Advances in Mathematics of Communications doi:10.3934/amc.2020033 Volume 14, No. 4, 2020, 603{611 DESIGNS FROM MAXIMAL SUBGROUPS AND CONJUGACY CLASSES OF REE GROUPS Jamshid Moori School of Mathematical Sciences, North-West University (Mafikeng) 2754, South Africa Bernardo G. Rodrigues School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal Durban 4000, South Africa Amin Saeidi School of Mathematical Sciences North-West University (Mafikeng) 2754, South Africa Seiran Zandi∗ School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal Durban 4000, South Africa (Communicated by Leo Storme) Abstract. In this paper, using a method of construction of 1-designs which are not necessarily symmetric, introduced by Key and Moori in [5], we de- termine a number of 1-designs with interesting parameters from the maximal 2 subgroups and the conjugacy classes of the small Ree groups G2(q). The 2 designs we obtain are invariant under the action of the groups G2(q). 1. Introduction The aim of the present paper is to construct designs from the maximal subgroups 2 and the conjugacy classes of the family of small Ree group G2(q), where q is an odd power of 3. The method that we use is one of the two methods introduced by Key and Moori in [4,5]. The first method called Method 1 in some papers con- cerns the construction of self-dual symmetric 1-designs from primitive permutation representations of finite simple groups. This method has been applied to several sporadic simple groups ([4,6,9, 10, 11, 12, 13, 15]) and to some other families of simple groups (see [14, 16, 17]). The second method introduced in [5] and henceforth called Method 2, outlines the construction of 1-designs which are not necessarily symmetric. The construction in this method uses a maximal subgroup M of a finite simple group G and a conjugacy class in G of some element x 2 M. In their recent paper [14], the authors constructed designs from Ree groups using Method 1. The 2020 Mathematics Subject Classification: 20D05, 05E15 , 05E20. Key words and phrases: Ree groups, design, conjugacy class, maximal subgroup, group action. The first author acknowledges support of NRF and NWU (Mafikeng). The second author acknowledges support of NRF through Grant Numbers 95725 and 106071:. The third author acknowledges support of NWU (Mafikeng) postdoctoral fellowship. The fourth author acknowledges support of NRF postdoctoral fellowship through Grant Num- ber 91495:. ∗ Corresponding author: Seiran Zandi. 603 604 Jamshid Moori, Bernardo G. Rodrigues, Amin Saeidi and Seiran Zandi present paper is concerned with the construction of 1-designs using Method 2. Our notation for designs is as in [1]. Let D = (P; B; I) be an incidence structure, i.e. a triple with point set P, block set B disjoint to P and incidence set I ⊆ P × B. If the ordered pair (p; B) 2 I, then we say that p is incident with B. It is often convenient to assume that the blocks in B are subsets of P so (p; B) 2 D if and only if p 2 B. For a positive integer t, we say that D is a t-design if every block B 2 B is incident with exactly k points and every t distinct points are together incident with λ blocks. In this case we write D = t-(v; k; λ) where v = jPj. We say that D is symmetric if it has the same number of points and blocks. In this paper, using Key-Moori Method 2, which we restate in Section 3 (Lemma 3.1), we construct a number of 1-designs from the maximal subgroups and conjugacy classes of elements 2 of the small Ree groups G2(q). We achieve this in Section 3 by proving a series of lemmas and propositions that lead to Theorem 3.12. Section 3 ends with Table 1, which gives the parameters of these designs. 2. Some results on Ree groups 2 The small Ree groups denoted by G2(q) where q is an odd power of 3 are a family of groups discovered by Ree in the 60s [18]. He showed that these groups 2 are simple except the first one G(3), which is isomorphic to P SL2(8):3. In [22], Wilson presented a simplified construction of the Ree groups, as the automorphism 2 of a 7-dimensional vector space over the field of q elements. Let G = G2(q) be a small Ree group (we always assume that q ≥ 27 to avoid the non-simple case). The order of G is q3(q3 + 1)(q − 1) and G acts doubly transitive on a set Ω of size q3 + 1. Moreover, every non-trivial element of G that fixes more than two points in Ω is an involution. A Sylow 3-subgroup P of G is a TI-subgroup, i.e. for g 62 NG(P ), g 3 we have P \ P = f1Gg. The group P is a 3-group of order q of nilpotence class 3 with jZ(P )j = q and jP 0j = q2. Both P 0 and P=P 0 are elementary abelian 3-groups. Moreover, all elements of order 3 lie in P 0. All non-identity elements of Z(P ) are conjugate in G and (2.1) P 0 n Z(P ) = bG [ (b−1)G; for b 2 P 0 n Z(P ). Also, we have P n P 0 is a union of three conjugacy classes of G. For more detailed information on Ree groups, we refer the reader to [14] and [20]. 2 Throughout the rest of this paper, we fix q and assume that G = G2(q). Our notation for groups is mainly from ATLAS [2]. The following theorem which can be found in [21] gives the classification of the maximal subgroups of G. Theorem 2.1. The maximal subgroups of G, up to conjugacy, are 1+1+1 (1): q :Cq−1; p (2): Cq− 3q+1:C6; p (3): Cq+ 3q+1:C6; (4): 2 × P SL2(q); 2 (5): (2 × D q+1 ):C3; 2 2 p (6): G2(q0), where q0 = p q. Proof. See [21, Theorem 4.2]. Notation. We use the following notation throughout the rest of the paper. Let p p 3 t1 = (q − 1)=2, t2 = (q + 1)=4, t3 = q − 3q + 1, t4 = q + 3q + 1 and t5 = q . Also set O(q) = ft1; t2; t3; t4; t5g. For 1 ≤ i ≤ 5, we denote by Bi the set of all subgroups Designs from maximal subgroups 605 of G of order ti. From [14], we collect the following results which will be useful for our discussions. Proposition 1. Let Bi be as above, and suppose that Bi 2 Bi are chosen arbitrarily for 1 ≤ i ≤ 5. Then (i) every element of Bi is a Hall subgroup of G; in particular every two elements of Bj for a fixed j are conjugate in G; ∼ (ii) NG(B1) = B1:2 = D2(q−1); ∼ 2 (iii) NG(B2) = (Q1 × (B2:2)):3 and NG(Q1) = NG(B2), where Q1 = 2 ; (iv) NG(B3) = B3:6; (v) NG(B4) = B4:6; (vi) NG(B5) = B5:(q − 1); (vii) if i 6= 5 then Bi is cyclic; and if f1Gg 6= S ≤ Bj then NG(S) = NG(Bj): Proof. All parts follow from [8] and [20]. Remark 1. If A 2 Bi for 2 ≤ i ≤ 5, then NG(A) is a maximal subgroup of G. If B 2 B1 then NG(B) lies in a maximal subgroup of the form 2 × P SL2(q). Lemma 2.2. Assume that Bi 2 Bi for 1 ≤ i ≤ 5 and x 2 G is non-trivial. Then the following statements hold. G 3 3 (i) if x 2 B1 then jx j = q (q + 1); G 3 2 (ii) if x 2 B2 then jx j = q (q − q + 1)(q − 1); G 3 p (iii) if x 2 B3 then jx j = q (q + 1)(q + 3q + 1)(q − 1); G 3 p (iv) if x 2 B4 then jx j = q (q + 1)(q − 3q + 1)(q − 1); (v) for x 2 B5 we have G 3 • if x 2 Z(B5) then jx j = (q + 1)(q − 1); 0 G 3 • if x 2 B5 n Z(B5) then jx j = q(q + 1)(q − 1)=2; 0 G 2 3 • if x 2 B5 n B5 then jx j = q (q + 1)(q − 1)=3; (vi) if o(x) = 2 then jxGj = q2(q2 − q + 1); (vii) if o(x) = 6 then jxGj = q2(q3 + 1)(q − 1)=2; (viii) if o(x) = q − 1 then jxGj = q3(q3 + 1); q+1 G 3 2 (ix) if o(x) = 2 then jx j = q (q − q + 1)(q − 1). G Proof. It follows by [14, Lemma 2.6] and the fact that jx j = jG : CG(x)j for all x 2 G. Lemma 2.3. Let H and K be distinct subgroups of G of equal order t 2 O(q). Then we have H \ K = f1Gg. Proof. See [14, Corollary 2.8]. Lemma 2.4. All subgroups of order t 2 O(q) are conjugate in G. Proof. It follows from part (a) of [19, Theorem 1.1]. Remark 2. Let G be a group and H be a subgroup of G. The subgroup H is called TI subgroup if for every g 2 G, H \ Hg = 1 or H \ Hg = H. Corollary 1. All subgroups of order t 2 O(q) are TI subgroups in G.