Multiplied Conﬁgurations Induced by Quasi Difference Sets

Quasi difference sets are introduced as a tool to produce partial linear spaces. We characterize geometry and automorphisms of conﬁgurations decomposable into components induced by quasi difference sets. In particular, we are interested in series of cyclically inscribed copies of a ﬁxed conﬁguration.


Introduction
There is well-known construction of a point-block geometry induced by some fixed subset D of a group G (cf. [2,14]). The idea is simple: points are elements of G, and blocks (lines) are the images of D under (left) translations. If every nonzero element of G can be presented in exactly λ ways as a difference of two elements of D, then D is called a difference set, cf. [3]. In this case we obtain a λ-design. Difference sets with λ = 1 are called Singer (or planar) difference sets and induce a linear spaces, in particular finite Desarguesian projective planes (see [5,13]). To get weaker geometries we admit sets with λ ∈ {0, 1} and call them quasi difference sets. This approach was used in [10] to study configurations that can be visualized as series of polygons, inscribed cyclically one into another. Classical Pappus configuration can be presented this way, for instance.  Some other variation on difference sets can be found in the literature, e.g. relative difference sets (cf. [4]), affine difference sets (cf. [6]), or partial difference sets (cf. [7]). Defining a partial difference set D we require that every nonzero element a of G can be presented as a corresponding difference in λ 1 ways if a ∈ D, and in λ 2 ways if a / ∈ D. Note that our quasi difference sets are not partial in this sense. One of the most important tasks in the theory of difference sets is to determine conditions of the existence, comp. [1,8]. These are not the questions considered in this paper. Instead, we are mainly interested in the geometry (in the rather classical style) of partial linear spaces determined by quasi difference sets.
We consider configurations which can be defined with the help of arbitrary quasi difference set. Elementary properties of these structures are discussed: we verify satisfiability of Veblen, Pappus, and Desargues axioms. A special emphasis is imposed on structures which arise from groups decomposed into a cyclic group C k and some other group G. These structures can be seen as multiplied configurations-series of cyclically inscribed configurations, each one isomorphic to the configuration associated with G. On the other hand, this construction is just a special case of the operation of "joining" two structures, corresponding to the operation of the direct sum of groups. In some cases corresponding decomposition can be defined within the resulting "join", in terms of the geometry of the considered structures. This definable decomposition enables us to characterize the automorphism group of such a "join". Some other techniques are used to determine the automorphism group of cyclically inscribed configuration. Roughly speaking, groups in question are semidirect products of some symmetric group and the group of translations of the underlying group.
The technique of quasi difference sets can be used to produce new configurations, so far not considered in the literature. Many of them seem to be of a real geometrical interest for their own. In the last section we apply our apparatus to get some new configurations arising from the well-known: cyclically inscribed Pappus or Fano configurations, multiplied Pappus configurations, a power of cyclic projective planes.

Basic Notions and Definitions
Let M = S, L, be an incidence point-line geometry. If p k for p ∈ S, k ∈ L then we say that " p is on the line k" or "k passes through the point p". M is a partial linear space iff there are at least two points on every line, there is a line through every point, and any two lines that share two or more points coincide. A partial linear space in which the rank of a point and the size of a line are equal is said to be a symmetric configuration or in short just a configuration. The set of all lines through a point p ∈ S is denoted by p * , and dually we write k * for the set of all points on a line k ∈ L. The rank of a point p is the number | p * |, and the size of a line k is the number |k * |. If p = q are two collinear points then we write p ∼ q and the line which joins these two points is denoted by p, q. We also write k l for the common point of two intersecting lines k and l.
An automorphism (or a collineation) of M is a pair ϕ = (ϕ , ϕ ) of bijections ϕ : S −→ S, ϕ : L −→ L such that for every a ∈ S, l ∈ L the conditions a l and ϕ (a) ϕ (l) are equivalent. A pair κ = (κ , κ ) of bijections κ : M −→ L, κ : L −→ M satisfying a l iff ϕ (l) ϕ (a) for every a ∈ S, l ∈ L is called a correlation of M. A substructure of M, whose points are all the points of M collinear with a ∈ S, and lines are all the lines of M which contain at least two points collinear with a is said to be the neighborhood of a point a and it is denoted by M (a) . Clearly, if ϕ = (ϕ , ϕ ) is an automorphism of M, then ϕ maps M (a) onto M (ϕ (a)) . M is said to be Veblenian iff any line that crosses two sides of a triangle meets also the third side of this triangle. We say that M is Desarguesian iff it satisfies Desargues axiom: if two triangles are perspective from a point, then they are perspective from a line.
In [10] quasi difference sets are defined to study series of cyclically inscribed ngons. We briefly recall this construction. Let G = G, ·, 1 be an arbitrary group and D ⊂ G. Let us introduce a point-line geometry . This yields that without loss of generality we can assume that 1 ∈ D. Let G D be the stabilizer of D in G. Then, the number of points in D(G, D) is |G|, the number of lines is |G| |G D | , the size of every line is |D|, and the rank of every point is |D| |G D | . It was proved in [10] that D(G, D) is a configuration iff for every c ∈ G, c = 1 there is at most one pair (a, b) ∈ D×D with ab −1 = c. Any D ⊆ G satisfying this condition is called a quasi difference set, for short QDS. In [10] we were mainly interested in the structures of the form D(C k ⊕ C n , D), where D = {(0, 0), (1, 0), (0, 1)}. In this paper we shall generalize this construction. Let us adopt the following convention: (a) means "coordinates" of the point a ∈ G, and [a] denotes "coordinates" of the line a · D ∈ G/D. Using this we get Hence, a is uniquely determined by [a], or in other words a · D = b · D holds only for a = b.

Generalities
Now, we are going to present some general properties of D = D(G, D). Every auto- We shall frequently refer to the pair f as to an automorphism of D. On the other hand, some of the automorphisms of D are determined by automorphisms of the underlying group G, namely  is an involutive correlation of the structure D = D(G, D). Consequently, D is self- . Thus κ is a correlation. Finally, assume that (a) κ((a)) = [a −1 ]. From (2.2) we obtain a 2 ∈ D.
The correlation defined by (3.1) will be referred to as the standard correlation of D(G, D). Immediate from (2.2) is the following As a straightforward consequence of Lemma 3.3 we get  For D ⊆ G we introduce the following condition: The next Lemma explains the meaning of the condition (3.5).
Proof Simple analysis based on Lemma 3.3 and (3.5).
Now we describe automorphisms of D(G, D) with a quasi difference set D satisfying (3.5). Let us begin with some rigidity properties. Proof Since D has a transitive group of automorphisms, we can assume that p = (1).
on l 2 are the unique points "between" l 1 and l 2 that are not collinear (cf. Lemma 3.6(ii)). We have Thus f preserves every line through p.
The case with the assumption (ii) can by proved in a similar way.
From Lemma 3.7 we get  Let G = i∈I G i , i.e. let G be the set of all functions g : I −→ {G i : i ∈ I } with g(i) ∈ G i . Then the product i∈I G i is the structure G, ·, 1 , where (g 1 · g 2 )(i) = g 1 (i) · i g 2 (i) for g 1 , g 2 ∈ G, and 1(i) = 1 i . It is just the standard construction of the direct product of groups. The set , and the standard inclusion ε j : G j −→ i∈I G i by the conditions (ε j (a))( j) = a and (ε j (a))(i) = 1 i for i = j and a ∈ G j . Recall that π j and ε j are group homomorphisms. We set i∈I From assumption we infer that g 1 = g 3 and g 2 = g 4 . If j 1 = j 2 ; analogously, we come to j 1 = j 3 and j 2 = j 4 . Then we obtain g 1 = g 3 and g 2 −1 = g 4 −1 , which yields our claim. From Proposition 3.4 we have the following:

Proposition 4.3 Let J be a nonempty proper subset of I . Then
is an involutory correlation of i∈I D i .
) iff the following holds: If we assume in Proposition 4.4 that every κ i is the standard correlation (ϕ i (a) = a −1 , cf. Proposition 3.2), then κ is also the standard correlation. Using (4.1) as in Proposition 4.4 one can also prove the following: But this is a rather trivial result, as i∈I τ a i = τ a . We have also some automorphisms of another type.

Proposition 4.6
Let β ∈ S n , x ∈ G n and h : G n −→ G n be the map defined by h ((x 1 , . . . ,

Cyclic Multiplying
Let C k be a cyclic group of the rank k. We use additive notation for these groups. In this section we consider configurations D(G, D r ), where r ≥ 2, G = C n 1 ⊕ · · · ⊕ C n r and Thus D(G, D r ) generalizes the construction of cyclically inscribed polygons considered in [10]. An interesting example of this type is D(C 3 ⊕ C 3 ⊕ C 3 , D 3 )-three copies of Pappus configuration cyclically inscribed, see Fig. 1 and a more general case considered in Sect. 6.2.  It is easy to check that f (−e i ) = −e α(i) . As every permutation is one of two permutations considered above, we get our claim.
(ii) Let α be a permutation such that α(0) = 0, then f is given by (4.5) and If α is a transposition. then f is given by (4.6), and thus f = ( f ) −1 . After simple calculation we get the claim. Note that the condition (3.5) does not hold for all cannonical quasi difference sets.  Next, we consider more general case. Namely, we describe the neighborhood of a point q in a configuration of the following form: where D is QDS in an abelian group G. Directly from definitions we calculate the following: Its points are q and the following:

Elementary Properties
Now we discuss some elementary axiomatic properties of D(G, D). Let D be QDS in an abelian group G. Using Lemma 3.6 we prove the following: The Pappus configuration can be considered as D(C 3 ⊕ C 3 , D 2 ), cf. [10,15] (see Fig. 4).

Then M = D(G, D) contains Pappus configurations.
Proof From the assumptions we get d 2 = d 4 . Note that incidences indicated in the following table hold in M: Then the map (ψ , ψ ) defined for the points by and for the lines by embeds the Pappus configuration into the structure M.

Examples
The goal of this section was to present some new and (we hope) interesting examples of configurations induced by quasi difference sets. Some of them are copies of one fixed configuration repeatedly inscribed, and the others are a join of a few well-known configurations. We determine automorphisms group of every example. However, in most cases we do not present all details of proofs, as they are very technical. Instead, we show only essential steps in the hope that they suffice to understand the idea and a specificity of a proof. Let G = G, +, 0 be an abelian group and D ⊂ G. Recall, cf. [14], that a multiplier α of the set D is an automorphism of G of the form x → α · x satisfying αD = q + D for some q ∈ G.
The following two Lemmas can be easily proved by analyzing the neighborhood of a point (cf. Fig. 5).
In both cases, h fixes all the lines through θ , so from Corollary 6.13 we get h = id, and thus g = G β . Finally, applying Lemma 6.8 we close the proof.

A Power of a Cyclic Projective Plane
Let P = D(C k , D) be a cyclic projective plane determined by a difference set D in the group C k . Then k = q 2 + q + 1, and q + 1 is both the size of a line and the degree of a point of P. Let us draw our attention to the following structure: P n := P ⊕ P ⊕ · · · ⊕ P n times (6.2) Note that P n = D((C k ) n , D), where D = D · · · D. Let us introduce a few auxiliary sets. Namely: The sets S i and J i consist of points and lines, respectively, which form a projective plane embedded in P n (θ) . There are n such planes with the common line [θ ], and the common point (θ ) in P n (θ) . Note that, the degree of the point (θ ) in P n (θ) equals qn + 1, and it is equal to the size of every line through (θ ).
By Lemma 3.3 and (4.1) we can prove the following lemmas: The next two Steps are immediate from Lemma 6.17 and Steps 1, 2: Step 4 Let (x) be a point of F n (θ) . Then x ∈ P i iff there are two distinct lines of the size 3 that pass through it. The set of points on the lines in J 0 ∪ J 0 is the set of points (x) with x / ∈ ∪ n i=1 P i .
Step 5 From the above it follows that F n (θ) contains n subconfigurations isomorphic to a Fano plane (cf. Fig. 6 with points marked by circles and squares). These are precisely substructures of the form S i , J i , . Intuitively, we can read Step 1 as "any two Fano subplanes of F n (θ) are joined by a line of the size 4". Analogously, Step 2 explains how lines of the size 3 join the Fano subplanes. In view of Steps 3 and 4 an automorphism F preserves the set n i=1 S i and permutes the Fano subplanes. So, F determines a permutation σ such that F maps the set S i onto S σ (i) and it maps the family J i onto J σ (i) for every i = 1, . . . , n.
Step 6 Obviously, F preserves the set of lines of the size 4 in F n (θ) . Since these lines are of the form [y] with y ∈ 2 3 , we can identify every such a line [y] with the set supp(y) ∈ ℘ 2 ({1, . . . , n}). Every point (x), where x ∈ 3 3 , is in F n the meet of three lines [y t ], y t ∈ 2 3 , and t = 1, 2, 3 iff supp(y t ) ⊂ supp(x). Therefore, lines {[y] : y ∈ 2 3 } together with their intersection points form the structure dual to combinatorial Grassmanian G 3 (n), cf. [12]. The map F determines a permutation F 0 of the lines in J 0 which, in view of the above, is an automorphism of G 3 (n). The automorphisms group of G 3 (n) is the group S n (comp. [11]), and thus there is σ ∈ S n which determines F 0 . It is seen (cf. Step 1) that σ = σ . Let G = G σ be the automorphism of F n determined by the permutation σ (cf. Lemma 4.6) and let ϕ = G −1 • F. Clearly, ϕ is an automorphism of F n , and ϕ maps every line in J 0 onto itself. Consequently, ϕ maps every family J i \ {[θ ]} onto itself and thus it leaves the line [θ ] invariant.
Step 7 From Step 3, the map ϕ preserves the family J 0 . Observing intersections of the lines of this family and the lines in the families J i (cf. Step 2) we get that every line through θ remains invariant under ϕ.
Step 8 Let F be an automorphism of M such that F leaves every line through a point a invariant. Then F F n (a) = id.
Step 9 Let a, b be two points of F n such that b is a point of a Fano subplane in F n (a) . If F is an automorphism of F n such that F F n (a) = id then F F n (b) = id.

Proposition 6.23
The group Aut(P n ) is isomorphic to S n ((C 3 ) n (C 13 ) n ).
Proof Let J 0 be the family of lines of the size 4 in P n (θ) (Fig. 7) and J 0 be the family of the lines of the size 3 in P n (θ) that are not in any of the J i . Step 1 Let y ∈ (C 13 ) n . Then [y] ∈ J 0 iff y ∈ 2 {α,β} , where α, β ∈ {1, 3, 9}. If Step 2 There are no lines of the size 3 in P n (θ) (i.e. J 0 = ∅).
Step 3 Let us consider the set J α := {[y] ∈ J i : y i = α ∈ −D and i = 1, . . . , n}. If F is an automorphism of P n leaving every line in J α invariant then F P n (θ) = id.
Step 5 Let F be an automorphism of M such that F leaves every line through a point a invariant. Then F P n (a) = id. Step 6 Let a, b be two points of P n such that b is a point of a projective subplane PG (2,3) in P n (a) . If F is an automorphism of P n such that F P n (a) = id then F P n (b) = id.
Let us come back the power P n of an arbitrary finite projective plane P = D(C k , D) induced by a difference set D in a cyclic group C k . Observing Propositions 6.22 appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.