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Gbrds with Block Size Three Over 2-Groups, Semi-Dihedral Groups and Nilpotent Groups

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Gbrds with Block Size Three Over 2-Groups, Semi-Dihedral Groups and Nilpotent Groups GBRDs with block size three over 2-groups, semi-dihedral groups and nilpotent groups R. Julian R. Abel Diana Combe School of Mathematics and Statistics The University of New South Wales NSW 2052, Australia [email protected] [email protected] Adrian M. Nelson William D. Palmer School of Mathematics and Statistics The University of Sydney NSW 2006, Australia [email protected] [email protected] Mathematics Subject Classifications: 05B05, 20D15. Submitted: Sep 9, 2009; Accepted: Jan 27, 2011; Published: Feb 14, 2011 Abstract There are well known necessary conditions for the existence of a generalized Bhaskar Rao design over a group G, with block size k = 3. We prove that they are sufficient for nilpotent groups G of even order, and in particular for 2-groups. In addition, we prove that they are sufficient for semi-dihedral groups. Key words: Generalized Bhaskar Rao design. 2-groups. Nilpotent groups. Semi- dihedral groups. Normal subgroups. Hall-Paige Conjecture. 1 Introduction 1.1 Definitions and Notation Throughout this paper G is a finite group written multiplicatively, 0 6∈ G is a zero symbol, and v,b,r,k,λ are positive integers with v ≥ 3. We denote the cyclic group of order n by C(n). A group is a p-group, if the order |G| = pr for some prime p and integer r. A group is elementary abelian if it is the direct product of cyclic groups of order p for some prime p. A group is nilpotent if it is the direct product of Pi where each Pi is a pi-group for some prime pi. The trivial group (or subgroup) is the group with only one element. the electronic journal of combinatorics 18 (2011), #P32 1 Groups with more than one element are non-trivial. A subgroup is a proper subgroup if it is strictly smaller than the whole group. There are several infinite families of groups of particularly importance in this paper. Each of them has a normal, cyclic subgroup of index 2 and this can lead to added compli- cations when using normal subgroup constructions of designs. We recall their definitions here and set up notation in which we parameterize each family by the group order. They are all non-abelian, except for the first one or two groups in each family. Definition 1. The dihedral group, D(2m) of order 2m. D(2m)= a, b : am =1, b2 =1, ba = a−1b ; m =1, 2,... Definition 2. The dicyclic group, Q(4m) of order 4m , sometimes called generalized quaternion. The group Q(8) is the quaternion group, and usually denoted Q. Q(4m)= a, b : a2m =1, b2 = am, ba = a−1b ; m =1, 2,... Definition 3. The semi-dihedral group, SD(8m), of order 8m, sometimes called quasi- dihedral or semi-hedral. SD(8m)= a, b : a4m =1, b2 =1, ba = a2m−1b ; m =1, 2, 3,... 1.2 Generalized Bhaskar Rao designs Definition 4. A generalized Bhaskar Rao design GBRD(v,b,r,k,λ; G) is a v × b array, each entry of which is either 0 or an element of G such that: 1. each row has r group element entries and each column has k group element entries; 2. for each pair of distinct rows (x1, x2,...,xb) and (y1,y2,...,yb) the list −1 xi yi : i =1, 2,...,b, xi =06 , yi =06 , λ contains each group element exactly |G| times. The parameters in a GBRD(v,b,r,k,λ; G) are not all independent of each other. Clearly λ ≡ 0 (mod |G|), and replacing the group entries by 1 and leaving the others 0, results in an incidence matrix for a BIBD(v,b,r,k,λ), (or an all 1’s matrix if v = k). λ(v−1) λv(v−1) G It is well known that r = k−1 and b = k(k−1) . We usually refer to a GBRD(v,k,λ; ), or to a GBRD(v,k,t|G|; G). Sometimes it is convenient to consider a more general form of a GBRD, where k is replaced by a finite set K of positive integers. Definition 5. A generalized Bhaskar Rao design GBRD(v,K,λ; G) is a rectangular array with v rows, each entry of which is either 0 or an element of G and such that for each column the number of group entries is an element of K and for each pair of distinct rows (x1, x2,...,xb) and (y1,y2,...,yb), where b is the number of columns, the list −1 xi yi : i =1, 2,...,b, xi =06 , yi =06 , λ contains each group element exactly |G| times. the electronic journal of combinatorics 18 (2011), #P32 2 Necessary conditions for the existence of a GBRD(v, 3,λ; G) are well known and given, for example, in Abel et al. [2]: Lemma 6. The following conditions are necessary for a GBRD(v, 3,λ; G): (i) λ ≡ 0 (mod |G|); (ii) λ(v − 1) ≡ 0 (mod 2); 0 (mod 6) if |G| is odd (iii) λv(v − 1) ≡ 0 (mod 24) if |G| is even; (iv) If v =3, and G has a non-trivial cyclic Sylow 2-subgroup, then λ ≡ 0 (mod 2|G|). It is known that, for v = 3 and λ = |G|, the necessary conditions in Lemma 6 are sufficient for the existence of a GBRD(v, 3,λ; G). This is a consequence of the recently proved long standing Hall-Paige conjecture (Evans [10], Wilcox [24], and Wilcox, Evans and Bray [6], New results). The Hall-Paige conjecture [15] concerns complete mappings of finite groups, and states that ‘a finite group has a complete mapping if and only if the Sylow 2-subgroup is trivial or non-cyclic’. For any group G, the existence of a GBRD(v, 3,λ; G) for v = 3 is equivalent to the existence of a complete mapping of the group. For more details about complete mappings, see Evans [11]. For more details of the Hall-Paige conjecture and recent extension to complete mappings of loops (algebraic structures similar to groups but not requiring the product to be associative) see Pula [20]. Abel et al. [2] show the necessary conditions in Lemma 6 are sufficient when v = 3 and λ> |G| and that this follows as a consequence of the Hall-Paige conjecture. The necessary conditions in Lemma 6 are known to be sufficient for the existence of a GBRD(v, 3,λ; G) for many families of groups G. Abel et al. [2] conjecture that the necessary conditions in Lemma 6 are always sufficient for the existence of a GBRD(v, 3,λ; G). Conjecture 7. For v ≥ 3, the necessary conditions in Lemma 6 are sufficient for the existence of a GBRD(v, 3,λ; G). We view Conjecture 7 as a generalization of the Hall-Paige conjecture. It is known that the necessary conditions are sufficient when the group is abelian, and the proof of this developed over some time, involved many people and was completed by Ge et al. [12]. They have been shown to be sufficient for any odd order nilpotent group by Palmer [19], for any dihedral group by Abel et al. [3], and for any sufficiently small group or any dicyclic group by Abel et al. [2]. Most recently Abel et al. [1] have shown them to be sufficient for pq groups and for groups of order ≤ 100 with the possible exception of some non-abelian groups with order |G|∈{32, 36, 48, 54, 60, 64, 72, 96}. (The details of the proofs given in [1] for |G|∈{56, 80} are corrected in this paper and given in Section 2.4.) We summarise the evidence for Conjecture 7 in the following theorem: Theorem 8. Let G be a finite group, and v ≥ 3, then in each of the following cases, the necessary conditions in Lemma 6 are sufficient for the existence of a GBRD(v, 3,λ; G): the electronic journal of combinatorics 18 (2011), #P32 3 (i) For v =3; (ii) For G abelian, or dihedral, or dicyclic; (iii) For G nilpotent of odd order; (iv) For G with |G| = pq for p, q primes; (v) For G with |G| ≤ 100 with the possible exception of |G|∈{ 32, 36, 48, 54, 60, 64, 72, 96}. The early results on groups of small order, say ≤ 8, necessarily required producing many explicit designs. Proving the existence of GBRDs over cyclic groups of order 2, 4 and 8 was a major challenge in dealing with cyclic groups in general (and ultimately all abelian groups). The case of the cyclic group of order 2 was difficult, and dealt with by Seberry [21]. The cyclic group of order 4 was dealt with partly by de Launey et al. [8]. The cyclic group of order 8 was considered by Seberry [22]. Some results for GBRDs over cyclic groups of even order were given by Bowler et al. [5]. The completion of the result for groups of order 4 and 8 had to wait until the proof for all abelian groups by Ge et al. [12]. Our aim in this paper was to show that the necessary conditions in Lemma 6 are sufficient for the existence of a GBRD(v, 3,λ; G) whenever G is a nilpotent group of even order, and hence for all nilpotent groups. The main difficulty we faced was to show this for arbitrary 2-groups. In proving our result we also show that the necessary conditions in Lemma 6 are sufficient for any semi-dihedral group.
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