Multiplied Configurations Induced by Quasi Difference Sets

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.

Introduction

There is well-known construction of a point-block geometry induced by some fixed subset D of a group \({\textsf {G}}\) (cf. [2, 14]). The idea is simple: points are elements of \({\textsf {G}}\), and blocks (lines) are the images of D under (left) translations. If every nonzero element of \({\textsf {G}}\) can be presented in exactly \(\lambda \) ways as a difference of two elements of D, then D is called a difference set, cf. [3]. In this case we obtain a \(\lambda \)-design. Difference sets with \(\lambda =1\) 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 \(\lambda \in \{0,1\}\) 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 \({\textsf {G}}\) can be presented as a corresponding difference in \(\lambda _1\) ways if \(a \in D\), and in \(\lambda _2\) ways if \(a \notin D\). 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 \({\textsf {G}}\). These structures can be seen as multiplied configurations—series of cyclically inscribed configurations, each one isomorphic to the configuration associated with \({\textsf {G}}\). 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 be an incidence point-line geometry. If for \(p\in S\), \(k\in {\mathcal {L}}\) then we say that “p is on the line k” or “k passes through the point p”. \({\mathfrak {M}}\) 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 \(p\in S\) is denoted by \({p}^{*}\), and dually we write \({k}^{*}\) for the set of all points on a line \(k\in {\mathcal {L}}\). The rank of a point p is the number \(|{p}^{*}|\), and the size of a line k is the number \(|{k}^{*}|\). If \(p\ne q\) are two collinear points then we write \(p \sim q\) and the line which joins these two points is denoted by . We also write for the common point of two intersecting lines k and l.

An automorphism (or a collineation) of \({\mathfrak {M}}\) is a pair \(\varphi =(\varphi ',\varphi '')\) of bijections \(\varphi ':S \longrightarrow S\), \(\varphi '':{\mathcal {L}}\longrightarrow {\mathcal {L}}\) such that for every \(a \in S\), \(l \in {\mathcal {L}}\) the conditions and are equivalent. A pair \(\varkappa = (\varkappa ',\varkappa '')\) of bijections \(\varkappa ':M \longrightarrow {\mathcal {L}}\), \(\varkappa '':{\mathcal {L}}\longrightarrow M\) satisfying iff for every \(a \in S\), \(l \in {\mathcal {L}}\) is called a correlation of \({\mathfrak {M}}\). A substructure of \({\mathfrak {M}}\), whose points are all the points of \({\mathfrak {M}}\) collinear with \(a\in S\), and lines are all the lines of \({\mathfrak {M}}\) which contain at least two points collinear with a is said to be the neighborhood of a point a and it is denoted by \({\mathfrak {M}}_{({a})}\). Clearly, if \(\varphi =(\varphi ',\varphi '')\) is an automorphism of \({\mathfrak {M}}\), then \(\varphi \) maps \({\mathfrak {M}}_{({a})}\) onto \({\mathfrak {M}}_{({\varphi '(a)})}\). \({\mathfrak {M}}\) 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 \({\mathfrak {M}}\) 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 \({{\textsf {G}}} = {\left\langle G,\cdot ,1 \right\rangle }\) be an arbitrary group and \(D\subset G\). Let us introduce a point-line geometry

$$\begin{aligned} \mathbf{D}({{\textsf {G}}},{D}) = {\left\langle G,G/D \right\rangle }. \end{aligned}$$
(2.1)

Every translation \(\tau _a:G\ni x\mapsto a\cdot x\in G\) is an automorphism of \(\mathbf{D}({{\textsf {G}}},{D})\). This yields that without loss of generality we can assume that \(1 \in D\). Let \(G_D\) be the stabilizer of D in G. Then, the number of points in \(\mathbf{D}({{\textsf {G}}},{D})\) is |G|, the number of lines is \(\frac{|G|}{|G_D|}\), the size of every line is |D|, and the rank of every point is \(\frac{|D|}{|G_D|}\).

It was proved in [10] that \(\mathbf{D}({{\textsf {G}}},{D})\) is a configuration iff for every \(c\in G\), \(c\ne 1\) there is at most one pair \((a,b)\in D\times D\) with \(a{b}^{{-1}}=c\). Any \(D\subseteq G\) satisfying this condition is called a quasi difference set, for short QDS. In [10] we were mainly interested in the structures of the form \(\mathbf{D}({C_k\oplus C_n},{ D})\), 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 \(a\in G\), and [a] denotes “coordinates” of the line \(a\cdot D\in G/D\). Using this we get

(2.2)

Note that if D is QDS then \(G_D=\{1\}\). Hence, a is uniquely determined by [a], or in other words \(a\cdot D=b\cdot D\) holds only for \(a=b\).

Generalities

Now, we are going to present some general properties of \({\mathfrak {D}} =\mathbf{D}({{\textsf {G}}},{D})\). Every automorphism \(\varphi =(\varphi ',\varphi '')\) of \({\mathfrak {D}}\) uniquely corresponds to a pair \(f=(f',f'')\) of bijections of G determined by

$$\begin{aligned} \varphi '((a))= (f'(a)), \; \varphi ''([b])= [f''(b)]. \end{aligned}$$

We shall frequently refer to the pair f as to an automorphism of \({\mathfrak {D}}\). On the other hand, some of the automorphisms of \({\mathfrak {D}}\) are determined by automorphisms of the underlying group \({\textsf {G}}\), namely

Fact 3.1

Let \(f\in {\mathrm{Aut}}({{\textsf {G}}})\). The map f determines an automorphism \(\varphi =(\varphi ',\varphi '')\) of \(\mathbf{D}({{\textsf {G}}},{D})\) iff \(f(D)=q\cdot D\) for some \(q\in G\). Then \(\varphi '((a))=(f(a))\) and \(\varphi ''([a])=[f(a)\cdot q]\) for every \(a \in G\) .

Proposition 3.2

Let D be QDS in an abelian group \({\textsf {G}}\). Then the map \(\varkappa \) defined by

$$\begin{aligned} \varkappa ((a))=[{a}^{{-1}}], \;\; \varkappa ([a])=({a}^{{-1}}) \end{aligned}$$
(3.1)

is an involutive correlation of the structure \({{\mathfrak {D}}} = \mathbf{D}({{\textsf {G}}},{D})\). Consequently, \({\mathfrak {D}}\) is self-dual. A point a of \({\mathfrak {D}}\) is selfconjugate under \(\varkappa \) iff \(a^2 \in D\).

Proof

Clearly, \(\varkappa \) is involutory. Let \(a,b\in G\). Then means that \({b}^{{-1}}\cdot a \in D\). This is equivalent to \({({a}^{{-1}})}^{{-1}}\cdot {b}^{{-1}}\in D\), i.e. . Thus \(\varkappa \) is a correlation. Finally, assume that . From (2.2) we obtain \(a^2\in D\). \(\square \)

The correlation defined by (3.1) will be referred to as the standard correlation of \(\mathbf{D}({{\textsf {G}}},{D})\). Immediate from (2.2) is the following

Lemma 3.3

Let \(D\subseteq G\), \(a,b\in G\), and \({{\mathfrak {D}}}=\mathbf{D}({{\textsf {G}}},{D})\).

  1. (i)

    The set of the lines of \({\mathfrak {D}}\) through the point (a) can be identified with \(a\cdot {D}^{{-1}}\), i.e.

    $$\begin{aligned} {(a)}^{*} = \{ [a\cdot {d}^{{-1}}]:d\in D \}. \end{aligned}$$
    (3.2)
  2. (ii)

    The points (a) and (b) are collinear in \({\mathfrak {D}}\) iff \({a}^{{-1}}\cdot b \in {D}^{{-1}} D\). If \({a}^{{-1}} b = {d_1}^{{-1}}d_2\) with \(d_1,d_2\in D\) then

    (3.3)
  3. (iii)

    The lines [a] and [b] of \({\mathfrak {D}}\) have a common point iff \({a}^{{-1}}\cdot b \in D{D}^{{-1}}\). If \({a}^{{-1}}\cdot b = d_1\cdot {d_2}^{{-1}}\) with \(d_1,d_2\in D\) then

    (3.4)

As a straightforward consequence of Lemma 3.3 we get

Proposition 3.4

Let \({{\textsf {G}}}={\left\langle G,\cdot ,1 \right\rangle }\) be a group, D be QDS in \({\textsf {G}}\) with \(1\in D\), and \(a_1,a_2\in G\). Points \((a_1)\) and \((a_2)\) can be joined with a polygonal path in \(\mathbf{D}({{\textsf {G}}},{D})\) iff there is a finite sequence \(q_1,\ldots ,q_s\) of elements of \({D}^{{-1}} D\) such that \(a_1=q_1\cdot \cdots \cdot q_s\cdot a_2\). Consequently, the connected component of the point (1) is isomorphic to \(\mathbf{D}({{\left\langle D \right\rangle }_{{\textsf {G}}}},{D})\), where \({\left\langle D \right\rangle }_{{\textsf {G}}}\) is the subgroup of \({\textsf {G}}\) generated by D. Every two connected components of any two points are isomorphic.

Corollary 3.5

\(\mathbf{D}({{\textsf {G}}},{D})\) is connected iff D generates the whole group \({\textsf {G}}\).

For \(D\subseteq G\) we introduce the following condition:

$$\begin{aligned} \begin{aligned}&\hbox {if } d_1,d_2,d_3,d_4\in D \hbox { and } d_1{d_2}^{{-1}}d_3{d_4}^{{-1}}\in D{D}^{{-1}} \hbox { then } \\&\quad d_1{d_2}^{{-1}}=1 \hbox { or } d_3{d_4}^{{-1}}=1 \hbox { or } d_1{d_4}^{{-1}}=1, \hbox { or } d_3{d_2}^{{-1}}=1. \end{aligned} \end{aligned}$$
(3.5)

The next Lemma explains the meaning of the condition (3.5).

Lemma 3.6

Assume that \({\textsf {G}}\) is an abelian group, and \(D\subset G\) is QDS satisfying (3.5). Let \(a\in G\) be a point of \(\mathbf{D}({{\textsf {G}}},{D})\), \(d_1,d_2\in D\), and \(b_1=[a{d_1}^{{-1}}]\), \(b_2=[a{d_2}^{{-1}}]\) be two distinct lines through a. Set \(i=1,2\).

  1. (i)

    If \(d'_i \in D\) then \(p_i=(a{d_i}^{{-1}}d'_i)\) is a point on \(b_i\), and \(p_1\sim p_2\) iff \(d'_1=d'_2\).

  2. (ii)

    For every point \(p_d=(a{d_1}^{{-1}}d)\) on \(b_1\) with \(d \in D\), \(d\ne d_2,d_1\) there is the unique point \(q=(a{d_2}^{{-1}}d)\) on \(b_2\) which completes points \(a,p_d\) to a triangle. The point \(p_{d_2}\) cannot be completed to a triangle this way.

  3. (iii)

    If c is a line of \(\mathbf{D}({{\textsf {G}}},{D})\) which crosses both of \(b_1\), \(b_2\) and misses a then \(c=[a{d_1}^{{-1}}{d_2}^{{-1}}d]\) for some \(d\in D\) with \(d\ne d_1,d_2\).

  4. (iv)

    For any two distinct lines \(c_1,c_2\) crossing both of \(b_1\), \(b_2\) and missing a, i.e. \(c_i=[a{d_1}^{{-1}}{d_2}^{{-1}}d'_i]\) for \(d'_i\in D\), we have

Proof

Simple analysis based on Lemma 3.3 and (3.5). \(\square \)

Now we describe automorphisms of \(\mathbf{D}({{\textsf {G}}},{D})\) with a quasi difference set D satisfying (3.5). Let us begin with some rigidity properties.

Lemma 3.7

Let \({{\mathfrak {D}}}=\mathbf{D}({{\textsf {G}}},{D})\), where D is QDS that satisfies (3.5) and \(p\in G\). Let f be a collineation of \({\mathfrak {D}}\) 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 \({\mathfrak {D}}\) has a transitive group of automorphisms, we can assume that \(p=(1)\). Let \(f=(f',f'')\) be a collineation of \({\mathfrak {D}}\) satisfying (i) for a line \(l_1=[{d_1}^{{-1}}]\) with \(d_1\in D\). Take \(d_2\in D\), \(d_2\ne d_1\) and a line \(l_2 = [{d_2}^{{-1}}]\). Then the points \(({d_1}^{{-1}}d_2)\) on \(l_1\) and \(({d_2}^{{-1}}d_1)\) on \(l_2\) are the unique points “between” \(l_1\) and \(l_2\) that are not collinear (cf. Lemma 3.6(ii)). We have \(f'({d_1}^{{-1}}d_2)=({d_1}^{{-1}}d_2)\) and . The only point in \({{\mathfrak {D}}}_{({p})}\) non-collinear with \(({d_1}^{{-1}}d_2)\) lies on \([{d_2}^{{-1}}]\); therefore, \(f''({d_2}^{{-1}})={d_2}^{{-1}}\). Thus f preserves every line through p.

The case with the assumption (ii) can by proved in a similar way. \(\square \)

From Lemma 3.7 we get

Corollary 3.8

Let f be a collineation which fixes a line l of \({\mathfrak {D}}\) point-wise. Under assumption (3.5) f fixes all points on every line which crosses l. Consequently, if \({\mathfrak {D}}\) is connected then \(f = {\mathrm {id}}\).

Corollary 3.9

Under assumption (3.5) every automorphism of \({\mathfrak {D}}\) which has a fixed point p is uniquely determined by its action on the lines through p. Consequently, the point stabilizer \({{\mathrm{Aut}}({{\mathfrak {D}}})}_{({p})}\) of the automorphism group of \({\mathfrak {D}}\) is isomorphic to a subgroup of the permutation group \(S_{|D|}\).

Products of Difference Sets

Let \({{\textsf {G}}}_i = {\left\langle G_i,\cdot _i,1_i \right\rangle }\) be a group for \(i\in I\), and \(1_i\in D_i\subset G_i\) for every \(i\in I\).

Let \(G=\prod _{i\in I} G_i\), i.e. let G be the set of all functions \(g:I\longrightarrow \bigcup \{ G_i:i\in I \}\) with \(g(i)\in G_i\). Then the product \(\prod _{i\in I}{{\textsf {G}}}_i\) is the structure \({\left\langle G,\cdot ,1 \right\rangle }\), where \((g_1\cdot g_2)(i)=g_1(i)\cdot _i g_2(i)\) for \(g_1,g_2\in G\), and \(1(i)=1_i\). It is just the standard construction of the direct product of groups. The set

$$\begin{aligned} \textstyle {\sum _{i\in I}G_i} = \{g\in G:g(i)\ne 1_i \text { for a finite number of }i\in I \} \end{aligned}$$

is a subgroup of \(\prod _{i\in I}{{\textsf {G}}}_i\), denoted by \(\sum _{i\in I}{{\textsf {G}}}_i\). If \(I=\{1,\ldots ,r\}\) is finite, then \({{\textsf {G}}}_1\oplus \cdots \oplus {{\textsf {G}}}_{r}:= \prod _{i\in I}{{\textsf {G}}}_i = \sum _{i\in I}{{\textsf {G}}}_i =: \sum _{i=1}^{r}{{\textsf {G}}}_i\).

For every \(j\in I\) we define the standard projection \(\pi _j:\prod _{i\in I}G_i \longrightarrow G_j\) by \(\pi _j(g)=g(j)\), and the standard inclusion \(\varepsilon _j:G_j\longrightarrow \sum _{i\in I}G_i\) by the conditions \((\varepsilon _j(a))(j)=a\) and \((\varepsilon _j(a))(i)=1_i\) for \(i\ne j\) and \(a\in G_j\). Recall that \(\pi _j\) and \(\varepsilon _j\) are group homomorphisms. We set \(\sum _{i\in I}D_i=\bigcup \{ \varepsilon _i(D_i):i\in I \}\). For a finite set \(I=\{ 1,\ldots r \}\) we write \(\sum _{i\in I}D_i = \sum _{i=1}^{r}D_i = D_1\uplus \cdots \uplus D_{r}\).

Proposition 4.1

If \(D_i\) is QDS in \({{\textsf {G}}}_i\) for every \(i\in I\), then \(\sum _{i\in I}D_i\) is QDS in \(\sum _{i\in I}{{\textsf {G}}}_i\).

Proof

We set \(D = \sum _{i\in I}D_i\). Let \(g_1,g_2,g_3,g_4\in D\) and assume that \(g_1{g_2}^{{-1}} = g_3{g_4}^{{-1}}\). Let \(g_i\in \varepsilon _{j_i}(D_{j_i})\). If \(j_1=j_2\); then \(\pi _j(g_1{g_2}^{{-1}}) = 1_j\) for every \(j\ne j_1\); thus \(\pi _j(g_3{g_4}^{{-1}})=1_j\), and thus \(j_3 = j_4=j_1\). From assumption we infer that \(g_1 = g_3\) and \(g_2 = g_4\). If \(j_1\ne j_2\); 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. \(\square \)

Let \({{\mathfrak {D}}}_i = \mathbf{D}({{{\textsf {G}}}_i},{D_i})\). We write

$$\begin{aligned} \textstyle {\sum _{i\in I}{{\mathfrak {D}}}_i}:= \mathbf{D}({\textstyle {\sum _{i\in I}{{\textsf {G}}}_i}},{\textstyle {\sum _{i\in I}D_i}}). \end{aligned}$$

For \(I = \{1,\ldots ,r\}\) we write also \(\sum _{i=1}^{r}{{\mathfrak {D}}}_i := \mathbf{D}({{{\textsf {G}}}_1},{D_1})\oplus \cdots \oplus \mathbf{D}({{{\textsf {G}}}_r},{D_r})\). Note that, for \(a,b\in \sum _{i\in I}{{\textsf {G}}}_i\), \(a\in b\cdot \sum _{i\in I} D_i\) iff \(a=b\cdot d\) for some \(d\in \bigcup \{ \varepsilon _i(D_i):i\in I \}\), i.e. iff there is \(i\in I\) such that \(a_i\in b_i\cdot D_i\) and \(a_j=b_j\) for all \(j\ne i\). So, we get

(4.1)

Let us consider two particular cases. First, let \({\mathfrak {D}}=\mathbf{D}({C_k},{\{0,1\}})\oplus \mathbf{D}({{{\textsf {G}}}'},{D'})\) and \(D=\{0,1\}\uplus D'\). Let \(a = (i,a')\in C_k\times G'\). Then the points of \(a\cdot D\) are points of the form \((i,a'\cdot p)\) with \(p\in D'\), and—one point— \((i+1,a')\). Somewhat informally we can say that the line with the coordinates \([i,a']\) consists of the points (ip), where and one “extra” point \((i+1,a')\). In other words, we have a function \(f_i\) which assigns a point of \({{\mathfrak {D}}}'= \mathbf{D}({{{\textsf {G}}}'},{D'})\) to every line of \({{\mathfrak {D}}}'\) such that the lines of \({{\mathfrak {D}}}\) are of the form \({i}\times l' \cup \{(i+1,f_i(l))\}\), where \(l'\) is a line of \({{\mathfrak {D}}}'\). Thus \({\mathfrak {D}}\) is a configuration consisting of k copies of \({{\mathfrak {D}}}'\) cyclically inscribed one into another. In the above construction, the function \(f_i\) is defined by \(f_i([a])=(a)\).

Now, let \({\mathfrak {D}} =\mathbf{D}({{{\textsf {G}}}_1},{D_1})\oplus \mathbf{D}({{{\textsf {G}}}_2},{D_2})\). The lines of \({\mathfrak {D}}\) are of the form \((a_1,a_2)+D_1\uplus D_2\), which, on the other hand, can be written as \({[a_1]}^{*}\times \{(a_2)\}\cup \{(a_1)\}\times {[a_2]}^{*}\). Recall, that the lines of the Segre product \({{\mathfrak {D}}}^{*}=\mathbf{D}({{{\textsf {G}}}_1},{D_1})\otimes \mathbf{D}({{{\textsf {G}}}_2},{D_2})\) (cf. [9]) are the sets of one of two forms: \({[a_1]}^{*}\times \{(a_2)\}\) or \(\{(a_1)\}\times {[a_2]}^{*}\). Therefore, the lines of \({\mathfrak {D}}\) are unions of some pairs of the lines of \({{\mathfrak {D}}}^{*}\).

From Proposition 3.4 we have the following:

Fact 4.2

Let \(D_i\) be QDS in a group \({{\textsf {G}}}_i\) such that \({\left\langle D_i \right\rangle }_{{{\textsf {G}}}_i} = G_i\) for all \(i\in I\). Then \(\sum _{i\in I}D_i\) generates \(\sum _{i\in I}{{\textsf {G}}}_i\). Consequently, if every one of the structures \({{\mathfrak {D}}}_i=\mathbf{D}({{{\textsf {G}}}_i},{D_i})\) is connected then \(\sum _{i\in I}{{\mathfrak {D}}}_i\) is connected as well.

Let \(J \subset I\); we extend the inclusions \(\varepsilon _i\) to the map \(\varepsilon _J :\sum _{j \in J} G_j \longrightarrow \sum _{i \in I} G_i\) by the condition

$$\begin{aligned} (\varepsilon _J(a))(i) = \left\{ \begin{array}{ll} 1_i &{}\quad \text {for } i \in I \setminus J \\ a(i) &{}\quad \text {for } i \in J \end{array} \right. \text { for arbitrary } a \in \textstyle {\sum _{j \in J} G_j}. \end{aligned}$$
(4.2)

Proposition 4.3

Let J be a nonempty proper subset of I. Then

$$\begin{aligned} \textstyle {\sum _{i \in I} {{\mathfrak {D}}}_i}\cong & {} \textstyle {\sum _{i \in J} {{\mathfrak {D}}}_i} \oplus \textstyle {\sum _{i \in I \setminus J} {{\mathfrak {D}}}_i}. \end{aligned}$$

Proposition 4.4

Let \({{\mathfrak {D}}}_i = \mathbf{D}({{{\textsf {G}}}_i},{D_i})\), where \(D_i\) is QDS in a group \({{\textsf {G}}}_i\) for all \(i\in I\). Assume that there is a pair of bijections \(\varphi '_i,\varphi _i'':G_i\longrightarrow G_i\) such that the pair \(\varkappa _i=(\varkappa '_i,\varkappa ''_i)\) of maps

$$\begin{aligned} \varkappa '_i:(a)\mapsto [\varphi '_i(a)], \;\; \varkappa ''_i:[a]\mapsto (\varphi ''_i(a))\text { for }a\in G_i \end{aligned}$$
(4.3)

is a correlation of \({{\mathfrak {D}}}_i\) for \(i\in I\). Set \(\varphi '=\varphi '_1\times \cdots \times \varphi '_r \) and \(\varphi ''=\varphi ''_1\times \cdots \times \varphi ''_r \). If \(\varphi '_i = \varphi ''_i\) for every \(i\in I\) (i.e. if \(\varkappa _i\) are involutory), then the pair \(\varkappa = (\varkappa ',\varkappa '')\) of maps

$$\begin{aligned} \varkappa ':(a)\mapsto [\varphi '(a)],\;\; \varkappa '':[a]\mapsto (\varphi ''(a))\text { for }a\in \textstyle {\sum _{i \in I} G_i} \end{aligned}$$
(4.4)

is an involutory correlation of \(\sum _{i\in I}{{\mathfrak {D}}}_i\).

Proof

By (4.1) we get iff the following holds: and \(\varphi ''_j(b_j)=\varphi '_j(a_j)\) for \(j\ne i\). Now the claim is evident. \(\square \)

If we assume in Proposition 4.4 that every \(\varkappa _i\) is the standard correlation (\(\varphi '_i(a) = {a}^{{-1}}\), cf. Proposition 3.2), then \(\varkappa \) is also the standard correlation. Using (4.1) as in Proposition 4.4 one can also prove the following:

Proposition 4.5

Let \({{\mathfrak {D}}}_i=\mathbf{D}({{{\textsf {G}}}_i},{D_i})\) and \(f_i=(f'_i,f''_i)\) be bijections of \(G_i\) such that \(f'_i:(a)\longmapsto (f'_i(a))\) and \(f''_i:[a]\longmapsto [f''_i(a)]\) yields a collineation of \({{\mathfrak {D}}}_i\) for \(i\in I\), and let \({{\mathfrak {D}}}=\sum _{i\in I}{{\mathfrak {D}}}_i\). We set \(F'=\prod _{i\in I}f'_i\), \(F''=\prod _{i\in I}f''_i\), and \(F=(F',F'')\). Then the pair F is a collineation of \({\mathfrak {D}}\) iff \(f'_i=f''_i\) for every \(i\in I\).

Obviously, the pair \((f'_i,f''_i)\), where \(f'_i=f''_i={\tau }_{a_i}\) and \(a_i \in G_i\), is a collineation of \({{\mathfrak {D}}}_i\); therefore, the pair \((\prod _{i \in I} f'_i , \prod _{i \in I} f''_i)\) is an automorphism of \({\mathfrak {D}}\). But this is a rather trivial result, as \(\prod _{i \in I} \tau _{a_i} = \tau _{a}\). We have also some automorphisms of another type.

Proposition 4.6

Let \(\beta \in S_n\), \(x\in G^n\) and \(h :G^n \longrightarrow G^n\) be the map defined by \(h((x_1,\ldots ,x_n))=(x_{\beta (1)},\ldots ,x_{\beta (n)})\). The pair \(F=(F',F'')\) with \(F'((x))=(h(x))\), \(F''([x])=[h(x)]\) is a collineation of

$$\begin{aligned} \underbrace{\mathbf{D}({{\textsf {G}}},{D})\oplus \mathbf{D}({{\textsf {G}}},{D})\oplus \cdots \mathbf{D}({{\textsf {G}}},{D})}_{n \; \mathrm {times}} = \mathbf{D}({{{\textsf {G}}}^n},{D^n}). \end{aligned}$$

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 \(\mathbf{D}({{\textsf {G}}},{\mathcal{D}_r})\), where \(r\ge 2\), \({{\textsf {G}}} = C_{n_1}\oplus \cdots \oplus C_{n_{r}}\) and \(\mathcal{D}_r = \{ e_0,e_1,\ldots ,e_r \}\) is QDS with \(e_0=(0,\ldots ,0)\) and \((e_i)_j=0\) for \(i,j=1,\ldots ,r\), \(i\ne j\), \((e_i)_i=1\). The set \(\mathcal{D}_r\) will be called canonical QDS. Note that \(\mathbf{D}({{\textsf {G}}},{\mathcal{D}_r}) \cong \sum _{i=1}^r\mathbf{D}({C_{n_i}},{\{ 0,1 \}})\) and in view of Proposition 4.3

$$\begin{aligned} \mathbf{D}({{\textsf {G}}},{\mathcal{D}_r}) \cong \mathbf{D}({C_{n_1}},{\{ 0,1 \}})\oplus \mathbf{D}({C_{n_2}\oplus \cdots \oplus C_{n_r}},{\mathcal{D}_{r-1}}). \end{aligned}$$

Thus \(\mathbf{D}({{\textsf {G}}},{\mathcal{D}_r})\) generalizes the construction of cyclically inscribed polygons considered in [10]. An interesting example of this type is \(\mathbf{D}({C_3\oplus C_3\oplus C_3},{\mathcal{D}_3})\)—three copies of Pappus configuration cyclically inscribed, see Fig. 1 and a more general case considered in Sect. 6.2.

Fig. 1
figure1

Multiplied Pappus configuration

Lemma 4.7

Let \({\mathfrak {M}}= \mathbf{D}({(C_k)^n},{ \mathcal{D}_n})\), \(\alpha \) be a permutation of the set \(\{0,\dots ,n\}\) and \(i=0,\dots ,n\).

  1. (i)

    A permutation \(\alpha \) induces a collineation \(f=(f',f'')\) of the structure \({\mathfrak {M}}\) such that \(f'(e_0)=e_0\) and \(f''(-e_i)=-e_{\alpha (i)}\) for all i.

  2. (ii)

    If \(f=(f',f'')\) is a collineation induced by \(\alpha \) such that \(f'(e_0)=e_0\), \(f''(-e_i)=-e_{\alpha (i)}\) for all i, then \(f' \tau _v (f')^{-1}= \tau _{f'(v)}\).

Proof

  1. (i)

    Assume that \(\alpha (0)=0\) and consider a map \(f': G \longrightarrow G\) defined by

    $$\begin{aligned} f'(x_1,\dots ,x_n)=(x_{\alpha (1)},\dots ,x_{\alpha (n)}). \end{aligned}$$
    (4.5)

    Then \(f'\in {\mathrm{Aut}}({{\textsf {G}}})\) and \(f'(\mathcal{D}_n)=\mathcal{D}_n\), and thus \(f'\) determines an automorphism of \({\mathfrak {M}}\). It is seen that \(f'(e_0)=e_0\). In view of Fact 3.1 we get \(f''(y_1,\dots ,y_n)=(y_{\alpha (1)},\dots ,y_{\alpha (n)})\), so \(f''(-e_i)=-e_{\alpha (i)}\).

    Let \(\alpha (0)=s\ne 0\) be a transposition and a map \(f': G \longrightarrow G\) be given by:

    $$\begin{aligned} f'(x_1,\dots ,x_n)=\left( x_1,\dots ,x_{s-1},-\sum _{i=1}^n x_i,x_{s+1},\dots ,x_n\right) . \end{aligned}$$
    (4.6)

    Again \(f'\in {\mathrm{Aut}}({{\textsf {G}}})\), and note that \(f'(\mathcal{D}_n)=-e_s+\mathcal{D}_n\), \(f'(e_0)=e_0\). From Fact 3.1\(f'\) induces a collineation of \({\mathfrak {M}}\) with \(f''(y_1,\dots ,y_n)=(y_1,\dots ,y_{s-1},-\sum _{i=1}^n y_i-1,y_{s+1},\dots ,y_n)\). It is easy to check that \(f''(-e_i) = -e_{\alpha (i)}\). As every permutation is one of two permutations considered above, we get our claim.

  2. (ii)

    Let \(\alpha \) be a permutation such that \(\alpha (0)=0\), then \(f'\) is given by (4.5) and \((f')^{-1}(x_1,\dots ,x_n)=(x_{\alpha ^{-1}(1)},\dots ,x_{\alpha ^{-1}(n)})\). Therefore \(f' \tau _v (f')^{-1}(x_1,\dots ,x_n)= (x_1,\dots ,x_n) + f'(v)= \tau _{f'(v)}(x_1,\dots ,x_n)\). If \(\alpha \) is a transposition. then \(f'\) is given by (4.6), and thus \(f'=(f')^{-1}\). After simple calculation we get the claim. \(\square \)

Lemma 4.8

Let \({\mathfrak {M}} = \mathbf{D}({(C_3)^n},{ \mathcal{D}_n})\). For every point o of \({\mathfrak {M}}\), any two distinct lines kl through o and every point p with there is the unique point q such that and \(p \sim q\).

Proof

As translations are transitive subgroup of \({\mathrm{Aut}}({\mathfrak {M}})\) assume that \(o=e_0\). Then, by Lemma 3.3, \(k=[-e_i]\), \(l=[-e_k]\) and \(p=(-e_k+e_j)\) for some \(i,j,k=0,\ldots ,n\), \(i\ne k\ne j\). Assume that q is on \([-e_k]\) and q is collinear with p. Using Lemma 3.3 we get

$$\begin{aligned} q = \left\{ \begin{array}{ll} (-e_i + e_j) &{}\quad \text { if }\; i \ne j, \\ (-e_i + e_k) &{}\quad \text { if }\; i = j. \end{array} \right. \end{aligned}$$
(4.7)

\(\square \)

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

Fact 4.9

The canonical QDS in \((C_k)^n\) satisfies (3.5) iff \(k>3\).

Theorem 4.10

Let \({\mathfrak {M}}= \mathbf{D}({(C_k)^n},{ \mathcal{D}_n})\) and \(k > 3\). The group \({\mathrm{Aut}}({\mathfrak {M}})\) is isomorphic to \(S_{n+1} \ltimes (C_k)^n\).

Proof

By Corollary 3.9 every collineation \(f=(f',f'')\) fixing \(e_0\) is uniquely determined by a permutation \(\alpha \) of lines through \(e_0\), i.e. \(f''(-e_i)=-e_{\alpha (i)}\). From Lemma 4.7(i) every permutation \(\alpha \) induces a collineation. So, there is an isomorphism \(\xi :f\mapsto \alpha \), and consequently \({{\mathrm{Aut}}({\mathfrak {M}})}_{({e_0})}\cong S_{n+1}\). Note that every automorphism of \({\mathfrak {M}}\) is a composition of two maps: a translation and a collineation fixing \(e_0\). Using Lemma 4.7(ii) we get \((\tau _u f_{\alpha })(\tau _v f_{\beta })= \tau _u \tau _{f_{\alpha }(v)} f_{\alpha \beta }\), which closes the proof. \(\square \)

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

$$\begin{aligned} {\mathfrak {M}} = \mathbf{D}({C_k},{\{ 0,1 \}})\oplus \mathbf{D}({{\textsf {G}}},{D}) = \mathbf{D}({C_k\oplus {{\textsf {G}}}},{\{ 0,1 \} \uplus D}), \end{aligned}$$

where D is QDS in an abelian group \({\textsf {G}}\). Directly from definitions we calculate the following:

Lemma 4.11

Let \(D = \{ d_0,\ldots ,d_n \}\) be QDS in an abelian group \({{\textsf {G}}} = {\left\langle G,+,\theta \right\rangle }\), and let \(q = (0,\theta ) \in C_k \times G\). Set \({\mathfrak {M}} = \mathbf{D}({C_k},{\{ 0,1 \}})\oplus \mathbf{D}({{\textsf {G}}},{D})\). The lines of \({\mathfrak {M}}\) through q are the following:

  1. (1)

    \(l_i = [0,- d_i]\) for \(i = 0,\ldots ,n\).

Each line \([0,-d_i]\) contains q and the following points:

  1. (a)

    \(q_{i,j} = (0,-d_i + d_j)\) for \(j=0,\ldots ,i\), \(i\ne j\);

  2. (b)

    \(p'_i = (1,-d_i)\).

  3. (2)

    \(l'' = [-1,\theta ] = [k-1,\theta ]\).

Its points are q and the following:

  1. (c)

    \(p''_i = (-1,d_i)\) for \(i=0,\ldots ,n\).

Then the points \(q_{i,j}\) form a substructure isomorphic under the map \((0,a) \longmapsto (a)\) to the neighborhood of \(\theta \) in \(\mathbf{D}({{\textsf {G}}},{D})\). Moreover, the following additional lines appear:

  1. (3)

    For every \(i,j=0,\ldots ,n\), \(i\ne j\) the line \(l''_{i,j} = [-1,-d_j + d_i]\) joins with .

  2. (4)

    For every ij as above, the line \(l'_{i,j} = l'_{j,i} = [1,-(d_i + d_j)]\) joins with .

The lines listed above are pairwise distinct.

  1. (i)

    If \(k > 3\), then no other line appears (Fig. 2).

  2. (ii)

    Let \(k = 3\). Then \(-1 = 2\) holds in \(C_k\), and then another connections are associated with triples \((d_i,d_j,d_r)\in D\) satisfying

    $$\begin{aligned} d_i + d_j + d_r = \theta . \end{aligned}$$
    (4.8)

Namely, let (4.8) be satisfied. Evidently, \(-(d_i+d_j) = d_r\).

  1. (5)

    The line \(l'_{i,j} = [1,-(d_j+d_i)] = [1,d_r]\) passes through and .

  2. (6)

    If, moreover, \(j\ne i\), then the above line passes through \(p'_i\) as well so, it coincides with the line defined in (3.2) (Fig. 3).

Fig. 2
figure2

The neighborhood of the point \((0,\theta )\) in a configuration \(\mathbf{D}({C_k},{\{ 0,1 \}})\oplus \mathbf{D}({{\textsf {G}}},{D})\) for \(k>3\)

Recall from [10] that a collineation of \(\mathbf{D}({C_k},{\{0,1\}})\) is simply an element of the dihedral group \(D_k\), i.e. it is any map \(\alpha _{\varepsilon ,q} :i\mapsto \varepsilon i + q\), where \(\varepsilon \in \{1,-1\}\). Proposition 4.5 determines all the automorphisms of \(\mathbf{D}({C_k\oplus {{\textsf {G}}}},{\{ 0,1 \} \uplus D})\) of the form \((i,a)\mapsto (\alpha _{\varepsilon ,q}(i),f'(a))\). Still, in this case we should look for automorphisms defined with more complicated formulas.

Fig. 3
figure3

The neighborhood of the point \((0,\theta )\) in a configuration \(\mathbf{D}({C_3},{\{ 0,1 \}})\oplus \mathbf{D}({{\textsf {G}}},{D})\)

Proposition 4.12

Let \({\mathfrak {M}}_0=\mathbf{D}({{\textsf {G}}},{D})\), \({\mathfrak {M}}=\mathbf{D}({C_k\oplus {{\textsf {G}}}},{\{ 0,1 \} \uplus D})\) and \(f=(f',f'')\in {\mathrm{Aut}}({\mathfrak {M}}_0)\). The following conditions are equivalent:

  1. (i)

    There is a collineation \(\varphi =(\varphi ',\varphi '')\) of \({\mathfrak {M}}\) such that \(\varphi '((0,a))=(0,f'(a))\) and \(\varphi ''([0,b])=[0,f''(b)]\).

  2. (ii)

    There is a sequence \(f_i\), \(i=0,\ldots ,k\), of collineations of \({\mathfrak {M}}_0\) defined recursively by the formulas: \(f_0=f\) and \(f'_{i+1}=f''_{i}\), where \(f_i=(f'_i,f''_i)\).

In the case (ii) we have \(\varphi '((i,a))=(i,f'_i(a))\) and \(\varphi ''([i,b])=[i,f''_i(b)]\).

Proof

It suffices to note that if (i) holds, then for every \(a\in G\), which gives , and thus \(\varphi '((1,a))=(1,f''(a))\). Therefore, \(f''\) (as a transformation of points) induces a collineation of \({\mathfrak {M}}_0\). \(\square \)

Elementary Properties

Now we discuss some elementary axiomatic properties of \(\mathbf{D}({{\textsf {G}}},{D})\). Let D be QDS in an abelian group \({{\textsf {G}}}\). Using Lemma 3.6 we prove the following:

Proposition 5.1

Under assumption (3.5) the structure \(\mathbf{D}({{\textsf {G}}},{D})\) is Veblenian.

Proposition 5.2

Under assumption (3.5) the structure \(\mathbf{D}({{\textsf {G}}},{D})\) is Desarguesian.

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

Corollary 5.3

For \(k > 3\) the structure \(\mathbf{D}({(C_k)^n},{ \mathcal{D}_n})\) is Veblenian and Desarguesian.

Proposition 5.4

The structure \({\mathfrak {M}} = \mathbf{D}({(C_3)^n},{ \mathcal{D}_n})\) is not Veblenian.

Proof

Take \(l_i=[-e_i]\) for \(i=1,2\), and \(k_1=[e_1+e_2]\), \(k_2=[-e_1-e_2+e_3]\). Then , and \(k_1\) crosses \(l_1\) in \((-e_1+e_2)\) and \(l_2\) in \((-e_2+e_1)\). Furthermore, the lines \(k_2\), \(l_1\) meet in \((-e_1+e_3)\), and \((-e_2+e_3)\) is the common point of \(l_2\), \(k_2\). Suppose that \(k_1\cap k_2\ne \emptyset \). Then \((e_1+e_2)+e_t = (-e_1-e_2+e_3)+e_s\) for some \(s,t=1,\ldots ,n\), \(s\ne t\), which implies a contradiction: \(e_1+e_2+e_3+e_s = e_t\). \(\square \)

Proposition 5.5

The structure \({\mathfrak {M}} = \mathbf{D}({(C_3)^n},{ \mathcal{D}_n})\) is Desarguesian.

Proof

Without loss of generality we can assume that \((e_0)\) is the perspective center of two triangles \(T_1\), \(T_2\) inscribed into three lines \(l_1,l_2,l_3\) of \({\mathfrak {M}}\) such that the corresponding pairs of their sides intersect each other. We do some calculations based on Lemma 3.3. Note that \(l_r = [-e_{i_r}]\) for \(r=1,2,3\), \(e_{i_r}\in \mathcal{D}_n\). By Lemma 4.8 and (4.7) the sides of our triangles are lines of the form \([-e_i-e_j+e_s]\) or \([e_i+e_j]\) for some \(e_i,e_j,e_s\in \mathcal{D}_n\), \(s\ne i,j\). The line \([e_{i_1}+e_{i_2}]\) does not meet any other line which crosses both \(l_1\) and \(l_2\). Thus the sides are lines of the type \([-e_i-e_j+e_s]\). Consequently, the vertices of \(T_1\) are \((-e_{i_1} + e_s)\), \((-e_{i_2} + e_s)\), \((-e_{i_1} + e_s)\), and the vertices of \(T_2\) are \((-e_{i_1} + e_t)\), \((-e_{i_2} + e_t)\), \((-e_{i_1} + e_t)\) for \(e_{i_r}\ne e_s,e_t\in \mathcal{D}_n\). Then the points of intersection of the corresponding sides of \(T_1\) and \(T_2\) are \(c_1 = (-e_{i_1}-e_{i_2}+e_s+e_t)\), \(c_2 = (-e_{i_2}-e_{i_3}+e_s+e_t)\), and \(c_3 = (-e_{i_3}-e_{i_1}+e_s+e_t)\). All these points are on the line \([-e_{i_1}-e_{i_2}-e_{i_3}+e_s+e_t]\), which proves the claim. \(\square \)

The Pappus configuration can be considered as \(\mathbf{D}({C_3\oplus C_3},{\mathcal{D}_2})\), cf. [10, 15] (see Fig. 4).

Proposition 5.6

Under assumption (3.5) the structure \(\mathbf{D}({{\textsf {G}}},{D})\) does not contain the Pappus configuration.

Proof

Assume that \(\mathbf{D}({{\textsf {G}}},{D})\) contains the Pappus configuration. Then ; ; \((1,0)\sim (1,2)\), and \((0,1)\sim (0,2)\), which contradicts Lemma 3.6(ii). \(\square \)

Fig. 4
figure4

Pappus configuration

Proposition 5.7

Let D be QDS in an abelian group \({\textsf {G}}\) such that \(1\in D\). Assume that there are \(d_1,d_2,d_3,d_4\in D\setminus \{1\}\) with \(d_1 \ne d_3\), \(d_1^2={d_2}^{{-1}}\), and \(d_3^2={d_4}^{{-1}}\). Then \({\mathfrak {M}}=\mathbf{D}({{\textsf {G}}},{D})\) contains Pappus configurations.

Proof

From the assumptions we get \(d_2 \ne d_4\). Note that incidences indicated in the following table hold in \({\mathfrak {M}}\):

  (1) \((d_1)\) \((d_3)\) \(({d_2}^{{-1}})\) \(({d_4}^{{-1}})\) \(({d_2}^{{-1}}{d_4}^{{-1}})\) \((d_1d_3)\) \(({d_2}^{{-1}}d_3)\) \(({d_4}^{{-1}}d_1)\)
[1] \(\times \) \(\times \) \(\times \)       
\({[}{d_2}^{{-1}}{d_4}^{{-1}}]\)     \(\times \) \(\times \) \(\times \)    
\({[}d_1d_3]\)        \(\times \) \(\times \) \(\times \)
\({[}{d_2}^{{-1}}]\) \(\times \)    \(\times \)     \(\times \)  
\({[}{d_4}^{{-1}}]\) \(\times \)     \(\times \)     \(\times \)
\({[}d_1]\)   \(\times \)   \(\times \)    \(\times \)   
\({[}d_1{d_4}^{{-1}}]\)   \(\times \)     \(\times \)    \(\times \)
\({[}d_3]\)    \(\times \)   \(\times \)   \(\times \)   
\({[}d_3{d_2}^{{-1}}]\)    \(\times \)    \(\times \)   \(\times \)  

Then the map \((\psi ',\psi '')\) defined for the points by

(a) (0, 0) (1, 0) (0, 1) (1, 2) (2, 1) (1, 1) (2, 2) (0, 2) (2, 0)
\(\psi '((a))\) (1) \((d_1)\) \((d_3)\) \(({d_2}^{{-1}})\) \(({d_4}^{{-1}})\) \(({d_2}^{{-1}}{d_4}^{{-1}})\) \((d_1d_3)\) \(({d_2}^{{-1}}d_3)\) \(({d_4}^{{-1}}d_1)\)

and for the lines by

[a] [0, 0] [1, 1] [2, 2] [0, 2] [2, 0] [1, 2] [1, 0] [2, 1] [0, 1]
\(\psi ''([a])\) [1] \([{d_2}^{{-1}}{d_4}^{{-1}}]\) \([d_1d_3]\) \({[}{d_2}^{{-1}}]\) \([{d_4}^{{-1}}]\) \([d_1]\) \({[}d_1{d_4}^{{-1}}]\) \([d_3]\) \([d_3{d_2}^{{-1}}]\)

embeds the Pappus configuration into the structure \({\mathfrak {M}}\). \(\square \)

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 \({{\textsf {G}}}={\left\langle G,+,0 \right\rangle }\) be an abelian group and \(\mathcal{D}\subset G\). Recall, cf. [14], that a multiplier \(\alpha \) of the set \(\mathcal{D}\) is an automorphism of \({\textsf {G}}\) of the form \(x\mapsto \alpha \cdot x\) satisfying \(\alpha \mathcal{D}=q+\mathcal{D}\) for some \(q\in G\).

Multi-Fano Configuration

The Fano configuration \({\mathfrak {F}}\) is a finite projective plane, so it can be obtained as \({\mathfrak {F}}=\mathbf{D}({C_7},{\{0,1,3\}})\) (cf. [5, 15]). Let us introduce the multi-Fano configuration \({{\mathfrak {F}}^+}=\mathbf{D}({C_k \oplus C_7},{\{ (0,0}),(0,1),(0,3),(1,0) \}) \cong \mathbf{D}({C_k},{\{ 0,1 \}})\oplus {\mathfrak {F}}\). We refer to the subconfiguration determined by points of the form \(\{ i \} \times C_7\) with fixed \(i=0,\ldots ,k-1\) as to a Fano’s part of \({{\mathfrak {F}}^+}\).

The following two Lemmas can be easily proved by analyzing the neighborhood of a point (cf. Fig. 5).

Lemma 6.1

If \(f \in {\mathrm{Aut}}(\mathfrak {F}^+)\), and xy are two points of \(\mathfrak {F}^+\), such that \(f(x)=y\), then f transforms the Fano’s part of \({{\mathfrak {F}}^+}_{({x})}\) into the Fano’s part of \({{\mathfrak {F}}^+}_{({y})}\).

Lemma 6.2

If \(f \in {\mathrm{Aut}}(\mathfrak {F}^+)\) and f preserves \(\{0\}\times C_7\), then f preserves \(\{i\}\times C_7\) for every \(i =0,\ldots ,k-1\).

Proposition 6.3

  1. (i)

    If \(7 \not \mid k\), then \({\mathrm{Aut}}(\mathfrak {F}^+)\cong C_k \oplus C_7\).

  2. (ii)

    If \(7 \mid k\), then \({\mathrm{Aut}}(\mathfrak {F}^+)\cong C_3 \ltimes (C_k \oplus C_7)\).

Fig. 5
figure5

The neighborhood of a point (ia) in multi-Fano configuration

Proof

Generally, \(\text {Tr}(C_k \oplus C_7) \subseteq {\mathrm{Aut}}(\mathfrak {F}^+)\). Let us take \(g= \tau _{-f(0,0)} \circ f\), where \(f \in {\mathrm{Aut}}(\mathfrak {F}^+)\). Then \(g((0,0))=(0,0)\). By Lemma 6.1, every collineation \(g \in {{\mathrm{Aut}}({\mathfrak {F}^+})}_{({(0,0)})}\) preserves the Fano substructure in the neighborhood of the point (0, 0). According to Lemma 6.2, the Fano substructure is preserved on every of i levels, where \(i= 0,1,\ldots ,k-1 \).

Denote the set \(\{0,1,3\}\) by \(\mathcal{D}\). Note that \(\mathcal{D}\) has two multipliers: \(2\mathcal{D}=6 +\mathcal{D}\) and \(4\mathcal{D}=4 +\mathcal{D}\). Thus, in view of Fact 3.1 maps \(g_1=(g_1',g_1'')\) with \(g_1'(x)=2x\), \(g_1''=\tau _6g_1'\), and \(g_2=(g_2',g_2'')\) with \(g_2'(x)=4x\), \(g_2''=\tau _4g_2'\) are collineations of \({\mathfrak {F}}\). Moreover, \(g_1, g_2\in {\mathrm{Aut}}({\mathfrak {F}})_{(0)}\). In view of Proposition 4.12, a map \(\varphi = (\varphi ',\varphi '')\), such that \(\varphi '(i,a)=(i,f'_i(a))\) and \(\varphi ''([i,b])=[i,f''_i(b)]\), where \((f_i',f_i'')=f_i \in {\mathrm{Aut}}({\mathfrak {F}})\), \(i\in C_k\), is a collineation of \(\mathfrak {F}^+\) iff \(f'_{i+1}=f''_i\). If we analyze all the elements of \({{\mathrm{Aut}}({\mathfrak {F}})}_{(0)}\) we note that \(g_1\), \(g_2\) are the unique two maps that can be extended to a collineation of \(\mathfrak {F}^+\), since translations are always collineations of \({\mathfrak {F}}\).

Set \(f'_0 = g'_1\), \(f''_0 = g''_1=\tau _6g'_1\). By induction we get \(f'_i = \tau _{6i}g'_1\), \(f''_i = \tau _{6(i+1)}g'_1\). In particular, \(f''_{k-1} = \tau _{6k}g'_1 = f'_0 = g'_1\). Therefore \(\tau _{6k} = {\mathrm {id}}\), i.e. \(6k\equiv 0\) holds in \(C_7\) and thus \(7 \mid k\). The same result we obtain for \((f'_0,f''_0)=(g_2',g_2'')\). To close the proof note that \(G = \{ g_1,g_2,{\mathrm {id}}\} \cong C_3\) and \(f\tau _{(j,b)}f^{-1}(i,a) = \tau _{f(j,b)}(i,a)\) for \(f \in G\). \(\square \)

Multi-Pappus Configuration

Since \(\mathbf{D}({(C_3)^2},{ \mathcal{D}_2})\) is simply the Pappus configuration, \(\mathbf{D}({(C_3)^n},{ \mathcal{D}_n})\) will be called the multi-Pappus configuration (there is \(\mathbf{D}({(C_3)^3},{ \mathcal{D}_3})\) shown in Fig. 1). Note that, in view of Fact 4.9, Proposition 4.10 cannot be used to characterize the automorphisms group of \(\mathbf{D}({(C_3)^n},{ \mathcal{D}_n})\).

Proposition 6.4

Let \({\mathfrak {M}} = \mathbf{D}({(C_3)^n},{\mathcal{D}_n})\) with \(n > 2\). Then \({\mathrm{Aut}}({\mathfrak {M}})\cong S_{n+1}\ltimes (C_3)^n\).

To prove Proposition 6.4 we use Lemmas 4.7, 4.8 and the following lemma:

Lemma 6.5

Let \(f = (f',f'')\in {\mathrm{Aut}}({\mathfrak {M}})\), p be a point of \({\mathfrak {M}}\) and l be a line through p. If \(f''\) fixes every line through p and \(f'\) fixes every point on l then \(f={\mathrm {id}}\).

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

Now we focus on

$$\begin{aligned} {\mathfrak {M}} = \mathbf{D}({(C_3)^k},{\mathcal{D}_k})\oplus \mathbf{D}({C_n},{D}), \end{aligned}$$
(6.1)

where \({\mathfrak {P}} = \mathbf{D}({C_n},{D})\) is the cyclic projective plane \({{\,\mathrm{PG}\,}}(2,q)\) with a prime power q, given by a difference set \(D = \{ d_0,d_1,\ldots ,d_q \}\) in the group \(C_n\), where \(n=q^2+q+1\) (cf. [2, 13]). The projective cyclic plane \({{\,\mathrm{PG}\,}}(2,q)\) can be determined by \(q+1\) distinct difference sets \(D^1,D^2,\ldots ,D^{q+1}\). Quite surprisingly, it turns out that \({\mathfrak {M}}^i = \mathbf{D}({(C_3)^k},{\mathcal{D}_k})\oplus \mathbf{D}({C_n},{D^i})\), \(i=1,2,\ldots ,q+1\) are not always pairwise isomorphic. Indeed, we will show that \(\mathbf{D}({C_3^k},{\mathcal{D}_k})\oplus \mathbf{D}({C_{13}},{\{0,1,3,9\}})\) is not isomorphic to \(\mathbf{D}({C_3^k},{\mathcal{D}_k})\oplus \mathbf{D}({C_{13}},{\{0,2,8,12\}})\). In view of Proposition 4.3

$$\begin{aligned} {\mathfrak {M}} \cong \underbrace{\mathbf{D}({C_3},{\{0,1\}}) \oplus \mathbf{D}({C_3},{\{0,1\}})\oplus \cdots \oplus \mathbf{D}({C_3},{\{0,1\}})}_{k \; \mathrm {times}} \oplus \mathbf{D}({C_n},{D}). \end{aligned}$$

Hence, in this case we can use Lemma 4.11. Every point \(p \in {\mathfrak {M}}\) can be written as \(p=(x_k,\ldots ,x_1,y)\), where \(x_k,\ldots ,x_1\in C_3\), \(y\in C_n\). Let \(\theta = (0,\ldots ,0) \in (C_3)^k\times C_n\). In this notation, the points and the lines of \({\mathfrak {M}}_{({\theta })}\) are the following:

$$\begin{aligned}&q_{i,j}=(0,\ldots ,0,-d_i+d_j),\\&p'_{m,i}=(0,\ldots ,1_m,0,\ldots ,-d_i),\quad p''_{m,i}=(0,\ldots ,2_m,0,\ldots ,d_i),\\&l_i = [0,\ldots ,0,-d_i],\qquad \qquad \qquad \quad l''_m = [0,\ldots ,2_m,0,\ldots ,0],\\&\text {for}\; d_i,d_j\in D; \; i,j=0,\ldots ,q; \; i\ne , j;\qquad m=1,\ldots ,k,\\&p'_{s,q+r}=(0,\ldots ,1_s,0,\ldots ,2_r,0,\ldots ,0),\quad p''_{s,q+r}=(0,\ldots ,2_s,0,\ldots ,1_r,0,\ldots ,0),\\&\text {for} \; s=2,\ldots ,k;\; r=1,\ldots ,s-q+2 . \end{aligned}$$

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

Lemma 6.6

Let \(F\in {{\mathrm{Aut}}({\mathfrak {M}})}_{({\theta })}\). Then, F leaves invariant the multi-Pappus subconfiguration \((C_3)^k \times \{0\} \cong \mathbf{D}({(C_3)^k},{\mathcal{D}_k})\) and the cyclic projective subplane \(\{(0,\ldots ,0)\}\times C_n \cong \mathbf{D}({C_n},{D})\). Moreover, F determines permutations \(\alpha \in S_q\) and \(\beta \in S_k\) such that \(F(l''_m)=l''_{\beta (m)}\), \(F(p'_{m,0})=p'_{\beta (m),0}\), and \(F(l_i) = l_{\alpha (i)}\) for \(m=1,\ldots ,k\), \(i=1,\ldots ,q\).

Lemma 6.7

Let \(\beta \in S_k\). We define the map \(G_{\beta }\) on \(C_3^k\oplus C_n\) by the formula \(G_{\beta }((x_k,\ldots ,x_1,y)) = (x_{\beta (k)},\ldots ,x_{\beta (1)},y)\). Then \(G_{\beta }\in {\mathrm{Aut}}({\mathfrak {M}})\), \(G_{\beta }(\theta ) = \theta \), and \(G_{\beta }(p'_{m,0})=p'_{\beta (m),0}\).

Lemma 6.8

Let \(G_{\beta }\) be the map defined in Lemma 6.7 and \(\mathcal{G}_0 = \{G_{\beta }:\beta \in S_k\}\), \(\mathcal{G} = \{\tau _a \circ g :g\in \mathcal{G}_0, a \in (C_3)^k \times C_n\}\). Then \(\mathcal{G}_0 \cong S_k\) and \(\mathcal{G}\cong S_k\ltimes (C_3^k\oplus C_n)\).

Lemma 6.9

If, under notation of Lemma 6.6 we assume \(\beta = {\mathrm {id}}\), then every point \(p''_{m,0}, p'_{s,q+r}, p''_{s,q+r}\) is fixed by F. Moreover, for \(\alpha \in S_q\) we have \(F(p'_{m,i})= p'_{m,\alpha (i)}\) for \(i=1,\ldots ,q\).

Lemma 6.10

Under assumptions of Lemma 6.9, and the condition

  1. (a)

    for every, except at most one, \(d_i\in D\) there exist \(d_j,d_r\in D\) such that

    $$\begin{aligned} d_i + d_j + d_r = 0 \end{aligned}$$

the permutation \(\alpha \) given in Lemma 6.6 satisfies the following: \(F(p''_{m,i})= p''_{m,\alpha (i)}\), \(F(q_{i,j})= q_{\alpha (i), \alpha (j)}\), for \(m=1,\ldots ,k\); \(i,j=1,\ldots ,q\), \(i\ne j\). Consequently, if \(\alpha = {\mathrm {id}}\), then F is the identity on \({\mathfrak {M}}_{({\theta })}\).

Let q be a point of \({\mathfrak {M}}\) and F be an automorphism of \({\mathfrak {M}}\) with \(F(q) = q\). Generalizing the notation of Lemma 6.6 we write for the permutation \(\alpha \) of \(\{ 1,\ldots ,q \}\) and for the permutation \(\beta \) of \(\{ 1,\ldots ,k \}\) determined by \(F\restriction {{\mathfrak {M}}_{({q})}}\).

Lemma 6.11

Let the condition (a) of Lemma 6.10 be satisfied. If \(F\in {\mathrm{Aut}}({\mathfrak {M}})\) preserves every line passing through q (in particular, \(F(q) = q\)), then and are identities, and thus F is the identity on \({\mathfrak {M}}_{({q})}\).

Lemma 6.12

Let the condition (a) of Lemma 6.10 be satisfied. Assume that \(F\in {\mathrm{Aut}}({\mathfrak {M}})\) fixes all the points of \({\mathfrak {M}}_{({q})}\). If \(q'\in {\mathfrak {M}}_{({q})}\), then F fixes the points of \({\mathfrak {M}}_{({q'})}\).

Since \({\mathfrak {M}}\) 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 \(F\in {\mathrm{Aut}}({\mathfrak {M}})\) and q is a point of \({\mathfrak {M}}\). If \(F(q) = q\) and F preserves every line through q, then \(F = {\mathrm {id}}\).

With the help of Lemma 6.6–Corollary 6.13 we determine automorphisms group of \({\mathfrak {M}} = \mathbf{D}({(C_3)^k},{\mathcal{D}_k})\oplus {\mathfrak {P}}\) in two particular cases: for \({\mathfrak {P}}={\mathfrak {F}}={{\,\mathrm{PG}\,}}(2,2)\) and for \({\mathfrak {P}}={{\,\mathrm{PG}\,}}(2,3)\). Let us start from \({\mathfrak {M}} = \mathbf{D}({(C_3)^k},{\mathcal{D}_k})\oplus {\mathfrak {F}}\). The obtained structure can be considered as a join of the multi-Pappus and the Fano configuration.

Proposition 6.14

Let \({\mathfrak {M}} = \mathbf{D}({C_3^k},{\mathcal{D}_k})\oplus \mathbf{D}({C_7},{\{ 0,1,3 \}})\) with \(k \ge 2\). Then the group \({\mathrm{Aut}}({\mathfrak {M}})\) is isomorphic to \(S_k\ltimes (C_3^k\oplus C_7)\).

Adopt \({\mathfrak {P}} = {{\,\mathrm{PG}\,}}(2,3)\) instead of Fano configuration. As \({{\,\mathrm{PG}\,}}(2,3)\) can be induced by four distinct difference sets \(D^1 = \{0,1,3,9\}\), \(D^2 = \{0,2,8,12\}\), \(D^3 = \{0,6,10,11\}\), \(D^4 = \{0,4,5,7\}\), we get

$$\begin{aligned} {\mathfrak {M}}^i = \mathbf{D}({C_3^k},{\mathcal{D}_k})\oplus \mathbf{D}({C_{13}},{D^i}), \end{aligned}$$

where \(i=1,2,3,4\). Note, that the condition (a) from Lemma 6.10 is satisfied for all sets \(D^i\), but in different ways: for every \(d_i\in D^1\) there exist \(d_j,d_r\in D^1\) such that \(d_i+d_j+d_r = 0\), and in \(D^2,D^3,D^4\) there is exactly one \(d_i\) for which do not exist such \(d_j,d_r\). So, the configurations \({\mathfrak {M}}^j\) are pairwise isomorphic for \(j=2,3,4\) (for \(\phi (x)=3x\) we get \(\phi (D^2)=D^3,\phi (D^3)=D^4,\phi (D^4)=D^2\)), but are not isomorphic to \({\mathfrak {M}}^1\). Although \({\mathfrak {M}}^1\) and \({\mathfrak {M}}^2\) are not isomorphic, they have isomorphic automorphisms groups. To justify this, we show the complete proof of the following Proposition:

Proposition 6.15

Let \({\mathfrak {M}}^i = \mathbf{D}({C_3^k},{\mathcal{D}_k})\oplus \mathbf{D}({C_{13}},{D^i})\), where \(i=1,2\) and \(D^1 = \{0,1,3,9\}\), \(D^2 = \{0,2,8,12\}\). Then, the group \({\mathrm{Aut}}({\mathfrak {M}}^i)\) is isomorphic to \(S_k\ltimes (C_3^k\oplus C_{13})\).

Proof

Let F be an automorphism of \({\mathfrak {M}}^i\) and \(g = \tau _{-F(\theta )}\circ F\). Then \(g(\theta ) = \theta \) and \(g\in {\mathrm{Aut}}({\mathfrak {M}}^i)\). In view of Lemma 6.6, g leaves the set \(\{p'_{m,0}:m=1,\ldots ,k\}\) invariant. We set \(h=g\circ G^{-1}_{\beta }\), where \(G_{\beta }\) is the map defined in Lemma 6.7 and . Then \(h\in {\mathrm{Aut}}({\mathfrak {M}}^i)\), \(h(\theta ) = \theta \), and . By Lemma 6.9h preserves the set \(\{l''_m:m=1,\ldots ,k\}\).

Let be the permutation determined by h, in accordance with Lemma 6.9. Then, from Lemma 6.10 we get \(h(q_{i,j}) = h_\alpha (q_{i,j}) := q_{\alpha (i),\alpha (j)}\) for all ij. Note that if \(\alpha \ne {\mathrm {id}}\), then \(h_\alpha \) does not preserve the collinearity in \({\mathfrak {M}}^1\). For example: \(q_{1,2}\), \(q_{0,2}\), \(q_{3,1}\), \(q_{2,1}\) are collinear, but \(q_{\alpha (1),\alpha (2)}\), \(q_{0,\alpha (2)}\), \(q_{\alpha (3),\alpha (1)}\), \(q_{\alpha (2),\alpha (1)}\) are not, unless \(\alpha = {\mathrm {id}}\). In \({\mathfrak {M}}^2\) for all \(m=1,\ldots ,k\) we have: \(p'_{m,1}\) are points of rank 5 on a line of rank 3, \(p'_{m,2}\) are points of rank 5 and there is no line of rank 3 passing through these points, and \(p'_{m,3}\) are points of rank 6. So, h fixes these points, and thus \(\alpha = {\mathrm {id}}\).

In both cases, h fixes all the lines through \(\theta \), so from Corollary 6.13 we get \(h = {\mathrm {id}}\), and thus \(g = G_{\beta }\). Finally, applying Lemma 6.8 we close the proof. \(\square \)

A Power of a Cyclic Projective Plane

Let \({\mathfrak {P}} = \mathbf{D}({C_k},\mathcal{D})\) be a cyclic projective plane determined by a difference set \(\mathcal D\) 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 \(\mathfrak P\). Let us draw our attention to the following structure:

$$\begin{aligned} {\mathfrak {P}}^n := \underbrace{\mathfrak P\oplus \mathfrak P\oplus \cdots \oplus \mathfrak P}_{n \; \mathrm {times}} \end{aligned}$$
(6.2)

Note that \({\mathfrak {P}}^n = \mathbf{D}({(C_k)^n},{D})\), where \(D = \mathcal{D}\uplus \cdots \uplus \mathcal{D}\). Let us introduce a few auxiliary sets. Namely:

$$\begin{aligned}&{\mathrm {supp}}(x) := \{ i \in \{ 1,\ldots ,n \} :x_i \ne 0 \} \text { for } x \in (C_k)^n,\\&\Xi _{\alpha }^{\gamma }:= \{x\in (C_k)^n:|{\mathrm {supp}}(x)|=\gamma \text { and if } i\in {\mathrm {supp}}(x) \text { then } x_i=\alpha \},\\&\Xi _{\{\alpha \},\{\beta \}}^{1}:= \Xi ^1_{\alpha } \cup \Xi ^1_{\beta },\\&\Xi _{\{\alpha ,\beta \}}^{2}:= \{x\in (C_k)^n:|{\mathrm {supp}}(x)|=2 ,\; \{ i,j \} = {\mathrm {supp}}(x) \implies \{x_i,x_j\}= \{\alpha ,\beta \}\},\\&P_i:= \{x\in (C_k)^n:{\mathrm {supp}}(x)= \{i\}, \text { or } x=\theta \},\\&S_i = \{(x):x \in P_i \},\;\; {\mathcal {J}}_i =\{ [x]:x \in P_i \}, \;\; i = 1,\ldots ,n. \end{aligned}$$

For \(i_1 \ne i_2\) we have \({\mathcal {J}}_{i_1} \cap {\mathcal {J}}_{i_2} = \{ [\theta ] \}\) and \({S}_{i_1} \cap {S}_{i_2} = \{ (\theta ) \}\). The sets \(S_i\) and \({\mathcal {J}}_i\) consist of points and lines, respectively, which form a projective plane embedded in \({{\mathfrak {P}}^n}_{({\theta })}\). There are n such planes with the common line \([\theta ]\), and the common point \((\theta )\) in \({{\mathfrak {P}}^n}_{({\theta })}\). Note that, the degree of the point \((\theta )\) in \({{\mathfrak {P}}^n}_{({\theta })}\) equals \(q n + 1\), and it is equal to the size of every line through \((\theta )\).

By Lemma 3.3 and (4.1) we can prove the following lemmas:

Lemma 6.16

The line [y] passes through \((\theta )\) iff \([y]\in {\mathcal {J}}_i\) for some i and \(y_i \in -{\mathcal {D}}\).

Lemma 6.17

Let \(x \in (C_k)^n\). The point (x) is a point of \({{\mathfrak {P}}^n}_{({\theta })}\) iff \(x \in P_i\) for some i (i.e. \(|{\mathrm {supp}}(x)| = 1\)) or \(x\in \Xi _{\{\alpha ,\beta \}}^{2}\) and \(\alpha \in -{\mathcal {D}}{\setminus \{ 0 \}}\), \(\beta \in {\mathcal {D}}{\setminus \{ 0 \}}\).

Lemma 6.18

If \(x \in P_i\) then either the line [x] passes through \((\theta )\) and its size in \({{\mathfrak {P}}^n}_{({\theta })}\) is \(qn+1\), or the size of [x] in \({{\mathfrak {P}}^n}_{({\theta })}\) is \(q+1\); then, in particular, [x] does not contain any point of \(P_j\) with \(j \ne i\).

Lemma 6.19

Let \(x\in (C_7)^n\). If the line [x] contains a point of \({{\mathfrak {P}}^n}_{({\theta })}\) (, i.e. it intersects a line of the form [y] defined in Lemma 6.16), then \(|{\mathrm {supp}}(x)| \le 3\). Moreover, if \(|{\mathrm {supp}}(x)| = 3\), then the size of [x] in \({{\mathfrak {P}}^n}_{({\theta })}\) is 2.

There are lines in \({{\mathfrak {P}}^n}_{({\theta })}\) joining points in \(P_i\) with points in \(\Xi _{\{\alpha ,\beta \}}^{2}\), where \(\alpha \in -{\mathcal {D}}{\setminus \{ 0 \}}\), \(\beta \in {\mathcal {D}}{\setminus \{ 0 \}}\). Namely:

Lemma 6.20

Let \((x)\in S_i\), \([y]\in {\mathcal {J}}_i\), and . For every \(j\ne i\) there are q lines of the size 2 in \({{\mathfrak {P}}^n}_{({\theta })}\), such that each of them joins (x) with one of the pairwise collinear points \((z^1),\ldots (z^q)\), where \(z^1_i=\cdots =z^q_i=x_i-y_i\), \(\{z^1_j,\ldots ,z^q_j\}= -{\mathcal {D}}\setminus \{0\}\), and \(z^1_s=\cdots =z^q_s=0\) for all \(s\ne i,j\); \(s = 1,\ldots ,n\).

Lemma 6.21

For any two points \(a,a'\) of \({\mathfrak {P}}^n\) there is a sequence \(b_0,\ldots ,b_m\) of points of \({\mathfrak {P}}^n\) such that \(b_0 = a\), \(b_m = a'\), and \(b_j\) is a point of a projective subplane in \({{{\mathfrak {P}}^n}_{({b_{j-1}})}}\) for \(j = 1,\ldots ,m\).

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

A Power of the Fano Plane

Let us put \({\mathfrak {P}} ={\mathfrak {F}}=\mathbf{D}({C_7},{\{0,1,3\}})\).

Fig. 6
figure6

The neighborhood of the point (0, 0) in \({\mathfrak {F}} \oplus {\mathfrak {F}}\)

Proposition 6.22

The group \({\mathrm{Aut}}({\mathfrak {F}}^n)\) is isomorphic to \( S_n \ltimes (C_7)^n \).

Proof

Let \({\mathcal {J}}''_0\) be the family of lines of the size 4 in \({{\mathfrak {F}}^n}_{({\theta })}\) and \({\mathcal {J}}'_0\) be the family of the lines of the size 3 in \({{\mathfrak {F}}^n}_{({\theta })}\) that are not in any of the \({\mathcal {J}}_i\). We need Lemma 4.6 and lemmas from Sect. 6.4.

Step 1 Let \(y \in (C_7)^n\). Then \([y]\in {\mathcal {J}}''_0\) iff \(y\in \Xi _{3}^{2}\).Let \(y\in \Xi _{3}^{2}\) and \({\mathrm {supp}}(y) = \{ i_1,i_2 \}\). The line [y] does not intersect \([\theta ]\), but [y] intersects every of the remaining two lines in \({\mathcal {J}}_{i_1}\) and in \({\mathcal {J}}_{i_2}\). Consequently, for any two \(i_1,i_2 \in \{ 1,\ldots ,n \}\) there is (exactly one) line in \({\mathcal {J}}''_0\) that crosses two lines in \({\mathcal {J}}_{i_1}\) and two lines in \({\mathcal {J}}_{i_2}\). No two distinct lines in \({\mathcal {J}}''_0\) intersect.

Step 2 If \(y\ne \theta \), then [y] is of the size 3 in \({{\mathfrak {F}}^n}_{({\theta })}\) iff either \(y\in \Xi _{\{1,3\}}^{2}\) or \(y\in P_i\) for some \(i\in \{1,\ldots ,n\}\). This gives, in particular, that \({\mathcal {J}}'_0= \{ [y]:y \in \Xi _{\{1,3\}}^{2} \}\). Let \({\mathrm {supp}}(y) = \{ i_1,i_2 \}\), \(y_{i_1} = 1\), and \(y_{i_2} = 3\). Lines [y] and \([\theta ]\) do not meet. The line [y] crosses two other lines in \({\mathcal {J}}_{i_2}\) and it crosses exactly one line in \({\mathcal {J}}_{i_1}\). Consequently, for every two \(i_1,i_2 \in \{ 1,\ldots ,n \}\) there is (exactly one) line in \({\mathcal {J}}'_0\) that crosses two lines in \({\mathcal {J}}_{i_2}\) and crosses exactly one line in \({\mathcal {J}}_{i_1}\). No two distinct lines in \({\mathcal {J}}'_0\) intersect.

Directly from Step 2 we get:

Step 3 A line l in \({{\mathfrak {F}}^n}_{({\theta })}\) belongs to \({\mathcal {J}}'_0\) iff the size of l equals 3 and no other line of the size 3 in \({{\mathfrak {F}}^n}_{({\theta })}\) crosses l.

The next two Steps are immediate from Lemma 6.17 and Steps 1, 2:

Step 4 Let (x) be a point of \({{\mathfrak {F}}^n}_{({\theta })}\). Then \(x \in P_i\) iff there are two distinct lines of the size 3 that pass through it. The set of points on the lines in \({\mathcal {J}}'_0\cup {\mathcal {J}}''_0\) is the set of points (x) with \(x \notin \cup _{i=1}^{n}P_i\).

Step 5 From the above it follows that \({{\mathfrak {F}}^n}_{({\theta })}\) 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 . Intuitively, we can read Step 1 as “any two Fano subplanes of \({{\mathfrak {F}}^n}_{({\theta })}\) 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 \(\bigcup _{i=1}^n S_i\) and permutes the Fano subplanes. So, F determines a permutation \(\sigma \) such that F maps the set \(S_i\) onto \(S_{\sigma (i)}\) and it maps the family \({\mathcal {J}}_i\) onto \({\mathcal {J}}_{\sigma (i)}\) for every \(i =1,\ldots ,n\).

Step 6 Obviously, F preserves the set of lines of the size 4 in \({{\mathfrak {F}}^n}_{({\theta })}\). Since these lines are of the form [y] with \(y\in \Xi _{3}^{2}\), we can identify every such a line [y] with the set . Every point (x), where \(x\in \Xi _3^3\), is in \({\mathfrak {F}}^n\) the meet of three lines \([y_t]\), \(y_t\in \Xi _3^2\), and \(t=1,2,3\) iff \({\mathrm {supp}}(y_t)\subset {\mathrm {supp}}(x)\). Therefore, lines \(\{[y]:y\in \Xi _3^2\}\) together with their intersection points form the structure dual to combinatorial Grassmanian \(\mathbf{G}_{3}({n})\), cf. [12]. The map F determines a permutation \(F_0\) of the lines in \({\mathcal {J}}''_0\) which, in view of the above, is an automorphism of \(\mathbf{G}_{3}({n})\). The automorphisms group of \(\mathbf{G}_{3}({n})\) is the group \(S_n\) (comp. [11]), and thus there is \(\sigma '\in S_n\) which determines \(F_0\). It is seen (cf. Step 1) that \(\sigma ' = \sigma \). Let \(G = G_{\sigma }\) be the automorphism of \({\mathfrak {F}}^n\) determined by the permutation \(\sigma \) (cf. Lemma 4.6) and let \(\varphi =G^{-1}\circ F\). Clearly, \(\varphi \) is an automorphism of \({\mathfrak {F}}^n\), and \(\varphi \) maps every line in \({\mathcal {J}}''_0\) onto itself. Consequently, \(\varphi \) maps every family \(\mathcal{J}_i \setminus \{ [\theta ] \}\) onto itself and thus it leaves the line \([\theta ]\) invariant.

Step 7 From Step 3, the map \(\varphi \) preserves the family \({\mathcal {J}}'_0\). Observing intersections of the lines of this family and the lines in the families \({\mathcal {J}}_i\) (cf. Step 2) we get that every line through \(\theta \) remains invariant under \(\varphi \).

Step 8 Let F be an automorphism of \({\mathfrak {M}}\) such that F leaves every line through a point a invariant. Then \(F\restriction {{{\mathfrak {F}}^n}_{({a})}}={\mathrm {id}}\).

Step 9 Let ab be two points of \({\mathfrak {F}}^n\) such that b is a point of a Fano subplane in \({{{\mathfrak {F}}^n}_{({a})}}\). If F is an automorphism of \({\mathfrak {F}}^n\) such that \(F\restriction {{{\mathfrak {F}}^n}_{({a})}}={\mathrm {id}}\) then \(F\restriction {{{\mathfrak {F}}^n}_{({b})}}={\mathrm {id}}\). \(\square \)

A Power of the Cyclic Projective Plane \({{\,\mathrm{PG}\,}}(2,3)\)

Let \({\mathfrak {P}} =\mathbf{D}({C_{13}},{\{0,1,3,9\}})\), i.e. let \(\mathfrak P\) be the cyclic projective plane \({{\,\mathrm{PG}\,}}(2,3)\). Note that the set \(\{0,1,3,9\}\) is fixed by the multiplier \(\alpha =3\). Thus, this multiplier yields an automorphism of \(\mathfrak P\) (cf. Fact 3.1). The map \(\mu _\alpha :C_{13} \ni x \longmapsto \alpha \cdot x\) generates the cyclic group consisting of \(\mu _1 = {\mathrm {id}}\), \(\mu _3\), and \(\mu _9\), which is isomorphic to \(C_3\). Every element of this group induces an automorphism of \({{\mathfrak {P}}}_{({\theta })}\).

Proposition 6.23

The group \({\mathrm{Aut}}({\mathfrak {P}}^n)\) is isomorphic to \(S_n\ltimes ( (C_3)^n \ltimes (C_{13})^n ) \).

Proof

Let \({\mathcal {J}}''_0\) be the family of lines of the size 4 in \({{\mathfrak {P}}^n}_{({\theta })}\) (Fig. 7) and \({\mathcal {J}}'_0\) be the family of the lines of the size 3 in \({{\mathfrak {P}}^n}_{({\theta })}\) that are not in any of the \({\mathcal {J}}_i\).

Step 1 Let \(y \in (C_{13})^n\). Then \([y]\in {\mathcal {J}}''_0\) iff \(y\in \Xi _{\{\alpha ,\beta \}}^{2}\), where \(\alpha ,\beta \in \{ 1,3,9 \}\). If \([y]\in {\mathcal {J}}''_0\) and \({\mathrm {supp}}(y) = \{ i_1,i_2 \}\) then [y] intersects two out of three lines in \({\mathcal {J}}_{i_1}\) and in \({\mathcal {J}}_{i_2}\), but do not intersects \([\theta ]\). Consequently, for any two \(i_1,i_2 \in \{ 1,\ldots ,n \}\) there are exactly nine lines in \({\mathcal {J}}''_0\) that cross two lines in \({\mathcal {J}}_{i_1}\) and two lines in \({\mathcal {J}}_{i_2}\). For every point \((x)\in {{\mathfrak {P}}^n}_{({\theta })}\) with \(x\notin P_i\) there are two lines \([y]\in {\mathcal {J}}''_0\) such that . Every line in \({\mathcal {J}}''_0\) intersects four other lines in \({\mathcal {J}}''_0\) and does not intersect remaining four lines from \({\mathcal {J}}''_0\).

Step 2 There are no lines of the size 3 in \({{\mathfrak {P}}^n}_{({\theta })}\) (i.e. \({\mathcal {J}}'_0=\emptyset \)).

Step 3 Let us consider the set \({\mathcal {J}}^{\alpha }:=\{[y]\in {\mathcal {J}}_i:y_i=\alpha \in -{\mathcal {D}}\text { and } i=1,\ldots ,n\}\). If F is an automorphism of \({\mathfrak {P}}^n\) leaving every line in \({\mathcal {J}}^{\alpha }\) invariant then \(F\restriction {{{\mathfrak P}^n}_{({\theta })}}={\mathrm {id}}\).

Step 4 Let \(\sigma \in S_n\). We define the map \(h_{\sigma }\) on \((C_{13})^n\) by the formula

$$\begin{aligned} h_{\sigma }((x_1,\ldots ,x_n)) = (x_{\sigma (1)},\ldots ,x_{\sigma (n)}). \end{aligned}$$

By Lemma 4.6 the map \(h_{\sigma }\) induces the collineation \(F_{\sigma }=(F',F'')\) of \({\mathfrak {P}}^n\). Moreover, \(F'(\theta ) = \theta \) and \(G'(P_i)= P_{\sigma (i)}\) for all \(i=1,\ldots ,n\).

Step 5 Let F be an automorphism of \({\mathfrak {M}}\) such that F leaves every line through a point a invariant. Then \(F\restriction {{{\mathfrak {P}}^n}_{({a})}}={\mathrm {id}}\).

Step 6 Let ab be two points of \({\mathfrak {P}}^n\) such that b is a point of a projective subplane \({{\,\mathrm{PG}\,}}(2,3)\) in \({{{\mathfrak {P}}^n}_{({a})}}\). If F is an automorphism of \({\mathfrak {P}}^n\) such that \(F\restriction {{{\mathfrak {P}}^n}_{({a})}}={\mathrm {id}}\) then \(F\restriction {{{\mathfrak {P}}^n}_{({b})}}={\mathrm {id}}\). \(\square \)

Fig. 7
figure7

The structure of lines of the size 4 in \({{\mathfrak {P}}^2}_{({\theta })}\)

Let us come back the power \({\mathfrak {P}}^n\) of an arbitrary finite projective plane \({\mathfrak {P}} = \mathbf{D}({C_k},\mathcal{D})\) induced by a difference set \(\mathcal D\) in a cyclic group \(C_k\). Observing Propositions 6.22 and 6.23 and their proofs we note that every automorphism F of \({\mathfrak {P}}^n\) is related to one of the following:

  • a multiplier of the set \(\mathcal D\), and then F leaves \({{\mathfrak {P}}^n}_{({\theta })}\) invariant,

  • a translation \(\tau \) of \((C_k)^n\), and then F maps \({{\mathfrak {P}}^n}_{({\theta })}\) onto \({{\mathfrak {P}}^n}_{({\tau (\theta )})}\),

  • a permutation \(\sigma \in S_n\), and then F permutes the projective subplanes isomorphic to \(\mathfrak P\).

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 \({\mathfrak {P}} = \mathbf{D}({C_k},\mathcal{D})\) be a finite projective plane induced by a difference set \(\mathcal D\) in a cyclic group \(C_k\). The group \({\mathrm{Aut}}({\mathfrak {P}}^n)\) is isomorphic to

$$\begin{aligned} S_n\ltimes \Bigl ( \bigl ((C_{r_1})^n\oplus \dots \oplus (C_{r_s})^n\bigr )\ltimes (C_{k})^n \Bigr ), \end{aligned}$$

where \(C_{r_1},\ldots C_{r_s}\) are the cyclic groups generated by the multipliers of the set \(\mathcal D\).

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Petelczyc, K., Prażmowski, K. Multiplied Configurations Induced by Quasi Difference Sets. Bull. Iran. Math. Soc. 47, 111–133 (2021). https://doi.org/10.1007/s41980-020-00370-0

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Keywords

  • Partial linear space
  • Difference set
  • Quasi difference set
  • Cyclic projective plane

Mathematics Subject Classification

  • 51D20
  • 51E30