Multiplied Configurations Induced by Quasi Difference Sets

Bulletin of the Iranian Mathematical Society (2021) 47:111–133 https://doi.org/10.1007/s41980-020-00370-0 ORIGINAL PAPER

Multiplied Conﬁgurations Induced by Quasi Difference Sets

Krzysztof Petelczyc1 · Krzysztof Pra˙zmowski1

Received: 1 October 2019 / Accepted: 25 February 2020 / Published online: 21 March 2020 © The Author(s) 2020

Abstract 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.

Keywords Partial linear space · Difference set · Quasi difference set · Cyclic projective plane

Mathematics Subject Classiﬁcation 51D20 · 51E30

1 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.

Communicated by Mohammad Koushesh.

B Krzysztof Petelczyc [email protected] Krzysztof Pra˙zmowski [email protected]

1 Faculty of Mathematics, University of Białystok, Ciołkowskiego 1 M, 15-245 Białystok, Poland 123 112 Bulletin of the Iranian Mathematical Society (2021) 47:111–133

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 λ1 ways if a ∈ D, and in λ2 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 satis- fiability of Veblen, Pappus, and Desargues axioms. A special emphasis is imposed on structures which arise from groups decomposed into a cyclic group Ck 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 geomet- rical 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.

2 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)ofM 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, 123 Bulletin of the Iranian Mathematical Society (2021) 47:111–133 113

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 n- gons. We briefly recall this construction. Let G = G, ·, 1 be an arbitrary group and D ⊂ G. Let us introduce a point-line geometry

D(G, D) = G, G/D . (2.1)

Every translation τa : G x → a · x ∈ G is an automorphism of D(G, D). This yields that without loss of generality we can assume that 1 ∈ D.LetG 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.AnyD ⊆ 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(Ck ⊕ Cn, D), where D ={(0, 0), (1, 0), (0, 1)}.Inthis 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

− (a) [b] iff a ∈ b · D iff b 1 · a ∈ D. (2.2)

Note that if D is QDS then G D ={1}. Hence, a is uniquely determined by [a],orin other words a · D = b · D holds only for a = b.

3 Generalities

Now, we are going to present some general properties of D = D(G, D). Every automorphism ϕ = (ϕ,ϕ) of D uniquely corresponds to a pair f = ( f , f ) of bijections of G determined by

ϕ((a)) = ( f (a)), ϕ([b]) =[f (b)].

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 Fact 3.1 Let f ∈ Aut(G). The map f determines an automorphism ϕ = (ϕ,ϕ) of D(G, D) iff f (D) = q · Dforsomeq∈ G. Then ϕ((a)) = ( f (a)) and ϕ([a]) = [ f (a) · q] for every a ∈ G. 123 114 Bulletin of the Iranian Mathematical Society (2021) 47:111–133

Proposition 3.2 Let D be QDS in an abelian group G. Then the map κ deﬁned by

− − κ((a)) =[a 1], κ([a]) = (a 1) (3.1) is an involutive correlation of the structure D = D(G, D). Consequently, D is self- dual. A point a of D is selfconjugate under κ iff a2 ∈ D.

Proof Clearly, κ is involutory. Let a, b ∈ G. Then (a) [b] means that b−1 · a ∈ D. −1 This is equivalent to (a−1) · b−1 ∈ D, i.e. κ([b]) = (b−1) [a−1]=κ((a)). Thus κ is a correlation. Finally, assume that (a) κ((a)) =[a−1].From(2.2) we obtain a2 ∈ 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 Lemma 3.3 Let D ⊆ G, a, b ∈ G, and D = D(G, D). (i) The set of the lines of D through the point (a) can be identiﬁed with a · D−1, i.e.

∗ − (a) ={[a · d 1]: d ∈ D}. (3.2)

−1 −1 −1 −1 (ii) The points (a) and (b) are collinear in D iff a · b ∈ D D. If a b = d1 d2 with d1, d2 ∈ D then

−1 −1 a, b =[a · d1 ]=[b · d2 ]. (3.3)

(iii) The lines [a] and [b] of D have a common point iff a−1 · b ∈ DD−1.Ifa−1 · b = −1 d1 · d2 with d1, d2 ∈ D then

a b = (a · d1) = (b · d2). (3.4)

As a straightforward consequence of Lemma 3.3 we get Proposition 3.4 Let G = G, ·, 1 be a group, D be QDS in G with 1 ∈ D, and a1, a2 ∈ G. Points (a1) and (a2) can be joined with a polygonal path in D(G, D) iff −1 there is a ﬁnite sequence q1,...,qs of elements of D D such that a1 = q1 ·····qs ·a2. Consequently, the connected component of the point (1) is isomorphic to D( D G, D), where D G is the subgroup of G generated by D. Every two connected components of any two points are isomorphic.

Corollary 3.5 D(G, D) is connected iff D generates the whole group G.

For D ⊆ G we introduce the following condition:

−1 −1 −1 if d1, d2, d3, d4 ∈ D and d1d2 d3d4 ∈ DD then (3.5) −1 −1 −1 −1 d1d2 = 1ord3d4 = 1ord1d4 = 1, or d3d2 = 1.

The next Lemma explains the meaning of the condition (3.5). 123 Bulletin of the Iranian Mathematical Society (2021) 47:111–133 115

Lemma 3.6 Assume that G is an abelian group, and D ⊂ GisQDS satisfying (3.5). −1 −1 Let a ∈ G be a point of D(G, D),d1, d2 ∈ D, and b1 =[ad1 ],b2 =[ad2 ] be two distinct lines through a. Set i = 1, 2. ∈ = ( −1 ) ∼ = (i) If di D then pi adi di is a point on bi , and p1 p2 iff d1 d2. −1 (ii) For every point pd = (ad1 d) on b1 with d ∈ D, d = d2, d1 there is the unique −1 point q = (ad2 d) on b2 which completes points a, pd to a triangle. The point

pd2 cannot be completed to a triangle this way. (iii) If c is a line of D(G, D) which crosses both of b1,b2 and misses a then c = −1 −1 [ad1 d2 d] for some d ∈ D with d = d1, d2. (iv) For any two distinct lines c1, c2 crossing both of b1,b2 and missing a, i.e. ci = [ −1 −1 ] ∈ = ( −1 −1 ). ad1 d2 di for di D, we have c1 c2 ad1 d2 d1d2 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. Lemma 3.7 Let D = D(G, D), where D is QDS that satisfies (3.5) and p ∈ G. Let f be a collineation of D such that f (p) = p. If f satisfies any of the following: (i) f fixes all points on a line through p, (ii) f preserves every line through p, then f fixes all the points on the lines through p.

Proof Since D has a transitive group of automorphisms, we can assume that p = (1). −1 Let f = ( f , f ) be a collineation of D satisfying (i) for a line l1 =[d1 ] with −1 −1 d1 ∈ D.Taked2 ∈ D, d2 = d1 and a line l2 =[d2 ]. Then the points (d1 d2) on l1 −1 and (d2 d1) on l2 are the unique points “between” l1 and l2 that are not collinear (cf. −1 −1 −1 −1 Lemma 3.6(ii)). We have f (d1 d2) = (d1 d2) and f (d2 d1) [ f (d2 )].The −1 −1 −1 only point in D(p) non-collinear with (d1 d2) lies on [d2 ]; therefore, f (d2 ) = −1 d2 . 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 Corollary 3.8 Let f be a collineation which ﬁxes a line l of D point-wise. Under assumption (3.5) f ﬁxes all points on every line which crosses l. Consequently, if D is connected then f = id.

Corollary 3.9 Under assumption (3.5) every automorphism of D which has a ﬁxed point p is uniquely determined by its action on the lines through p. Consequently, the point stabilizer Aut(D)(p) of the automorphism group of D is isomorphic to a subgroup of the permutation group S|D|.

4 Products of Diﬀerence Sets

Let Gi = Gi , ·i , 1i be a group for i ∈ I , and 1i ∈ Di ⊂ Gi for every i ∈ I . 123 116 Bulletin of the Iranian Mathematical Society (2021) 47:111–133 = : −→ { : ∈ } Let G i∈I Gi , i.e. let Gbe the set of all functions g I Gi i I with ( ) ∈ G , ·, ( · )( ) = g i Gi . Then the product i∈I i is the structure G 1 , where g1 g2 i g1(i) ·i g2(i) for g1, g2 ∈ G, and 1(i) = 1i . It is just the standard construction of the direct product of groups. The set ={ ∈ : ( ) = ∈ } i∈I Gi g G g i 1i for a finite number of i I G G ={ ,..., } G ⊕ is a subgroup of i∈I i, denoted by i∈I i .IfI 1 r is finite, then 1 ···⊕G := G = G =: r G r i∈I i i∈I i i=1 i . ∈ π : −→ For every j I we define the standard projection j i∈I Gi G j by π ( ) = ( ) ε : −→ j g g j , and the standard inclusion j G j i∈I Gi by the conditions (ε j (a))( j) = a and (ε j (a))(i) = 1ifor i = j and a ∈ G j . Recall that π j and ε j = {ε ( ): ∈ } are group homomorphisms. We set i∈I Di i Di i I . For a finite set r I ={1,...r} we write ∈ Di = = Di = D1 ··· Dr . i I i 1 Proposition 4.1 If Di is QDS in Gi for every i ∈ I , then ∈ Di is QDS in ∈ Gi . i I i I Proof = , , , ∈ −1 = We set D i∈I Di .Letg1 g2 g3 g4 D and assume that g1g2 −1 ∈ ε ( ) = π ( −1) = = g3g4 .Letgi ji D ji .If j1 j2; then j g1g2 1 j for every j j1; thus −1 π j (g3g4 ) = 1 j , and thus j3 = j4 = j1. From assumption we infer that g1 = g3 and g2 = g4.If j1 = j2; analogously, we come to j1 = j3 and j2 = j4. Then we −1 −1 obtain g1 = g3 and g2 = g4 , which yields our claim.

Let Di = D(Gi , Di ). We write D := ( G , ). i∈I i D i∈I i i∈I Di ={ ,..., } r D := (G , ) ⊕···⊕ (G , ) For I 1 r we write also i=1 i D 1 D1 D r Dr .Note , ∈ G ∈ · = · ∈ {ε ( ): ∈ } that, for a b i∈I i , a b i∈I Di iff a b d for some d i Di i I , i.e. iff there is i ∈ I such that ai ∈ bi · Di and a j = b j for all j = i. So, we get

(a) [b] iff (ai ) [bi ] in Di for some i ∈ I and a j = b j for i = j ∈ I . (4.1)

Let us consider two particular cases. First, let D = D(Ck, {0, 1}) ⊕ D(G , D ) and D ={0, 1}D .Leta = (i, a ) ∈ Ck × G . Then the points of a · D are points of the form (i, a · p) with p ∈ D, and—one point—(i + 1, a). Somewhat informally we can say that the line with the coordinates [i, a] consists of the points (i, p), where (p) [a ] and one “extra” point (i +1, a ). In other words, we have a function fi which assigns a point of D = D(G, D) to every line of D such that the lines of D are of the form i ×l ∪{(i +1, fi (l))}, where l is a line of D . Thus D is a conﬁguration consisting of k copies of D cyclically inscribed one into another. In the above construction, the function fi is deﬁned by fi ([a]) = (a). Now, let D = D(G1, D1) ⊕ D(G2, D2). The lines of D are of the form (a1, a2) + ∗ ∗ D1 D2, which, on the other hand, can be written as [a1] ×{(a2)}∪{(a1)}×[a2] . ∗ Recall, that the lines of the Segre product D = D(G1, D1) ⊗ D(G2, D2) (cf. [9]) are ∗ ∗ the sets of one of two forms: [a1] ×{(a2)} or {(a1)}×[a2] . Therefore, the lines of D are unions of some pairs of the lines of D∗. From Proposition 3.4 we have the following: 123 Bulletin of the Iranian Mathematical Society (2021) 47:111–133 117

QDS G = ∈ Fact 4.2Let Di be inagroup i such that Di Gi Gi for all i I. G Then i∈I Di generates i∈I i. Consequently, if every one of the structures D = (G , ) D i D i Di is connected then i∈I i is connected as well. ⊂ ε ε : −→ Let J I ; we extend the inclusions i to the map J j∈J G j i∈I Gi by the condition 1 for i ∈ I \ J (ε (a))(i) = i for arbitrary a ∈ G . (4.2) J a(i) for i ∈ J j∈J j

Proposition 4.3 Let J be a nonempty proper subset of I . Then D =∼ D ⊕ D . i∈I i i∈J i i∈I \J i

Proposition 4.4 Let Di = D(Gi , Di ), where Di is QDS inagroupGi for all i ∈ I. ϕ,ϕ : −→ κ = Assume that there is a pair of bijections i i Gi Gi such that the pair i (κ, κ) i i of maps

κ : ( ) →[ϕ( )], κ :[ ] → (ϕ( )) ∈ i a i a i a i a for a Gi (4.3)

D ∈ ϕ = ϕ ×···×ϕ ϕ = ϕ ×···×ϕ is a correlation of i for i I . Set 1 r and 1 r .If ϕ = ϕ ∈ κ κ = (κ, κ) i i for every i I (i.e. if i are involutory), then the pair of maps κ : ( ) →[ϕ( )], κ :[ ] → (ϕ( )) ∈ a a a a for a i∈I Gi (4.4) D is an involutory correlation of i∈I i .

Proof By (4.1) we get κ([b]) = (ϕ(b)) [ϕ(a)]=κ((a)) iff the following holds: κ([ ]) = (ϕ( )) [ϕ( )]=κ(( )) ϕ( ) = ϕ ( ) = i bi i bi ai i ai and j b j j a j for j i.Nowthe claim is evident.

κ ϕ( ) = If we assume in Proposition 4.4 that every i is the standard correlation ( i a a−1, cf. Proposition 3.2), then κ is also the standard correlation. Using (4.1)asin Proposition 4.4 one can also prove the following: D = (G , ) = ( , ) Proposition 4.5 Let i D i Di and fi fi fi be bijections of Gi such that : ( ) −→ ( ( )) :[ ] −→ [ ( )] D ∈ fi a fi a and fi a fi a yields a collineation of i for i I, D = D = = = ( , ) and let i∈I i .WesetF i∈I fi ,F i∈I fi , and F F F . D = ∈ Then the pair F is a collineation of iff fi fi for every i I. ( , ) = = τ ∈ Obviously, the pair fi fi , where fi fi ai and ai Gi , is a collineation D ( , ) D of i ; therefore, the pair i∈I fi i∈I fi is an automorphism of . But this is a τ = τ rather trivial result, as i∈I ai a. We have also some automorphisms of another type.

n n n Proposition 4.6 Let β ∈ Sn,x ∈ G and h : G −→ G be the map deﬁned by h((x1,...,xn)) = (xβ(1),...,xβ(n)). The pair F = (F , F ) with F ((x)) = (h(x)), 123 118 Bulletin of the Iranian Mathematical Society (2021) 47:111–133

Fig. 1 Multiplied Pappus conﬁguration

F([x]) =[h(x)] is a collineation of

(G, ) ⊕ (G, ) ⊕··· (G, ) = (Gn, n). D D D D D D D D n times

4.1 Cyclic Multiplying

Let Ck be a cyclic group of the rank k. We use additive notation for these groups. In this (G, D ) ≥ G = ⊕···⊕ section we consider conﬁgurations D r , where r 2, Cn1 Cnr and Dr ={e0, e1,...,er } is QDS with e0 = (0,...,0) and (ei ) j = 0fori, j = 1,...,r, ∼ i= j, (ei )i = 1. The set Dr will be called canonical QDS. Note that D(G, Dr ) = r ( , { , }) i=1 D Cni 0 1 and in view of Proposition 4.3

(G, D ) =∼ ( , { , }) ⊕ ( ⊕···⊕ , D ). D r D Cn1 0 1 D Cn2 Cnr r−1

Thus D(G, Dr ) generalizes the construction of cyclically inscribed polygons considered in [10]. An interesting example of this type is D(C3 ⊕ C3 ⊕ C3, D3)—three copies of Pappus conﬁguration cyclically inscribed, see Fig. 1 and a more general case considered in Sect. 6.2.

n Lemma 4.7 Let M = D((Ck) , Dn), α be a permutation of the set {0,...,n} and i = 0,...,n. (i) A permutation α induces a collineation f = ( f , f ) of the structure M such that f (e0) = e0 and f (−ei ) =−eα(i) for all i. (ii) If f = ( f , f ) is a collineation induced by α such that f (e0) = e0,f (−ei ) = −1 −eα(i) for all i, then f τv( f ) = τ f (v). Proof (i) Assume that α(0) = 0 and consider a map f : G −→ G deﬁned by

f (x1,...,xn) = (xα(1),...,xα(n)). (4.5) 123 Bulletin of the Iranian Mathematical Society (2021) 47:111–133 119

Then f ∈ Aut(G) and f (Dn) = Dn, and thus f determines an automorphism of M. It is seen that f (e0) = e0. In view of Fact 3.1 we get f (y1,...,yn) = (yα(1),...,yα(n)),so f (−ei ) =−eα(i). Let α(0) = s = 0 be a transposition and a map f : G −→ G be given by: n f (x1,...,xn) = x1,...,xs−1, − xi , xs+1,...,xn . (4.6) i=1

Again f ∈ Aut(G), and note that f (Dn) =−es + Dn, f (e0) = M ( ,..., ) = e0. From Fact 3.1 f induces a collineation of with f y1 yn ( ,..., , − n − , ,..., ) (− ) = y1 ys−1 i=1 yi 1 ys+1 yn . It is easy to check that f ei −eα(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 ( )−1( ,..., ) = ( ,..., ) τ ( )−1( ,..., ) = f x1 xn xα−1(1) xα−1(n) . Therefore f v f x1 xn (x1,...,xn)+ f (v) = τ f (v)(x1,...,xn).Ifα is a transposition. then f is given by (4.6), and thus f = ( f )−1. After simple calculation we get the claim.

n Lemma 4.8 Let M = D((C3) , Dn). For every point o of M, any two distinct lines k, l through o and every point p with o = p k there is the unique point q such that o = q l and p ∼ q.

Proof As translations are transitive subgroup of Aut(M) assume that o = e0. Then, by Lemma 3.3, k =[−ei ], l =[−ek] and p = (−ek +e j ) for some i, j, k = 0,...,n, i = k = j. Assume that q is on [−ek] and q is collinear with p. Using Lemma 3.3 we get (−e + e ) if i = j, q = i j (4.7) (−ei + ek) if i = j.

Note that the condition (3.5) does not hold for all cannonical quasi difference sets.

n Fact 4.9 The canonical QDS in (Ck) satisﬁes (3.5) iff k > 3.

n Theorem 4.10 Let M = D((Ck) , Dn) and k > 3. The group Aut(M) is isomorphic n to Sn+1 (Ck) .

Proof By Corollary 3.9 every collineation f = ( f , f ) ﬁxing e0 is uniquely determined by a permutation α of lines through e0, i.e. f (−ei ) =−eα(i).From Lemma 4.7(i) every permutation α induces a collineation. So, there is an isomorphism ∼ ξ : → α (M) = + f , and consequently Aut (e0) Sn 1. Note that every automorphism of M is a composition of two maps: a translation and a collineation ﬁxing e0.Using Lemma 4.7(ii) we get (τu fα)(τv fβ ) = τuτ fα(v) fαβ , which closes the proof. 123 120 Bulletin of the Iranian Mathematical Society (2021) 47:111–133

Next, we consider more general case. Namely, we describe the neighborhood of a point q in a conﬁguration of the following form:

M = D(Ck, {0, 1}) ⊕ D(G, D) = D(Ck ⊕ G, {0, 1}D), where D is QDS in an abelian group G. Directly from deﬁnitions we calculate the following:

Lemma 4.11 Let D ={d0,...,dn} be QDS in an abelian group G = G, +,θ , and let q = (0,θ)∈ Ck × G. Set M = D(Ck, {0, 1}) ⊕ D(G, D). The lines of M through q are the following:

(1) li =[0, −di ] for i = 0,...,n.

Each line [0, −di ] contains q and the following points:

(a) qi, j = (0, −di + d j ) for j = 0,...,i, i = j; = ( , − ) (b) pi 1 di . (2) l =[−1,θ]=[k − 1,θ]. Its points are q and the following: = (− , ) = ,..., (c) pi 1 di for i 0 n. Then the points qi, j form a substructure isomorphic under the map (0, a) −→ (a) to the neighborhood of θ in D(G, D). Moreover, the following additional lines appear: , = ,..., = =[−, − + ] = (3) For every i j 0 n, i j the line li, j 1 d j di joins pi (−1, di ) [−1,θ]=l with q j,i = (0, −d j + di ) [0, −d j ]=l j . , = =[, −( + )] = (4) For every i j as above, the line li, j l j,i 1 di d j joins pi ( , − ) [ , − ]= = ( , − ) [ , − ]= 1 di 0 di li with p j 1 d j 0 d j l j . The lines listed above are pairwise distinct. (i) If k > 3, then no other line appears (Fig. 2). (ii) Let k = 3. Then −1 = 2 holds in Ck, and then another connections are associated with triples (di , d j , dr ) ∈ D satisfying

di + d j + dr = θ. (4.8)

Namely, let (4.8) be satisﬁed. Evidently, −(di + d j ) = dr . =[ , −( + )]=[ , ] = (− , ) [− ,θ]= (5) The lineli, j 1 d j di 1 dr passes through pr 1 dr 1 = ( , − ) [ , − ]= l and p j 1 d j 0 d j l j . = (6) If, moreover, j i, then the above line passes through pi as well so, it coincides with the line deﬁned in (3.2)(Fig. 3).

Recall from [10] that a collineation of D(Ck, {0, 1}) is simply an element of the dihedral group Dk, i.e. it is any map αε,q : i → εi + q, where ε ∈{1, −1}. Propo- sition 4.5 determines all the automorphisms of D(Ck ⊕ G, {0, 1}D) of the form (i, a) → (αε,q (i), f (a)). Still, in this case we should look for automorphisms deﬁned with more complicated formulas. 123 Bulletin of the Iranian Mathematical Society (2021) 47:111–133 121

q=(0, )

|| l l|| q p|| q 0 l|| q p|| i l|| q || l|| q p j l|| q

| | | p l | p j 0 l j,i l|| l | 0 p i lj li

Fig. 2 The neighborhood of the point (0,θ)in a conﬁguration D(Ck , {0, 1}) ⊕ D(G, D) for k > 3

q=(0, )

|| l l|| q || p 0 || | q l l q p|| i l|| q || l|| q p j l|| q || prji =(-1,-(d +d ))

| l j,i l|

| | | p j p 0 l l|| l | 0 p i lj li

Fig. 3 The neighborhood of the point (0,θ)in a conﬁguration D(C3, {0, 1}) ⊕ D(G, D)

Proposition 4.12 Let M0 = D(G, D), M = D(Ck ⊕ G, {0, 1}D) and f = ( f , f ) ∈ Aut(M0). The following conditions are equivalent: (i) There is a collineation ϕ = (ϕ,ϕ) of M such that ϕ((0, a)) = (0, f (a)) and ϕ([0, b]) =[0, f (b)]. (ii) There is a sequence fi ,i = 0,...,k, of collineations of M0 deﬁned recursively = = = ( , ) by the formulas: f0 f and fi+1 fi , where fi fi fi . 123 122 Bulletin of the Iranian Mathematical Society (2021) 47:111–133

ϕ(( , )) = ( , ( )) ϕ([ , ]) =[, ( )] In the case (ii) we have i a i fi a and i b i fi b .

Proof It sufﬁces to note that if (i) holds, then (1, a) [0, a] for every a ∈ G, which gives ϕ((1, a)) ϕ([0, a]) =[0, f (a)], and thus ϕ((1, a)) = (1, f (a)). Therefore, f (as a transformation of points) induces a collineation of M0.

5 Elementary Properties

Now we discuss some elementary axiomatic properties of D(G, D).LetD be QDS in an abelian group G. Using Lemma 3.6 we prove the following:

Proposition 5.1 Under assumption (3.5) the structure D(G, D) is Veblenian.

Proposition 5.2 Under assumption (3.5) the structure D(G, D) is Desarguesian.

As a consequence of Fact 4.9 and Propositions 5.1, 5.2 we get

n Corollary 5.3 For k > 3 the structure D((Ck) , Dn) is Veblenian and Desarguesian.

n Proposition 5.4 The structure M = D((C3) , Dn) is not Veblenian.

Proof Take li =[−ei ] for i = 1, 2, and k1 =[e1 + e2], k2 =[−e1 − e2 + e3]. Then (e0) l1, l2, and k1 crosses l1 in (−e1 + e2) and l2 in (−e2 + e1). Furthermore, the lines k2, l1 meet in (−e1 + e3), and (−e2 + e3) is the common point of l2, k2. Suppose that k1 ∩ k2 =∅. Then (e1 + e2) + et = (−e1 − e2 + e3) + es for some s, t = 1,...,n, s = t, which implies a contradiction: e1 + e2 + e3 + es = et .

n Proposition 5.5 The structure M = D((C3) , Dn) is Desarguesian.

Proof Without loss of generality we can assume that (e0) is the perspective center of two triangles T1, T2 inscribed into three lines l1, l2, l3 of M such that the corresponding pairs of their sides intersect each other. We do some calculations based on Lemma 3.3. =[− ] = , , ∈ D Note that lr eir for r 1 2 3, eir n. By Lemma 4.8 and (4.7) the sides of our triangles are lines of the form [−ei −e j +es] or [ei +e j ] for some ei , e j , es ∈ Dn, = , [ + ] s i j. The line ei1 ei2 does not meet any other line which crosses both l1 and l2. Thus the sides are lines of the type [−ei −e j +es]. Consequently, the vertices of T1 are (− + ) (− + ) (− + ) (− + ) (− + ) ei1 es , ei2 es , ei1 es , and the vertices of T2 are ei1 et , ei2 et , (− + ) = , ∈ D ei1 et for eir es et n. Then the points of intersection of the corresponding = (− − + + ) = (− − + + ) sides of T1 and T2 are c1 ei1 ei2 es et , c2 ei2 ei3 es et , and = (− − + + ) [− − − + + ] c3 ei3 ei1 es et . All these points are on the line ei1 ei2 ei3 es et , which proves the claim.

The Pappus conﬁguration can be considered as D(C3 ⊕ C3, D2),cf.[10,15](see Fig. 4).

Proposition 5.6 Under assumption (3.5) the structure D(G, D) does not contain the Pappus conﬁguration. 123 Bulletin of the Iranian Mathematical Society (2021) 47:111–133 123

Fig. 4 Pappus conﬁguration

Proof Assume that D(G, D) contains the Pappus conﬁguration. Then (0, 0), (1, 0), (0, 1) [0, 0]; (0, 0), (1, 2), (0, 2) [0, 2]; (1, 0) ∼ (1, 2), and (0, 1) ∼ (0, 2), which contradicts Lemma 3.6(ii).

Proposition 5.7 Let D be QDS in an abelian group G such that 1 ∈ D. Assume that , , , ∈ \{ } = 2 = −1 2 = −1 there are d1 d2 d3 d4 D 1 with d1 d3,d1 d2 , and d3 d4 . Then M = D(G, D) contains Pappus conﬁgurations.

Proof From the assumptions we get d2 = d4. Note that incidences indicated in the following table hold in M:

−1 −1 −1 −1 −1 −1 (1)(d1)(d3)(d2 )(d4 )(d2 d4 )(d1d3)(d2 d3)(d4 d1)

[1]××× −1 −1 [d2 d4 ]××× [d1d3]××× −1 [d2 ]× × × −1 [d4 ]× × × [d1]×× × −1 [d1d4 ]× × × [d3]××× −1 [d3d2 ]× × ×

Then the map (ψ,ψ) deﬁned for the points by

(a)(0, 0)(1, 0)(0, 1)(1, 2)(2, 1)(1, 1)(2, 2)(0, 2)(2, 0)

−1 −1 −1 −1 −1 −1 ψ ((a)) (1)(d1)(d3)(d2 )(d4 )(d2 d4 )(d1d3)(d2 d3)(d4 d1)

123 124 Bulletin of the Iranian Mathematical Society (2021) 47:111–133 and for the lines by

[a][0, 0][1, 1][2, 2][0, 2][2, 0][1, 2][1, 0][2, 1][0, 1]

−1 −1 −1 −1 −1 −1 ψ ([a]) [1][d2 d4 ][d1d3][d2 ][d4 ][d1][d1d4 ][d3][d3d2 ] embeds the Pappus conﬁguration into the structure M.

6 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.

6.1 Multi-Fano Configuration

The Fano configuration F is a finite projective plane, so it can be obtained as F = D(C7, {0, 1, 3}) (cf. [5,15]). Let us introduce the multi-Fano configuration + ∼ F = D(Ck ⊕ C7, {(0, 0), (0, 1), (0, 3), (1, 0)}) = D(Ck, {0, 1})⊕F. We refer to the subconfiguration determined by points of the form {i}×C7 with fixed i = 0,...,k −1 as to a Fano’s part of F+. The following two Lemmas can be easily proved by analyzing the neighborhood of a point (cf. Fig. 5). Lemma 6.1 If f ∈ Aut(F+), and x, y are two points of F+, such that f (x) = y, then + + f transforms the Fano’s part of F (x) into the Fano’s part of F (y).

+ Lemma 6.2 If f ∈ Aut(F ) and f preserves {0}×C7, then f preserves {i}×C7 for every i = 0,...,k − 1.

+ ∼ Proposition 6.3 (i) If 7 k, then Aut(F ) = Ck ⊕ C7. + ∼ (ii) If 7 | k, then Aut(F ) = C3 (Ck ⊕ C7).

+ Proof Generally, Tr(Ck ⊕ C7) ⊆ Aut(F ). Let us take g = τ− f (0,0) ◦ f , where f ∈ Aut(F+). Then g((0, 0)) = (0, 0). By Lemma 6.1, every collineation g ∈ + Aut(F )((0,0)) preserves the Fano substructure in the neighborhood of the point (0, 0). 123 Bulletin of the Iranian Mathematical Society (2021) 47:111–133 125

Fig. 5 The neighborhood of a point (i, a) in multi-Fano conﬁguration

According to Lemma 6.2, the Fano substructure is preserved on every of i levels, where i = 0, 1,...,k − 1. Denote the set {0, 1, 3} by D. Note that D has two multipliers: 2D = 6 + D and 4D = 4 + D. Thus, in view of Fact 3.1 maps g1 = (g , g ) with g (x) = 2x, 1 1 1 g = τ6g , and g2 = (g , g ) with g (x) = 4x, g = τ4g are collineations of F. 1 1 2 2 2 2 2 Moreover, g1, g2 ∈ Aut(F)(0). In view of Proposition 4.12,amapϕ = (ϕ ,ϕ ), such ϕ( , ) = ( , ( )) ϕ([ , ]) =[, ( )] ( , ) = ∈ (F) that i a i fi a and i b i fi b , where fi fi fi Aut , ∈ F+ = (F) i Ck, is a collineation of iff fi+1 fi . If we analyze all the elements of Aut (0) we note that g1, g2 are the unique two maps that can be extended to a collineation of F+, since translations are always collineations of F. = = = τ = τ = τ Set f0 g1, f0 g1 6g1. By induction we get fi 6i g1, fi 6(i+1)g1. = τ = = τ = ≡ In particular, fk−1 6k g1 f0 g1. Therefore 6k id, i.e. 6k 0 holds in C7 | ( , ) = ( , ) and thus 7 k. The same result we obtain for f0 f0 g2 g2 . To close the proof ∼ −1 note that G ={g1, g2, id} = C3 and f τ( j,b) f (i, a) = τ f ( j,b)(i, a) for f ∈ G.

6.2 Multi-Pappus Configuration

2 n Since D((C3) , D2) is simply the Pappus conﬁguration, D((C3) , Dn) will be called 3 the multi-Pappus conﬁguration (there is D((C3) , D3) shown in Fig. 1). Note that, in view of Fact 4.9, Proposition 4.10 cannot be used to characterize the automorphisms n group of D((C3) , Dn).

n ∼ n Proposition 6.4 Let M = D((C3) , Dn) with n > 2. Then Aut(M) = Sn+1 (C3) . To prove Proposition 6.4 we use Lemmas 4.7, 4.8 and the following lemma:

Lemma 6.5 Let f = ( f , f ) ∈ Aut(M), p be a point of M and l be a line through p. If f ﬁxes every line through p and f ﬁxes every point on l then f = id. 123 126 Bulletin of the Iranian Mathematical Society (2021) 47:111–133

6.3 Join of the Multi-Pappus Configuration and a Cyclic Projective Plane

Now we focus on k M = D((C3) , Dk) ⊕ D(Cn, D), (6.1) where P = D(Cn, D) is the cyclic projective plane PG(2, q) with a prime power q, 2 given by a difference set D ={d0, d1,...,dq } in the group Cn, where n = q +q + 1 (cf. [2,13]). The projective cyclic plane PG(2, q) can be determined by q + 1dis- tinct difference sets D1, D2,...,Dq+1. Quite surprisingly, it turns out that Mi = k i D((C3) , Dk) ⊕ D(Cn, D ), i = 1, 2,...,q + 1 are not always pairwise isomor- ( k, D ) ⊕ ( , { , , , }) phic. Indeed, we will show that D C3 k D C13 0 1 3 9 is not isomorphic to ( k, D ) ⊕ ( , { , , , }) D C3 k D C13 0 2 8 12 . In view of Proposition 4.3

M =∼ ( , { , }) ⊕ ( , { , }) ⊕···⊕ ( , { , }) ⊕ ( , ). D C3 0 1 D C3 0 1 D C3 0 1 D Cn D k times

Hence, in this case we can use Lemma 4.11. Every point p ∈ M can be written as p = k (xk,...,x1, y), where xk,...,x1 ∈ C3, y ∈ Cn.Letθ = (0,...,0) ∈ (C3) × Cn. In this notation, the points and the lines of M(θ) are the following:

qi, j = (0,...,0, −di + d j ), = ( ,..., , ,...,− ), = ( ,..., , ,..., ), pm,i 0 1m 0 di pm,i 0 2m 0 di =[ ,..., , − ], =[ ,..., , ,..., ], li 0 0 di lm 0 2m 0 0 for di , d j ∈ D; i, j = 0,...,q; i =, j; m = 1,...,k, = ( ,..., , ,..., , ,..., ), = ( ,..., , ,..., , ,..., ), ps,q+r 0 1s 0 2r 0 0 ps,q+r 0 2s 0 1r 0 0 for s = 2,...,k; r = 1,...,s − q + 2.

Using Lemma 4.6 and Proposition 4.5 we prove the following Lemmas:

Lemma 6.6 Let F ∈ Aut(M)(θ). Then, F leaves invariant the multi-Pappus sub- k ∼ k conﬁguration (C3) ×{0} = D((C3) , Dk) and the cyclic projective subplane ∼ {(0,...,0)}×Cn = D(Cn, D). Moreover, F determines permutations α ∈ Sq β ∈ ( ) = ( ) = ( ) = and Sk such that F lm lβ(m),F pm,0 pβ(m),0, and F li lα(i) for m = 1,...,k, i = 1,...,q. β ∈ k ⊕ Lemma 6.7 Let Sk. We deﬁne the map Gβ on C3 Cn by the formula Gβ ((xk,...,x1, y)) = (xβ(k),...,xβ(1), y). Then Gβ ∈ Aut(M),Gβ (θ) = θ, and ( ) = Gβ pm,0 pβ(m),0.

Lemma 6.8 Let Gβ be the map deﬁned in Lemma 6.7 and G0 ={Gβ : β ∈ Sk}, G ={τ ◦ : ∈ G , ∈ ( )k × } G =∼ G =∼ ( k ⊕ ) a g g 0 a C3 Cn . Then 0 Sk and Sk C3 Cn . Lemma 6.9 If, under notation of Lemma 6.6 we assume β = id, then every point , , α ∈ ( ) = pm,0 ps,q+r ps,q+r is ﬁxed by F. Moreover, for Sq we have F pm,i pm,α(i) for i = 1,...,q. 123 Bulletin of the Iranian Mathematical Society (2021) 47:111–133 127

Lemma 6.10 Under assumptions of Lemma 6.9, and the condition

(a) for every, except at most one, di ∈ D there exist d j , dr ∈ D such that

di + d j + dr = 0

α ( ) = the permutation given in Lemma 6.6 satisfies the following: F pm,i pm,α(i), F(qi, j ) = qα(i),α( j),form = 1,...,k; i, j = 1,...,q, i = j. Consequently, if α = id, then F is the identity on M(θ). Let q be a point of M and F be an automorphism of M with F(q) = q. Generalizing α β the notation of Lemma 6.6 we write F(q) for the permutation α of {1,...,q} and F(q) for the permutation β of {1,...,k} determined by F M(q). Lemma 6.11 Let the condition (a) of Lemma 6.10 be satisfied. If F ∈ Aut(M) pre- β α serves every line passing through q (in particular, F(q) = q), then F(q) and F(q) are identities, and thus F is the identity on M(q). Lemma 6.12 Let the condition (a) of Lemma 6.10 be satisfied. Assume that F ∈ Aut(M) fixes all the points of M(q).Ifq ∈ M(q), then F fixes the points of M(q). Since M is connected, combining Lemmas 6.10 and 6.12 we obtain Corollary 6.13 Let the condition (a) of Lemma 6.10 be satisfied. Assume that F ∈ Aut(M) and q is a point of M.IfF(q) = q and F preserves every line through q, then F = id. With the help of Lemma 6.6–Corollary 6.13 we determine automorphisms group k of M = D((C3) , Dk) ⊕ P in two particular cases: for P = F = PG(2, 2) and for k P = PG(2, 3). Let us start from M = D((C3) , Dk) ⊕ F. The obtained structure can be considered as a join of the multi-Pappus and the Fano configuration. M = ( k, D ) ⊕ ( , { , , }) ≥ Proposition 6.14 Let D C3 k D C7 0 1 3 with k 2. Then the group (M) ( k ⊕ ) Aut is isomorphic to Sk C3 C7 . Adopt P = PG(2, 3) instead of Fano configuration. As PG(2, 3) can be induced by four distinct difference sets D1 ={0, 1, 3, 9}, D2 ={0, 2, 8, 12}, D3 ={0, 6, 10, 11}, D4 ={0, 4, 5, 7}, we get

Mi = ( k, D ) ⊕ ( , i ), D C3 k D C13 D where i = 1, 2, 3, 4. Note, that the condition (a) from Lemma 6.10 is satisﬁed for i 1 1 all sets D , but in different ways: for every di ∈ D there exist d j , dr ∈ D such 2 3 4 that di + d j + dr = 0, and in D , D , D there is exactly one di for which do not j exist such d j , dr . So, the conﬁgurations M are pairwise isomorphic for j = 2, 3, 4 (for φ(x) = 3x we get φ(D2) = D3,φ(D3) = D4,φ(D4) = D2), but are not isomorphic to M1. Although M1 and M2 are not isomorphic, they have isomorphic automorphisms groups. To justify this, we show the complete proof of the following Proposition: 123 128 Bulletin of the Iranian Mathematical Society (2021) 47:111–133

Mi = ( k, D ) ⊕ ( , i ) = , 1 = Proposition 6.15 Let D C3 k D C13 D , where i 1 2 and D { , , , } 2 ={ , , , } (Mi ) ( k ⊕ 0 1 3 9 ,D 0 2 8 12 . Then, the group Aut is isomorphic to Sk C3 C13). i Proof Let F be an automorphism of M and g = τ−F(θ) ◦ F. Then g(θ) = θ and ∈ (Mi ) { : = ,..., } g Aut . In view of Lemma 6.6, g leaves the set pm,0 m 1 k invariant. −1 β We set h = g ◦ Gβ , where Gβ is the map defined in Lemma 6.7 and β = g(θ) ∈ Sk. β Then h ∈ Aut(Mi ), h(θ) = θ, and h(θ) = id. By Lemma 6.9 h preserves the set {l : m = 1,...,k}. m α Let α = h(θ) be the permutation determined by h, in accordance with Lemma 6.9. Then, from Lemma 6.10 we get h(qi, j ) = hα(qi, j ) := qα(i),α( j) for all i, j.Note 1 that if α = id, then hα does not preserve the collinearity in M . For example: q1,2, q0,2, q3,1, q2,1 are collinear, but qα(1),α(2), q0,α(2), qα(3),α(1), qα(2),α(1) are not, unless α = M2 = ,..., id. In for all m 1 k we have: pm,1 are points of rank 5 on a line of rank 3, pm,2 are points of rank 5 and there is no line of rank 3 passing through these α = points, and pm,3 are points of rank 6. So, h fixes these points, and thus id. In both cases, h fixes all the lines through θ, so from Corollary 6.13 we get h = id, and thus g = Gβ . Finally, applying Lemma 6.8 we close the proof.

6.4 A Power of a Cyclic Projective Plane

Let P = D(Ck, D) be a cyclic projective plane determined by a difference set D in 2 the group Ck. Then k = q + 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:

Pn := P ⊕ P ⊕···⊕P (6.2) n times

n n Note that P = D((Ck) , D), where D = D···D. Let us introduce a few auxiliary sets. Namely:

n supp(x) := {i ∈{1,...,n}: xi = 0} for x ∈ (Ck) , γ n α := {x ∈ (Ck) :|supp(x)|=γ and if i ∈ supp(x) then xi = α}, 1 1 1 {α},{β} := α ∪ β , 2 n {α,β} := {x ∈ (Ck) :|supp(x)|=2, {i, j}=supp(x) ⇒ { xi , x j }={α, β}}, n Pi := {x ∈ (Ck) : supp(x) ={i}, or x = θ}, Si ={(x): x ∈ Pi }, Ji ={[x]: x ∈ Pi }, i = 1,...,n.

= J ∩ J ={[θ]} ∩ ={(θ)} J For i1 i2 we have i1 i2 and Si1 Si2 .ThesetsSi and i consist of points and lines, respectively, which form a projective plane embedded in n P (θ). There are n such planes with the common line [θ], and the common point (θ) n n in P (θ). Note that, the degree of the point (θ) in P (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: 123 Bulletin of the Iranian Mathematical Society (2021) 47:111–133 129

Lemma 6.16 The line [y] passes through (θ) iff [y]∈Ji for some i and yi ∈−D. n n Lemma 6.17 Let x ∈ (Ck) . The point (x) is a point of P (θ) iff x ∈ Pi for some i 2 (i.e. |supp(x)|=1)orx∈ {α,β} and α ∈−D\{0}, β ∈ D\{0}. n Lemma 6.18 If x ∈ Pi then either the line [x] passes through (θ) and its size in P (θ) n is qn + 1, or the size of [x] in P (θ) is q + 1; then, in particular, [x] does not contain any point of Pj with j = i. n n Lemma 6.19 Let x ∈ (C7) .Iftheline[x] contains a point of P (θ) (, i.e. it intersects a line of the form [y] deﬁned in Lemma 6.16), then |supp(x)|≤3. Moreover, if n |supp(x)|=3, then the size of [x] in P (θ) is 2.

n 2 There are lines in P (θ) joining points in Pi with points in {α,β}, where α ∈ −D\{0}, β ∈ D\{0}. Namely:

Lemma 6.20 Let (x) ∈ Si , [y]∈Ji , and (x) [y]. For every j = i there are q lines of n the size 2 in P (θ), such that each of them joins (x) with one of the pairwise collinear ( 1),...( q ) 1 = ··· = q = − { 1,..., q }=−D \{ } points z z , where zi zi xi yi , z j z j 0 , and 1 =···= q = = , = ,..., zs zs 0 for all s i j; s 1 n. n Lemma 6.21 For any two points a, a of P there is a sequence b0,...,bm of points Pn = = Pn of such that b0 a, bm a , and b j is a point of a projective subplane in (b j−1) for j = 1,...,m.

On this level of generality not much more could be said. Now we consider a power of PG(2, 2) and PG(2, 3).

6.4.1 A Power of the Fano Plane

Let us put P = F = D(C7, {0, 1, 3}). n n Proposition 6.22 The group Aut(F ) is isomorphic to Sn (C7) . Proof J Fn J Let 0 be the family of lines of the size 4 in (θ) and 0 be the family of n the lines of the size 3 in F (θ) that are not in any of the Ji . We need Lemma 4.6 and lemmas from Sect. 6.4. ∈ ( )n [ ]∈J ∈ 2 ∈ 2 ( ) ={ , } Step 1 Let y C7 . Then y 0 iff y 3.Let y 3 and supp y i1 i2 . The line [y] does not intersect [θ],but[y] intersects every of the remaining two lines J J , ∈{ ,..., } in i1 and in i2 . Consequently, for any two i1 i2 1 n there is (exactly one) J J J line in 0 that crosses two lines in i1 and two lines in i2 . No two distinct lines in J 0 intersect. = θ [ ] Fn ∈ 2 ∈ Step 2 If y , then y is of the size 3 in (θ) iff either y {1,3} or y Pi ∈{,..., } J ={[ ]: ∈ 2 } for some i 1 n . This gives, in particular, that 0 y y {1,3} .Let ( ) ={ , } = = [ ] [θ] [ ] supp y i1 i2 , yi1 1, and yi2 3. Lines y and do not meet. The line y J J crosses two other lines in i2 and it crosses exactly one line in i1 . Consequently, for , ∈{ ,..., } J every two i1 i2 1 n there is (exactly one) line in 0 that crosses two lines in J J J i2 and crosses exactly one line in i1 . No two distinct lines in 0 intersect. 123 130 Bulletin of the Iranian Mathematical Society (2021) 47:111–133

Fig. 6 The neighborhood of the point (0, 0) in F ⊕ F

Directly from Step 2 we get: Fn J Step 3 A line l in (θ) belongs to 0 iff the size of l equals 3 and no other line of the size3inFn(θ) crosses l. The next two Steps are immediate from Lemma 6.17 and Steps 1, 2: n Step 4 Let (x) be a point of F (θ). Then x ∈ Pi iff there are two distinct lines of the J ∪ J size 3 that pass through it. The set of points on the lines in 0 0 is the set of points ( ) ∈∪/ n x with x i=1 Pi . Step 5 From the above it follows that Fn(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 Si , Ji , . Intuitively, we can read Step 1 as “any two Fano subplanes of Fn(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 n automorphism F preserves the set i=1 Si and permutes the Fano subplanes. So, F determines a permutation σ such that F maps the set Si onto Sσ(i) and it maps the family Ji onto Jσ(i) for every i = 1,...,n. Step 6 Obviously, F preserves the set of lines of the size 4 in Fn(θ). Since these [ ] ∈ 2 [ ] lines are of the form y with y 3, we can identify every such a line y with ( ) ∈ ℘ ({ ,..., }) ( ) ∈ 3 Fn the set supp y 2 1 n . Every point x , where x 3,isin the meet [ ] ∈ 2 = , , ( ) ⊂ ( ) of three lines yt , yt 3, and t 1 2 3 iff supp yt supp x . Therefore, {[ ]: ∈ 2} lines y y 3 together with their intersection points form the structure dual to combinatorial Grassmanian G3(n),cf.[12]. The map F determines a permutation J ( ) F0 of the lines in 0 which, in view of the above, is an automorphism of G3 n . The automorphisms group of G3(n) is the group Sn (comp. [11]), and thus there is σ ∈ Sn which determines F0. It is seen (cf. Step 1) that σ = σ .LetG = Gσ be the automorphism of Fn determined by the permutation σ (cf. Lemma 4.6) and let ϕ = −1 ◦ ϕ Fn ϕ J G F. Clearly, is an automorphism of , and maps every line in 0 onto 123 Bulletin of the Iranian Mathematical Society (2021) 47:111–133 131 itself. Consequently, ϕ maps every family Ji \{[θ]} onto itself and thus it leaves the line [θ] invariant. ϕ J Step 7 From Step 3, the map preserves the family 0. Observing intersections of the lines of this family and the lines in the families Ji (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 n a invariant. Then F F (a) = id. n n Step 9 Let a, b be two points of F such that b is a point of a Fano subplane in F (a). n n n If F is an automorphism of F such that F F (a) = id then F F (b) = id.

6.4.2 A Power of the Cyclic Projective Plane PG(2, 3)

Let P = D(C13, {0, 1, 3, 9}), i.e. let P be the cyclic projective plane PG(2, 3).Note that the set {0, 1, 3, 9} is ﬁxed by the multiplier α = 3. Thus, this multiplier yields an automorphism of P (cf. Fact 3.1). The map μα : C13 x −→ α · x generates the cyclic group consisting of μ1 = id, μ3, and μ9, which is isomorphic to C3.Every element of this group induces an automorphism of P(θ).

n n n Proposition 6.23 The group Aut(P ) is isomorphic to Sn ((C3) (C13) ). Proof J Pn J Let 0 be the family of lines of the size 4 in (θ) (Fig. 7) and 0 be the family n of the lines of the size 3 in P (θ) that are not in any of the Ji . ∈ ( )n [ ]∈J ∈ 2 α, β ∈{, , } Step 1 Let y C13 . Then y 0 iff y {α,β}, where 1 3 9 .If [ ]∈J ( ) ={ , } [ ] J y 0 and supp y i1 i2 then y intersects two out of three lines in i1 and J [θ] , ∈{ ,..., } in i2 , but do not intersects . Consequently, for any two i1 i2 1 n there J J J are exactly nine lines in 0 that cross two lines in i1 and two lines in i2 . For every ( ) ∈ Pn ∈/ [ ]∈J ( ) [ ] point x (θ) with x Pi there are two lines y 0 such that x y .Every J J line in 0 intersects four other lines in 0 and does not intersect remaining four lines J from 0 . n Step 2 There are no lines of the size 3 in P (θ) (i.e. J =∅). α 0 Step 3 Let us consider the set J := {[y]∈Ji : yi = α ∈−D and i = 1,...,n}.If n α n F is an automorphism of P leaving every line in J invariant then F P (θ) = id. n Step 4 Let σ ∈ Sn. We deﬁne the map hσ on (C13) by the formula

hσ ((x1,...,xn)) = (xσ(1),...,xσ(n)).

By Lemma 4.6 the map hσ induces the collineation Fσ = (F , F ) of Pn. Moreover, F (θ) = θ and G (Pi ) = Pσ(i) for all i = 1,...,n. Step 5 Let F be an automorphism of M such that F leaves every line through a point Pn = a invariant. Then F (a) id. Step 6 Let a, b be two points of Pn such that b is a point of a projective subplane ( , ) Pn Pn Pn = PG 2 3 in (a).IfF is an automorphism of such that F (a) id then Pn = F (b) id.

n Let us come back the power P of an arbitrary ﬁnite projective plane P = D(Ck, D) induced by a difference set D in a cyclic group Ck. Observing Propositions 6.22 123 132 Bulletin of the Iranian Mathematical Society (2021) 47:111–133

(0,0)

(1,12) [9,1] [3,1] (1,4) (10,1) (4,1) (1,10) [1,1] (10,3) (12,1) (3,4) (3,12) [3,3] (12,3) (4,3) [1,3] (3,10) (9,4)

[4,0] [9,9] (10,9) (12,9) [9,3] (9,12) (9,10)

[10,0] [0,12] [0,10] [1,9] [0,4] [12,0] [3,9]

(4,9)

Fig. 7 The structure of lines of the size 4 in P2(θ) and 6.23 and their proofs we note that every automorphism F of Pn is related to one of the following: n • a multiplier of the set D, and then F leaves P (θ) invariant, n n n • a translation τ of (Ck) , and then F maps P (θ) onto P (τ(θ)), • a permutation σ ∈ Sn, and then F permutes the projective subplanes isomorphic to P. Moreover, a composition of the two ﬁrst automorphisms is not commutative, which also does not commute with the third one. We claim that

Conjecture 6.24 Let P = D(Ck, D) be a ﬁnite projective plane induced by a difference n set D in a cyclic group Ck. The group Aut(P ) is isomorphic to ( )n ⊕···⊕( )n ( )n , Sn Cr1 Crs Ck

,... D where Cr1 Crs are the cyclic groups generated by the multipliers of the set .

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