a2 a2
It follows that |T| < 4|Out(T)|2 when m = 2, and |T| < 6|Out(T)| when m = 3. By [16, Lemma 2.3], T is isomorphic to one of following groups:
L2(q) for q = 5,7,8, 9,11,13,16, 27, or L3(4).
However, from the facts |Out(L3(4))| = 12, |Out(L2(q))| = 2 for q G {5,7, 8,11,13, 16, 27} and |Out(L2(9))| = 4 that |T| > 4|Out(T)|2 > 6|Out(T)|, a contradiction. □
3.2 Twisted wreath product action
Proposition 3.2. Let D be a 2-(v, k, A) design with A > (r, A)2. If G is a flag-transitive automorphism group of D, then G is not of twisted wreath product type.
Proof. By Lemma 2.3, G is primitive on P. Suppose G has a twisted wreath product action. Then
G = T twro P = qB x P
where P is a transitive permutation group on {1,..., m} with m > 6 (see [7, Theorem 4.7B(iv)]), Q = Pi and M = Soc(G) = QB = Ti x • • • x Tm = Tm. Put £ = {Ti,..., Tm}, where Ti is identified with the group {(1,1,..., t,..., 1; 1) | t G T} = T where t is in the i-th position. Then G acts on £ (see [12]). Moreover, the set P of points can be identified with G\P, the set of right cosets of P in G, so that G acts transitively on P. Define a = P, so that Ga = P and v = |P| = |T|m.
Similarly to the case of simple diagonal action, let r1 = aTl = {P (t, 1,..., 1; 1) 11 G T} so that |r11 = |T|, and define r = aT* for 1 < i < m. Clearly r n r^ = {a} for i = j.
Choose an orbit A of Ga in P — {a} such that |A n r1| = d = 0. Let m1 = |Ga : NGa (I11)|. Since Ga = P and Gs is transitive on £, it follows that m1 < m, and
thus |A| = m1d < m|T|. Lemma 2.2(iii) implies (^xy | m1d, and then r < (r, A)m1d < (r, A)m|T|. On the other hand, by Av < r2 and A > (r, A)2, we have A|T|m < r2 < ((r, A)m|T|)2. It follows that
60m-2 < |T|m-2 < m2. Thus, m < 2. However, this contradicts the fact that m > 6.	□
3.3 Product action
Proposition 3.3. Let D be a 2-(v, k, A) design with A > (r, A)2 admitting a flag-transitive automorphism group G and G is of product type. Then Soc(G) = T1 x T2 (where T = T is a nonabelian simple group) and G has rank 3.
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Suppose that G has a product action on P. Then there is a group K with a primitive action (of almost simple or diagonal type) on a set r of size v0 > 5, where
P = rm, G < Km X Sm = K I Sm and m > 2.
The proof of Proposition 3.3 follows from the next two lemmas.
Lemma 3.4. If G acts flag-transitively on a 2-(v, k, A) design with A > (r, A)2 and G is of product type, then m = 2.
Proof. Let H = KI Sm, and let Sm act on M = {1,2,..., m}. As G is flag-transitive, by Lemma 2.2(iii) we get [Ga : Gaß] > (Tyy for any two distinct points a, ß. Since H > G, it follows that
[Ha : Haß] > [Ga : Gaß] > (TÄ) = (rAAl(k -1). (3.1)
Let a = (7, y, ..., 7), y g r, ß = (S, y, ..., Y), Y = S € r and let B = Km be the base group of H. Then Ba = KY x • • • x KY, Baß = KYg x KY x • • • x KY. Now Ha = Ky ^ Sm, and Haß > KYs x (KY ^ Sm_i). Suppose K has rank s on r with s > 2. We can choose a S satisfying [KY : KYs ] < vr-f, so that
H : H 1= lH°! < KI" • m! <mv0 - 1
H : Haß 1 = IHaßI < |K7,IIKy|m_1 • (m - 1)! < m s - 1 and hence by Equation (3.1),
Ö^W-T) < [Ga:Ga«]< m^-L.	,3.2,
So
VOm-1 < m (r'A)(k - 1).	(3.3) vo - 1 A(s - 1)
Now (k - 1)2 < (r - 1)(k - 1) < r(k - 1) = Av. Thus
m (r A) m v"_ < mvo2 —4— < mvo2 . A1
Hence m < 2, or m = 3 and v0 < 9.
3
2	3v 2
If m = 3, from Equation (3.3) we have v2 + v0 + 1 < , so that v0 = 5 or 6 and
s = 2. Now, from (k - 1)2 < Av we have (k a1^ < A. On the other hand, Equation (3.3)
v0
and A > (r, A)2 imply v2 + vo + 1 < 3(k-1}, so that A < ( S+k-++1 . It follows that
X 2	(v0 +v0 + 1)
(k - 1)% A< 9(k - 1)2 v3 " " (v2 + vo + 1)2 ,
where v0 = 5 or 6. Now G < K I S3 < SVo I S3 implies that G is a {2,3,5}-group, so by flag-transitivity, k divides |GB |, and hence the only primes dividing k are 2, 3 or 5. The only integers v0, k, A satisfying these conditions are v0 = 5, k = 32, A = 8 or 9, by using the software package GAP [9]. Then r = 32 or 36 which contradicts the condition
A > (r, A)2. Hence m = 2.	□
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Lemma 3.5. If G acts flag-transitively on 2-(v, k, A) design with A > (r, A)2 and G is of product type, then G is a point-primitive rank 3 group and v is an odd number
Proof. Since G is of product type, then m = 2 by Lemma 3.4. From Equation (3.3), we have
, 1 ^ 2(r,A)(k - 1) 2(r,A)vn 2vq
V0 + 1 - A(s -1) <A2(7-I) - —1
This implies s = 2. It follows that K acts 2-transitively on r, and H = K I S2 has rank 3 with subdegrees 1, 2(v0 - 1), (v0 - 1)2.
Now G - H, so each subdegree of H is the sum of some subdegrees of G and so
(Txy I 2(vq - 1). If (Ta) = 2(vq - 1), then ^ - vq - 1, so that
r2 - (r, A)2(v0 - 1)2 < (r, A)2v2 - Av
which is a contradiction. Thus (Ta) = 2(v0 - 1), by Equation (3.2), we obtain that G must have a subdegree 2(v0 - 1) and it follows that G induces a 2-transitive group G - K on r. We conclude that G itself has rank 3 on P with subdegrees: 1, 2(v0 - 1), (v0 - 1)2. Therefore, (Ta) I (v0 - 1)2, i.e., 2 | v0 - 1. So v = v2 is an odd number.	□
Proof of Theorem 1.2. Follows immediately from Propositions 3.1, 3.2 and 3.3.	□
4 Proof of Theorem 1.3
In this section, we will apply Theorem 1.2 to symmetric designs. For this purpose, we first give some basic facts on rank 3 permutation groups and symmetric designs. Lemma 4.1 first appears in [16, Lemma 1.5] with a sketch of proof. Since it is an important result for symmetric designs, we provide its proof here for completeness.
Lemma 4.1. Let G be a finite imprimitive rank 3 permutation group on a set P. Let P = {a} U X U Y be the decomposition into Ga-orbits. Assume |X| - |Y|. Then Q = {a} U X is a block of the action of G. Set Q = {Qg | g G G}, then G acts 2-transitively on Q.
Proof. Let M be a maximal subgroup of G containing Ga. Then Ga < M and M is not transitive on P (otherwise, we have G = MGa = M a contradiction). From Ga < M, we have Ma = Ga n M = Ga and
1 + |X| + |Y| = |G : Ga| = |G : Ma| = |G : M||M : Ma| > 2|M : Ma|. (4.1)
Let R = {ax | x G M}. Since both X and Y are orbits of Ga, and Ga < M, there exists m G M\Ga such that am G X or Y. If am G X, then X = amG" C aM = R, so that Q C R. We argue that Q = R. For if Q C R, then there exists m' G M\Ga such that y = am' G R \ Q C Y, from which it follows that Y C R. Hence {a} U X U Y = PC R, and thus P = R which contradicts the fact that M is intransitive. Therefore, Q = R = {a} U X. Similarly, if am G Y, we have R = {a} U Y. Next, we prove that R = {a} U Y, and R = Q is a block.
If R = {a} U Y, since M is transitive on R, we have |R| = |M : Ma| = 1 + |Y|. Equation (4.1) implies that 1 + |X| + |Y| > 2(1 + |Y|), so that |x| > 1 + |Y| > |Y| which contradicts the assumption. Thus we must have Q = {a} U X = R. Since Q = R =
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{ax | x G M}, if Q9 n Q = 0 for some g G G, then there exist x,y G M such that axg = ay, so that xgy-1 G Ga < M and g G M. Hence Q9 = Q and Q is a block.
Since Q = {a} U X is a block and a G Q, then Ga < Gq, where Gq is the stabilizer of the block Q. Let Q = {Q9 | g G G} be a block system of imprimitivity of G. As Ga is transitive on Y = P - Q, it follows that Gq acts transitively on Q\{Q}, and thus G acts 2-transitively on Q.	□
Lemma 4.2 ([16, Lemma 1.6]). Let D = (P, B) be a symmetric 2-(v, k, A) design and G < Aut(D) be a point-primitive rank 3 group. Then G is also a block-primitive rank 3 group if one of the following holds for (G, P):
(a)	The permutation group is of product or affine type.
(b)	The group G is almost simple and G has no 2-transitive representation of degree d, such that d properly divides v.
Now we begin the proof of Theorem 1.3.
Proof of Theorem 1.3. Assume D is a 2-(v,k, A) symmetric design with A > (r, A)2, which admits a flag-transitive automorphism group G. By Theorem 1.2, G is one of the following: (i) affine type, (ii) almost simple type, or (iii) product type with Soc(G) = T x T where T is a nonabelian simple group and G is a primitive rank 3 group. So we only need to prove that case (iii) cannot occur.
Suppose for the contrary that G has a product action on the set of points. Here G is a point-primitive rank 3 group, so we know from Lemma 4.2 that G is also a block-primitive rank 3 group. By Lemma 2.4, we have m =1. This contradicts the fact that m = 2 (see Lemma 3.4). Hence G is of affine or almost simple type.	□
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