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Cyclic Affine Planes
Abstract
Let Π be an affine plane which admits a collineation τ such that the cyclic group generated by τ leaves one point (say X ) fixed, and is transitive on the set of all other points of Π. Such “cyclic affine planes” have been previously studied, especially in India, and the principal result relevant to the present discussion is the following theorem of Bose [ 2 ]: every finite Desarguesian affine plane is cyclic. The converse seems quite likely true, but no proof exists. In what follows, we shall prove several properties of cyclic affine planes which will imply that for an infinite number of values of n there is no such plane with n points on a line.
... An affine difference set D gives rise to an affine plane (and hence to a projective plane Π q (D)) as follows: Points of the plane are the elements of G, together with a special point O (the origin), lines through O are the cosets of N , the remaining lines are of the form Dg, g ∈ G. We refer the reader to [9], [10] and [11] for further information about affine difference sets. ...
... Remark 4.3. For 11 ≤ q ≤ 133, we are able to verify by a computer aided search that there exists a balanced coloring for an arbitrary projective plane of order q with color classes of size 11 and 12, namely the number of colors needed is at most q 2 +q−18 11 . One should repeat the steps in the proof of Theorem 1.6 but use the concrete expected value instead of Condition (1) and use the stronger inequalities in Condition (2) and (3). ...
... One should repeat the steps in the proof of Theorem 1.6 but use the concrete expected value instead of Condition (1) and use the stronger inequalities in Condition (2) and (3). Finally, suppose that the number of color classes is at most q 2 +q−18 11 and q ≤ 10. Then we have less than 11 colors, thus every line has a pair of monochromatic points by the pigeon-hole principle. ...
In this paper, we study vertex colorings of hypergraphs in which all color class sizes differ by at most one (balanced colorings) and each hyperedge contains at least two vertices of the same color (rainbow-free colorings). For any hypergraph H, the maximum number k for which there is a balanced rainbow-free k-coloring of H is called the balanced upper chromatic number of the hypergraph. We confirm the conjecture of Araujo-Pardo, Kiss and Montejano by determining the balanced upper chromatic number of the desarguesian projective plane for all q. In addition, we determine asymptotically the balanced upper chromatic number of several families of non-desarguesian projective planes and also provide a general lower bound for arbitrary projective planes using probabilistic methods which determines the parameter up to a multiplicative constant.
... is analogous to the standard proof of Singer's Theorem 3.1 [104]. This result of Bose [12] was the starting point for the investigation of cyclic affine planes, that is, those affine planes of order n which admit a cyclic group of order n 2 − 1; they have been studied extensively beginning with the work of Hoffman [61] who already stated the PPC for this case. The two papers of Bose [12] and of Hoffman [61] started the theory of affine difference sets in much the same way as the work of Singer [104], followed by that of Hall [53], started the theory of planar difference sets. ...
... This result of Bose [12] was the starting point for the investigation of cyclic affine planes, that is, those affine planes of order n which admit a cyclic group of order n 2 − 1; they have been studied extensively beginning with the work of Hoffman [61] who already stated the PPC for this case. The two papers of Bose [12] and of Hoffman [61] started the theory of affine difference sets in much the same way as the work of Singer [104], followed by that of Hall [53], started the theory of planar difference sets. Interestingly, the results -and to a considerable extent also the methods -for both the affine and the planar case of the PPC are quite parallel. ...
... As with planar difference sets, multipliers are a central tool in the theory of affine difference sets. Hoffman [61] proved the affine analogue of Hall's multiplier theorem for planar difference sets: every prime divisor p of n is a multiplier of every cyclic affine difference set of order n. This result remains true for abelian affine difference sets, as a special case of the multiplier theorem of Elliott and Butson [33] for relative difference sets. ...
Let Π be a finite projective plane admitting a large abelian collineation group. It is well known that this situation may be studied by algebraic means (via a representation by suitable types of difference sets), namely using group rings and
algebraic number theory and leading to rather strong nonexistence results. What is less well-known is the fact that the abelian group (and sometimes its group ring) can also be used in a much more geometric way; this will be the topic of the present survey.
In one direction, abelian collineation groups may be applied for the construction of interesting geometric objects such as unitals, arcs and (hyper-)ovals, (Baer) subplanes, and projective triangles. On the other hand, this approach makes it sometimes possible to provide simple geometric proofs for non-trivial structural restrictions on the given collineation group, avoiding algebraic machinery.
... See [20], Theorem 6.2. The case X = 1 is already in [23]. ...
... The result is for two reasons particularly interesting. First, there is no affine plane with a cyclic collineation group acting regularly on all points (by [23]) excepting the plane of order 2; but any plane over a commutative division ring admits a group regular on the points and regular on all lines except for those of one parallel class. (Omitting the last condition, this holds of course for every translation plane.) ...
... The special case d = 2 is due to Bose [4]. Using Corollary 2.11, we therefore obtain a series of cyclic difference sets which for d = 2 are the affine difference sets of [4] which have also been studied by Hoffman [23]. The general case has first been obtained by Butson Delsarte, Goethals and Seidel have shown in [15], Theorem 5.1 that the existence of a cyclic relative difference set with parameters (4.3) is equivalent to that of a "negacyclic" conference matrix. ...
A (group) divisible design is a tactical configuration for which the v points are split into m classes of n each, such that points have joining number λ (resp. λ 2 ) if and only if they are in the same (resp. in different) classes. We are interested in such designs with a nice automorphism group. We first investigate divisible designs with equally many points and blocks admitting an automorphism group acting regularly on all points and on all blocks, i.e., with a Singer group (Singer [ 50 ] obtained the first result in this direction for the finite projective spaces).
As in the case of block designs, one may expect a divisible design with a Singer group to be equivalent to some sort of difference set; as it turns out, one here obtains a generalisation of the relative difference sets of Butson and Elliott [ 11 ] and [ 20 ].
... An affine difference set D gives rise to an affine plane (and hence to a projective plane Π q (D)) as follows: Points of the plane are the elements of G, together with a special point O (the origin), lines through O are the cosets of N, the remaining lines are of the form Dg, g ∈ G. We refer the reader to [5,11] and [10] for further information about affine difference sets. ...
In this paper, we study vertex colorings of hypergraphs in which all color class sizes differ by at most one (balanced colorings) and each hyperedge contains at least two vertices of the same color (rainbow-free colorings). For any hypergraph H, the maximum number k for which there is a balanced rainbow-free k-coloring of H is called the balanced upper chromatic number of the hypergraph. We confirm the conjecture of Araujo-Pardo, Kiss and Montejano by determining the balanced upper chromatic number of the desarguesian projective plane PG(2,q) for all q. In addition, we determine asymptotically the balanced upper chromatic number of several families of non-desarguesian projective planes and also provide a general lower bound for arbitrary projective planes using probabilistic methods which determines the parameter up to a multiplicative constant.
... Let Σ be an affine plane of order n admitting a collineation τ such that the cyclic group G generated by τ leaves one point (say ∞) fixed and acts regularly on the set of all remaining points of Σ. Such an affine plane is called a cyclic affine plane; see [2]. The order of the collineation τ is v = n 2 − 1. ...
... If r=k, and s=l, a Type I difference set is the (v, k, 1) difference set of Bruck [3]. Hoffmann [6] has considered Type II difference sets for which r=k+l, and T is cyclic. If \D R (L t )\ =k(k-1) for l<i<s a Type I difference set is the difference system of Vajda [11]. ...
In the past three decades the problem of generating (balanced incomplete block) designs by difference sets has received much attention. Bose [2] gave the two "fundamental theorems of the method of differences". Bose, Sprott [9], Lehmer [7], Chowla [4], Takeuchi [10] and others have given specific classes of difference sets.
... The above embedding result prompted Hoffman [4] to introduce the following Definition. ...
We are already familiar with (υ, k, λ)-difference sets and (υ, k, λ)-designs. In this paper, we will introduce a new class of difference sets and designs: (υ, k, [λ1, λ2, … , λm])-difference sets and (υ, k, [λ1,λ2, … , λm])-designs. We will mainly study designs with a relationship we call λ-equivalence and use them to produce other designs. Some existence or nonexistence theorems will be given. © 2002 Wiley Periodicals, Inc. J Combin Designs 11: 1–23, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/jcd.10031
... Theorem 7.6(ii) and (iii) are equivalent to Jungnickel's result [28] that semi-regular RDSs, with parameters ( p r , p r , p r , 1) and (2 r , 2 r , 2 r , 1) respectively, exist for the stated groups and subgroups. (Nonexistence results for RDSs show that no other abelian group and subgroup can be substituted in Theorem 7.6(ii) for r=1 [26] or r=2 [36], or in Theorem 7.6(iii) for any r [24].) Theorem 7.6(iv) is due to Arasu and Sehgal [3], as described in Section 2. ...
We present a recursive construction for difference sets which unifies the Hadamard, McFarland, and Spence parameter families and deals with all abelian groups known to contain such difference sets. The construction yields a new family of difference sets with parameters (v, k, λ,n)=(22d+4(22d+2−1)/3, 22d+1(22d+3+1)/3, 22d+1(22d+1+1)/3, 24d+2) ford⩾0. The construction establishes that a McFarland difference set exists in an abelian group of order 22d+3(22d+1+1)/3 if and only if the Sylow 2-subgroup has exponent at most 4. The results depend on a second recursive construction, for semi-regular relative difference sets with an elementary abelian forbidden subgroup of orderpr. This second construction deals with all abelian groups known to contain such relative difference sets and significantly improves on previous results, particularly forr>1. We show that the group order need not be a prime power when the forbidden subgroup has order 2. We also show that the group order can grow without bound while its Sylowp-subgroup has fixed rank and that this rank can be as small as 2r. Both of the recursive constructions generalise to nonabelian groups.
... 2.2 (A.J. Hoffman [4]). Let D be an affine difference set of order n in an abelian group. ...
Let D be an affine difference set of order n in an abelian group G relative to a subgroup N. We denote by π(s) the set of primes dividing an integer s(>0) and set H∗=H∖{ω}, where H=G/N and ω=∏σ∈Hσ. In this article, using D we define a map g from H to N satisfying for τ,ρ∈H∗,g(τ)=g(ρ) iff {τ,τ−1}={ρ,ρ−1} and show that ordo(σ)(m)/ordo(g(σ))(m)∈{1,2} for any σ∈H∗ and any integer m>0 with π(m)⊂π(n). This result is a generalization of J.C. Galati’s theorem on even order n [J.C. Galati, A group extensions approach to affine relative difference sets of even order, Discrete Mathematics 306 (2006) 42–51] and gives a new proof of a result of Arasu–Pott on the order of a multiplier modulo exp(H) ([K.T. Arasu, A. Pott, On quasi-regular collineation groups of projective planes, Designs Codes and Cryptography 1 (1991) 83–92] Section 5).
... The following result for multipliers of abelian affine RDSs was first proved in the cyclic case by Hoffman [9]. The version stated here appears in [5]. ...
It is shown that a group extensions approach to central relative (k+1,k-1,k,1)-difference sets of even order leads naturally to the notion of an “affine” planar map; a notion analogous to the well-known planar map corresponding to a splitting relative (m,m,m,1)-difference set. Basic properties of affine planar maps are derived and applied to give some new results regarding abelian relative (k+1,k-1,k,1)-difference sets of even order and to give new proofs, in the even order case, for some known results. The paper concludes with computational non-existence results for 10,000k⩽100,000.
Let p be a prime and Vm the m-dimensional vector space over GF(p). If M and N are m ×n and m × l matrices over GF(p) the selfmap x→Mx +N is called affine. The problem which arose in the context of finite algebras is the characterization of cyclic affine permutations. They exist if and only if m = 1 (translations) or p = m = 2 (6 cases). This is derived from somewhat more general conditions. The shape of affine permutations with p cycles of length pm-t is delimited.
A multiplier theorem in (J. Combinatorial Theory Ser. A, in press) is extended to cyclic group divisible difference sets (GDDSs) of small size. A multiplier theorem for abelian difference sets in (Proc. Amer. Math. Soc. 68 (1978), 375–379) is extended to abelian GDDSs. A remark on the existence of cyclic affine planes is made based on a previously proved multiplier theorem.
This paper provides new exponent and rank conditions for the existence of abelian relative (p
a,p
b,p
a,p
a−b)-difference sets. It is also shown that no splitting relative (22c,2d,22c,22c−d)-difference set exists if d > c and the forbidden subgroup is abelian. Furthermore, abelian relative (16, 4, 16, 4)-difference sets are studied in detail; in particular, it is shown that a relative (16, 4, 16, 4)-difference set in an abelian group G ≇ Z8 × Z4 × Z2 exists if and only if exp(G) ≤ 4 or G = Z8 × (Z2)3 with N ≅ Z2 × Z2.
An abstract is not available.
We state and prove a multiplier theorem for a central element A of ZG, the group ring over Z of a group G. This generalizes most previously known multiplier theorems for difference sets and divisible difference sets. We also provide applications to show that our theorem provides new multipliers and establish the nonexistence of a family of divisible difference sets which correspond to elliptic semiplanes admitting a regular collineation group. © 1995 John Wiley & Sons, Inc.
The sporadic complete 12-arc in PG(2, 13) contains eight points from a conic. In PG(2,q) with q>13 odd, all known complete k-arcs sharing exactly ½(q+3) points with a conic have size at most ½(q+3)+2, with only two exceptions, both due to Pellegrino, which are complete (½(q+3)+3) arcs, one in PG(2, 19) and another in PG(2, 43). Here, three further exceptions are exhibited, namely a complete (½(q+3)+4)-arc in PG(2, 17), and two complete (½(q+3)+3)-arcs, one in PG(2, 27) and another in PG(2, 59). The main result is Theorem 6.1 which shows the existence of a (½(qr+3)+3)-arc in PG(2,qr) with r odd and q≡3 (mod 4) sharing ½(qr+3) points with a conic, whenever PG(2,q) has a (½(qr+3)+3)-arc sharing ½(qr+3) points with a conic. A survey of results for smaller q obtained with the use of the MAGMA package is also presented. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 25–47, 2010
Generalizing a result of Jungnickel for affine difference sets, we present an existence test for arbitrary divisible difference
setsD which is analogous to the well-known and powerful test of Mann for ordinary difference sets. Several applications show that
this approach is of interest also in the general case.
Dierolf and Moscatelli [3] gave an example of a Fr6chet space whose topology is defined by a sequence of norms but whose bidual does not admit a continuous norm. It is perhaps more surprising to realize that there are even K6the spaces which exhibit the same phenomenon. The purpose of this note is to construct in quite an elementary fashion a K6the space whose bidual admits no continuous norm. In fact, we shall first derive a necessary and sufficient condition in terms of A for the bidual 2 (A)" of a K/Sthe space 2 (A) to admit a continuous norm. We shall then demonstrate by an example how this condition can be violated. We use standard terminology and notation from [4]. A = (aj, k)j,k~ N always denotes an infinite matrix with 0 < aj, k < aj,k+l, sup aj,k > 0 for allj and k. For 0 < p < + oo we put k
In this article we study abelian affine difference sets in connection with the related group extensions and give some results on their orders.
In this paper, we shall prove several non-existence results for divisible difference sets, using three approaches:(i)
character sum arguments similar to the work of Turyn [25] for ordinary difference sets,
(ii)
involution arguments and
(iii)
multipliers in conjunction with results on ordinary difference sets.
Among other results, we show that an abelian affine difference set of odd orders (s not a perfect square) inG can exist only if the Sylow 2-subgroup ofG is cyclic. We also obtain a non-existence result for non-cyclic (n, n, n, 1) relative difference sets of odd ordern.
In this paper, we prove the following theorem: Suppose there exists a cyclic affine plane of even order n.Then (a) either n=2 orn≡0 (mod4), and (b) for each prime divisor p of n, we have either for each prime q|n2−1or for some positive integer r (which depends on p), n+1|pr+1 and n−1|pr−1, according as expn2−1(p) is odd or even. For p=2, the former condition cannot hold and hence the latter one holds making expn+1(2) even. As a corollary, we prove that if there exists a cyclic affine plane of order n≡4 (mod8), then (i) n must be a square, (ii) n≡1 (mod3) and (iii) each prime divisor of n+1 is ≡1 (mod4).(For an integer a, if t is any integer with(t,a)=1, expa(t) would mean the smallest positive integer l such that tl≡1 (moda).Here is the Legendre symbol.
The existence of a cyclic affine plane implies the existence of a Paley type difference set. We use the existence of this difference set to give the following condition on the existence of cyclic affine planes of order n: If n ≡ 8 mod 16 then n − 1 must be a prime. We discuss the structure of the Paley type difference set constructed from the plane.
Generalizing a result of Ko and Ray-Chaudhuri (Discrete Math.39 (1982), 37–58), we show the following: Assume the existence of an affine difference set in G relative to N of even order n≠2. If G is of the form G = N ⊕ H, where N is abelian, then n is actually a multiple of 4, say n = 4k, and there exists a (4k − 1, 2k − 1, k − 1)-Hadamard difference set in N. More detailed considerations lead to variations of this result (under appropriate assumptions) which yield even stronger non-existence theorems. In particular, we show the non-existence of abelian affine difference sets of order n ≡ 4 mod 8 (with the exception n = 4) and of nilpotent affine difference sets of order n ≡ 2 mod 4 (n ≠ 2). The latter result is the first general non-existence theorem in the non-abelian case.
We prove an exponent bound for relative difference sets corresponding to symmetric nets. We discuss the case λ = 1 which gives some restrictions on possible automorphism groups of projective planes.
We investigate quasiregular collineation groups G of type (d) in the Dembowski-Piper classification. We prove that the Sylow 2-subgroup of G as well as the Sylow 2-subgroup of its multiplier group have to be cyclic. We use these results to obtain new necessary conditions on the existence of affine difference sets.
An n-subsetD of a group G of order
is called an affine difference set of G relativeto a normal subgroup N of G of order
if the list of differences
containseach element of G-N exactly once and no elementof N. It is a well-known conjecture that if Dis an affine difference set in an abelian group G,then for every prime p, the Sylow p-subgroupof G is cyclic. In Arasu and Pott [1], it was shownthat the above conjecture is true when
. In thispaper we give some conditions under which the Sylow p-subgroupof G is cyclic.
A multiplier theorem in (J. Combinatorial Theory Ser. A, in press) is extended to cyclic group divisible difference sets (GDDSs) of small size. A multiplier theorem for abelian difference sets in (Proc. Amer. Math. Soc.68 (1978), 375–379) is extended to abelian GDDSs. A remark on the existence of cyclic affine planes is made based on a previously proved multiplier theorem.
Let it be required to arrange v elements into v sets such that every set contains exactly k distinct elements and such that every pair of sets has exactly elements in common . This combinatorial problem is studied in conjunction with several similar problems, and these problems are proved impossible for an infinitude of v and k . An incidence matrix is associated with each of the combinatorial problems, and the problems are then studied almost entirely in terms of their incidence matrices. The techniques used are similar to those developed by Bruck and Ryser for finite projective planes [3]. The results obtained are of significance in the study of Hadamard matrices [6;8], finite projective planes [9], symmetrical balanced incomplete block designs [2; 5], and difference sets [7].