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ISSN 1061-9208, Russian Journal of Mathematical Physics, Vol. 15, No. 1, 2008, pp. 66-76.

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Pleiades Publishing, Ltd., 2008.

Faithful Representations of Groups by Automorphisms of Topologies
A. V. Koganov
Institute of Systems Research (VNIISI), Russian Academy of Sciences, Moscow, 117312 Russia
Received August 3, 2007

Abstract. It is proved that every (in general, infinite) group is the full automorphism group of some topology. DOI: 10.1134/S1061920808010081

1. INTRODUCTION In the pap er, we generalize the well-known theorem claiming that every finite group is the full automorphism group of some finite graph [16]. Namely, we claim that every (in general, infinite) group is the full automorphism group of some top ology. In the system of definitions we use, the automorphisms and the isomorphisms of a top ology coincide with homeomorphisms, i.e., with top ological mappings (we sometimes treat homeomorphisms in a more general way). Moreover, we consider a sp ecial generalization of the notion of top ology obtained by weakening the base system of axioms and producing the so-called inductor spaces. The class of these spaces includes not only the ordinary top ological spaces but also finite and infinite directed graphs, and also sp ecial spaces that are neither top ological spaces nor graphs. The application of this notion enables one to immediately generalize the ab ove theorem on automorphisms of graphs to infinite groups and graphs. We also describ e the standard actions of some geometric symmetry groups as full automorphism groups of inductor spaces constructed from the corresp onding geometric spaces. In particular, we construct an inductor space on the set of p oints of a finite-dimensional linear space in such a way that the symmetry group of the inductor space coincides with the Lorentz group provided that the dimension exceeds two. Here the top ology induced on every hyp erplane cutting out isotropic cones is homeomorphic to the Euclidean one. This result admits a purely geometric formulation. If the dimension of the space is greater than two, then the full automorphism group of the system of cones obtained by parallel shifts of the same spherical cone coincides with the standard action of the Lorentz group (on the space of the corresp onding dimension) extended by arbitrary translations, uniform dilations, and also rotations and reflections (Euclidean isometries) in the hyp erplane of the spherical section of the cones. The proof uses an earlier result of the author [8] claiming that the affine automorphism group of this system of cones coincides with the group of the ab ove action. We also claim that, if the dimension exceeds two, the automorphisms of the structure are affine. Representations in the form of automorphisms of an inductor space can also b e constructed for the main Euclidean isometries, namely, translations, rotations, and rotations with reflections [11]. The corresp onding results are not included in the present pap er. The author expresses his gratitude to A. I. Shtern, A. Yu. Lemin, S. Yu. Vladimirov, M. I. Graev, A. P. Levich, Yu. B. Kotov, V. V. Smolyaninov, A. P. Chernyaev, and A. I. Lobanov for the interest to the present pap er, help, and useful remarks. 2. INDUCTOR SPACES To solve the problem to represent groups by automorphisms of top ologies or graphs, it is convenient to introduce an ob ject generalizing these notions and defining a structure on an abstract p oint set such that the automorphisms of the structure form a representation of the given group. To this end, we introduce the class of inductor spaces. Definition 2.1. By an induction relation I on a set T we mean an arbitrary set of pairs of the form [t, V ]I , where t T, V T . We refer to elements of the induction relation [t, V ]I as inductor pairs, the p oint t is called the center of induction or the center of the pair, and the set V is called the inductor of the pair or of the p oint t.
Supported by RFBR under grant no. 07-01-00101-a.

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Definition 2.2. By an inductor space we mean a set T with an induction relation I satisfying the following axioms. AI1. Axiom of membership. If [t, V ]I I , then t V . AI2. Axiom of transitivity. If [t, V ]I I, [x, W ]I I, x V , then [t, V W ]I I . In [79], inductor spaces were introduced with an extended family of axioms. However, investigations show that the other axioms reduce the class of ob jects too much and lead to technical complications in the proofs of some theorems. However, the theorems proved b elow remain valid in the extended axiomatics. The corresp onding proofs are presented in [9]. From the p oint of view of top ology, it is natural to interpret an inductor as a neighb orhood of the center of induction of a given induction pair. Corresp ondingly, one can sp eak ab out the convergence of sequences and on limit p oints of subsets on T if an induction relation is given. The axiom of transitivity is in this interpretation a weakened axiom of arbitrary union of op en sets. From the p oint of view of graph theory, one can interpret an inductor as a subset of vertices from which there is a path to the center of induction along the arrows of the graph. Corresp ondingly, the axiom of transitivity can b e interpreted as the p ossibility to augment the corresp onding paths on the graph. We sometimes use the abbreviated terms "I-relation," "I-pair," and "I-space." On an inductor space, we refer to the induction relation as the induction, the set T is called the support of induction, and its elements are referred to as points of the space. This term arose in the use of I-spaces as supp orts of distributed processes in mathematical models, where inductors of a p oint play the role of domains of influence on the p oint [8, 9]. This interpretation is inessential for the purp oses of the present pap er. In the general case, inductor spaces are neither graphs nor top ological spaces. For the corresp onding examples, see [8, 9]. However, top ological spaces and graphs are simplest sp ecial cases of I-spaces. For exact definitions, see [8, 9] and the definitions given b elow. 3. RELATIONS OF INDUCTOR SPACES TO GRAPHS AND TOPOLOGICAL SPACES A top ology given by a family of op en subsets on a set of p oints T defines an inductor space formed of all p ossible I-pairs of the form [x, V ] ,where V , x V . In this case, to any top ological neighb orhood of an arbitrary p oint there corresp onds an I-pair with the center of induction at a given p oint. To any op en set, the set of I-pairs corresp onding to diverse centers is assigned. Here it should b e taken into account that, when constructing the top ology from a base family of neighb orhoods of the p oints, one uses a broader class of generating op erations (arbitrary unions and finite intersections) than that for the inductor spaces (with transitive unions only). Therefore, the base family of I-pairs generating the top ology in the form of an inductor space corresp onds in the general case to an extended family of base neighb orhoods. An imp ortant difference b etween the general construction of an induction from a top ology is that a p oint can b elong to an inductor which is not a prop er neighb orhood of the p oint, i.e., there is no I-pair for which the given p oint is the center of induction in the inductor. However, when passing from top ology to an induction in the ab ove way, this situation cannot occur. A directed graph with a set of vertices T (defined as a given subset S T в T of the ordered pairs of vertices ("arrows") (x, y ) = (initial vertex, ending vertex)) can b e describ ed as an inductor space on the set of vertices as p oints (the base I-pairs are of the form [x, V ]S , where x is a vertex and V is the set of vertices issuing arrows ending at x, V = {y |(y, x) S }). The other elements of induction are defined by the base elements by the axiom of transitivity. Definition 3.1. The ab ove constructions of inductions from top ologies and graphs are said to be canonical. Corollary 3.1. For a graph, a vertex y enters some inductor of a vertex x if and only if there is a path from y to x along the arrows. Corollary 3.2. Both for topologies and for graphs, under the canonical imprinting (passage to the corresponding inductions ), the intersection of any two inductors of a point gives an inductor of the same point. A more general property holds : if some inductor of a point x belongs to an inductor V of some point y , then any inductor of x intersects V by an inductor of x. Other versions of corresp ondence are p ossible. Their common prop erty is the p ossibility to uniquely recover the top ology or a graph from the induction, and conversely. A version of passage
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from a top ology to an I-space by using closures of op en sets as inductors was suggested in [79]. This was convenient to describ e models of distributed processes. In this version, limit p oints for sets and sequences must b e defined by using interiors of inductors as neighb orhoods of p oints. We do not consider this version in the present pap er. Definition 3.2. By an alternative passage from a graph to an induction we mean base I-pairs [x, {x; y }]Sa , (x, y ) S . These must also b e completed by I-pairs obtained by the closure according to the axiom of transitivity. In general, this version gives more inductors of a p oint than the canonical one. Corollary 3.2 can fail for the alternative passage, but Corollary 3.1 is preserved. Some top ological notions can b e extended to general inductor spaces. Definition 3.3. A p oint is said to b e interior for some subset of the supp ort of induction if this subset contains some inductor of the p oint. The other p oints of this subset are said to b e boundary. A p oint is limit for a subset of the supp ort if any inductor of the p oint contains an element of the subset. A p oint is limit for a sequence of p oints in the supp ort of induction if any inductor of the p oint contains infinitely many terms of the sequence. A p oint is a limit of a sequence if any its inductor contains all terms of the sequence starting from some of them. 4. AUTOMORPHISMS OF INDUCTOR SPACES The class of inductor spaces can b e equipp ed with the notion of isomorphism. Definition 4.1. Two I-spaces [T, I ], [T ,I ] are said to b e isomorphic if there is a bijection of the supp orts (an isomorphism ) h : T T generating a bijection of the inductions H : I I , where H [t, V ]I =[h(t),h V ]I . An isomorphism of an I-space onto itself (T = T , I = I ) is called an automorphism. Definition 4.2. The automorphisms of any I-space [T, I ] form a group with resp ect to the op eration of sup erp osition. Denote it by aut[T, I ] (the automorphism group ). The unit element is the identity automorphism E (x) = x, x T . The element inverse to an arbitrary automorphism h is the bijection h-1 . Lemma 4.1. Homeomorphisms of topologies and isomorphisms of graphs are equivalent to isomorphisms of inductor spaces under the canonical mapping. Proof. If there is a homeomorphism of two top ologies, then the image and the preimage of any op en set in one of the top ologies is op en in the other. Under the canonical mapping into an inductor space, the inductors are bijective images of op en sets and the corresp onding centers of induction are bijective images of p oints of these sets. Thus, every homeomorphism of two top ologies is an isomorphism of their canonical imprints. Conversely, an isomorphism of two canonical imprints of top ologies is a bijection of the sets of their supp orts under which every inductor of one of the inductions is the image (and the preimage) of an inductor of the other induction. Hence, every op en set of one of the top ologies is the image and the preimage of an op en set of the other, and hence the mapping is a homeomorphism. Thus, we are done for top ologies. An isomorphism of two graphs is a bijection of the sets of their vertices under which the arrows of one of the graphs are unique images and preimages of the arrows of the other graph. This means that the minimal neighb orhood of any vertex with resp ect to the incoming arrows of one of the graphs corresp onds to the minimal neighb orhood of the image of this vertex with resp ect to the incoming arrows of the other graph. Since these neighb orhoods form a generating system of I-pairs of each of the inductions, it follows that an isomorphism of graphs is an isomorphism of their canonical imprints. Conversely, if there is an isomorphism of canonical inductor imprints of two graphs, then it defines a bijection of the supp orts of these I-spaces, and the supp orts coincide with the sets of vertices. Every isomorphism of inductor spaces preserves an emb edding of inductors of any p oint b ecause it is a bijection. Hence, if a p oint has an inductor that is minimal with resp ect to inclusion, then it is mapp ed onto the minimal inductor of the image of the p oint (b oth under the direct and the inverse mapping). Hence, an isomorphism of canonical imprints of two graphs assigns to incoming arrows (of any vertex) of one of the graphs incoming arrows (of the image of this vertex) on the other graph. This means that the image and the preimage of any arrow is an arrow of the corresp onding orientation, which defines an isomorphism of graphs b ecause the mapping is bijective.
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Remark 4.1. Lemma 4.1 remains valid for the alternative passage from a graph to an I-space b ecause the images of the arrows in this graph are minimal inductors of their issuing vertices. In this case, a p oint can have arbitrarily many minimal inductors. For a canonical mapping (of a graph of a top ology), at most one minimal inductor can occur. 5. RIGID I-SPACES, TOPOLOGIES, AND GRAPHS Definition 5.1. An inductor space is said to b e rigid if its automorphism group consists of the identity automorphism only, aut[T, I ] = {E } . The rigid spaces form an infinite class. This enables one to extend inductor spaces in such a way that their automorphism groups are preserved or even strictly reduced. Below we suggest a standard construction of infinitely many pairwise nonisomorphic rigid spaces which is used in the proof of the main theorems. Consider an arbitrary transfinite order typ e p. To this typ e, there corresp onds a well-ordered set of the related cardinality #p, which is a transfinite sequence of order typ e p of pairwise distinct elements, (5.1) S (p) = (a1 , a2 ,... ,ap ) . If p is a limit ordinal, then the last term in (5.1) must b e understood conditionally, as a restriction of the sequence. Assign to the sequence (5.1) the following inductor space SHp = [S (p),H (p)] with the supp ort S (p) and the induction H (p), (5.2) H (p) = {[ai , {a1 ; a2 ; ... ; ai }]H | i p} . In the induction (5.2), the only inductor of any p oint i is the set S (i) S (p) of the elements in (5.1) that do not exceed i in this ordering. The spaces SH (p) are not isomorphic for distinct ordinals p b ecause there is no order-preserving bijection for transfinite sequences of distinct order typ es. For the same reason, the automorphism group of SH (p) is trivial. Every initial segment S (i) in S (p) (an inductor) can b e strictly monotonically mapp ed only onto itself, and thus aut(SH (p)) = aut[S (p),H (p)] = {E }. Lemma 5.1. For any cardinality m, there is a rigid inductor space whose support is of cardinality m. To any ordinal p of cardinality m, there corresponds a rigid I-space nonisomorphic to any space of this kind for other ordinals. (The proof was given ab ove.) Remark 5.1. In what follows, we need a set of G copies of some rigid space that are isomorphic but distinguishable. To this end, we introduce a parameter g G. Copies of the space are denoted by SHp,g = [S (p, g),H (p, g)] and their p oints by a(g)i , in accordance with (5.1). One can pass from rigid I-spaces to rigid top ologies and graphs. Definition 5.2. A graph is said to b e rigid if its automorphism group contains the identity automorphism only. Lemma 5.2. For any cardinality m, there is a rigid graph with support of cardinality m. To any ordinal p of cardinality m, there corresponds a rigid graph nonisomorphic to these graphs for other ordinals. Proof. The ab ove I-space SHp is the canonical imprint of the graph SH Gp with the set of vertices S (p) and the arrows of the form (ai ,aj ), where 1 i j p. The assertion of the lemma follows from Lemma 4.1 and Lemma 5.1. Definition 5.3. A top ological space is said to b e rigid if the group of its automorphisms (selfhomeomorphisms) contains the identity automorphism only. Lemma 5.3. For any cardinality m, there is a rigid topological space with support of cardinality m. To any ordinal p of cardinality m, there corresponds a rigid topological space not homeomorphic to the topological spaces corresponding to the other ordinals. Proof. The inductors of the ab ove I-space SHp give a base system of neighb orhoods for the canonical imprint of the top ological space SH Tp constructed on the p oint set S (p) by using the op en sets of the form S (j ), 1 j p. The set S (i) is the minimal neighb orhood of the p oint ai and the minimal inductor of the canonical imprint of ai . Under an automorphism of inductions, the minimal inductor of a p oint passes to the minimal inductor of its image. Therefore, every automorphism of SH Tp is an automorphism of SHp . Thus, the assertion of the lemma follows from Lemma 4.1 and Lemma 5.1.
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6. REPRESENTATIONS OF GROUPS BY AUTOMORPHISMS OF I-SPACES, TOPOLOGICAL SPACES, AND GRAPHS As was proved in [8, 9], every group defines an infinite class of inductor spaces whose automorphism groups are (algebraically) isomorphic to the given group. We refer to these spaces as imprints of the group (giving the class of imprints), and their automorphisms can b e regarded as representations of the group. The induced action of automorphisms on the linear space of functions defined on the supp ort of an I-space gives a ("multiplicative") representation into an algebra of linear op erators. To obtain a similar result for automorphism groups of top ological spaces and graphs, b elow we use the construction of inductor imprints of groups. Definition 6.1. An inductor space is said to b e an inductor imprint (an I-imprint, an imprint ) of some group if the full automorphism group of the space is isomorphic to the given group. Theorem 6.1. Every group G admits an inductor imprint. Proof. Let us use the notation Sp , Hp , and SHp introduced in the proof of Lemma 5.3. Denote by m the cardinality of the set of elements of the group G. Let Q = {Yg |g G {0}} b e a family of disjoint sets (layers), the cardinality of each of the layers Yg being equal to m. Choose an ordinal p of cardinality m. Define a bijection r : G Sp . On any set Yg Q, we introduce a bijection hg : Yg G. In this case, a bijection r hg = rg : Yg Sp is well defined. Let us equip every layer Yg , g G, with the induction Ig = Hp with resp ect to the order rg . Define the induction I0 = {[x, {x}]I |x Y0 } on the set Y0 ; we refer to I0 as a loop induction because I0 canonically corresp onds to a graph in which every vertex gives an arrow to itself and there are no other arrows. Write T = Q. The induction I on T , along with the ab ove inductors in Ig |gG {0}, contains all inductors of the form [x, {x; z }]I , x Y0 , z = h-1 (h0 (x)g-1 ), g G. This corresp onds to arrows of g the graph J joining the p oint x of the layer Y0 (as the entrance p oint) to the p oint z of the layer Yg (as the issuing p oint), and z corresp onds on the layer Yg to the right multiplication by g-1 G of the image of the p oint x in the system of mappings h of these layers onto the group G. Let us show that aut[T, I ] G. The induction on every layer Yg , g G, is rigid. Hence, under any automorphism, this layer can b e mapp ed only onto a layer of the same form by preserving the order of r . The layer Y0 can b e mapp ed only onto itself, and admits arbitrary self-bijections with resp ect to the induction I0 . However, these self-bijections are limited by the graph J . The only admissible auto-bijections must take the layers Yg , g G, bijectively onto one another by preserving the arrows in J . Let us show that every bijection of this kind corresp onds to the right multiplication of the indices of the layers by some element of the group. Let v aut[T, I ], x Y0 . For any x, there is an f G, f = (h0 (x))-1 h0 (v (x)), for which v (x) = h-1 (h0 (x)f ). Consider an 0 arbitrary layer Yg , g G. Supp ose that v Yg = Yq . There is a unique arrow (x, zg ), zg Yg , in the graph J , namely, zg = h-1 (h0 (x)g-1 ). Since v is an automorphism, it follows that J contains g an arrow (v (x),v (zg )), where v (zg ) = h-1 (h0 (v (x))q -1 ) = h-1 (h0 (x)fq -1 ). Since v : Yg Yq is q q an isomorphism, it follows that r (hg (zg )) = r (h0 (x)g-1 ) = r (hq (v (zg ))) = r (h0 (x)fq -1 ). Since r is bijective, we have h0 (x)g-1 = h0 (x)fq -1 and q = gf . Since the entire layer Yg is taken onto the layer Yq , it follows that the element f of the group is the same for any x Y0 , namely, f = g-1 q . The automorphism v on the layer Y0 is of the form v Y0 = h-1 ((h0 Y0 )f ). If v1 ,v2 aut[T, I ], and if the elements f1 ,f2 G corresp ond to v1 0 and v2 by the ab ove formula, then v2 v1 Y0 = h-1 ((h0 Y0 )f1 f2 ). This formula establishes an 0 isomorphism aut[T, I ] G. Thus, the automorphisms define a representation of the group. Remark 6.1. If one identifies the elements of the layers Yg with the elements of the group G with resp ect to the bijections hg and if e is the identity element of G, then, on the layers with the indices g G, the graph J writes out a p ermutation of the elements of the layer Ye which arises under the right action of the element g-1 on the group. Therefore, an automorphism is admissible only if a self-action of the group corresp onds to the automorphism. This excludes the outer automorphisms of the group from aut[T, I ]. In fact, the ab ove imprint of the group in the finite case corresp onds to the representation of all p ermutations of the elements of the group arising under a multiplication by an element of the group in the form of a graph. Claim 6.1. The nonisomorphic imprints of a given group form an infinite class.
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Proof. The assertion follows from Lemma 5.1. Consider the union of an I-space T and a rigid space X disjoint from T and such that X has no isomorphic emb eddings in T . If now new inductors are added, then aut(T ) aut(T X ). At the same time, adding nonisomorphic spaces X gives nonisomorphic unions. Theorem 6.2. Every group has an imprint in the class of graphs (in general, infinite ). Proof. Consider the construction in the proof of Theorem 6.1. By Lemma 5.2, the rigid layers Yg can b e constructed in the form of graphs. Their union with the graph Y0 by using the graph J gives a desired graph with the automorphism group aut[T, I ] G. Remark 6.2. If a group is finite, then the imprint constructed ab ove is a finite graph. Thus, the ab ove construction gives another proof of the theorem concerning representations of finite groups [2]. Theorem 6.3. Every group G has an imprint in the class of topological spaces. Proof. Consider the construction in the proof of Theorem 6.1. By Lemma 5.3, the rigid layers Yg can b e constructed in the form of top ological spaces. The layer Y0 can then b e equipp ed with the discrete top ology. The arrows of the graph J are transformed into op en sets not canonically. This is needed to prevent the occurrence of new neighb orhoods on the layers Yg . Denote by Wg (x) the set of elements on Yg which do not exceed x in the ordering of Sp . An arrow (x, y ), where x Yg and y Y0 , is transformed to the op en set Vx,y = {y } Wg (x). In this case, the layers remain rigid b ecause they keep the intrinsic top ology of Lemma 5.3. New neighb orhoods of p oints occur on the layer Y0 due to unions of op en sets Vx,y . However, the corresp ondence of the p oints of the layers is preserved as in the graph J , b ecause Vx,y is the minimal neighb orhood of the p oint y having an intersection with the layer containing x. Therefore, under any automorphism, Vx,y comes to a neighb orhood of the same form. These neighb orhoods of the p oint y Y0 (on every layer) uniquely define the arrow (x, y ). We obtain a top ological space with the automorphism group aut[T, I ] G. Remark 6.3. If a group is finite, then the imprint thus obtained is a finite top ological space. 7. REPRESENTATIONS OF AN ACTION OF A GROUP BY AUTOMORPHISMS OF I-SPACES, TOPOLOGICAL SPACES, AND GRAPHS Definition 7.1. By an action W : G of some group G on a given set W we mean a family Q of self-bijections of W , q Q q : W W , which is closed under sup erp osition and group isomorphic (with resp ect to the sup erp osition) to the group G, i.e., there is a bijection P : Q G such that q, q Q P (q (q )) = P (q )P (q ). Definition 7.2. An imprint ) of an action of is an invariant set of all which (H -1 в P ) (HW The set in the subscript to this set. inductor space [T, I ] is said to b e an inductor imprint (an I-imprint, an a group G on a set W if there are an injection H : W T such that HW automorphisms of the space T and a bijection P : (aut[T, I ]/HW ) G for в aut[T, I ]/HW ) = (W : G). In this case, we write aut[T, I ]/HW (W : G). under the slash stands for the restriction of the action of the automorphisms

Theorem 7.1. Every action of a group G on a set W has an inductor imprint. Proof. The construction of the imprint is similar to the construction in Theorem 6.1. The layers Yg corresp ond to elements g of the group G. The cardinality of the ordinal p and of every layer Yg , g G {0}, is equal to m = #W . Introduce some bijections r : W Sp and hg : Yg W . The layers Yg , g G, have rigid induction Hp in the ordering hg . The layer Y0 is equipp ed with the loop induction. Denote by x : g = y , x, y W , g G, the action of an element of a group on an element of the set. The graph J is constructed on T = Yg |g G by the arrows (z, x), where x Y0 and z = z (g, x) = h-1 (h0 x : g-1 ) Yg . To an arrow (z, x), the inductor pair [x, {x; z }]I corresp onds. g This completes the construction of the induction I . If v aut[T, I ] and x Y0 , then the value v (x) must satisfy the condition hg (z (g, x)) = hq (z (q, v (x))) for some layers Yg and Yq . In this case, there is an f = g-1 q for which v (x) = x : f . Here the entire layer Yg is bijectively mapp ed onto the layer Yq b ecause the induction Hp is rigid. Therefore, for any x Y0 , the action of v aut[T, I ] is
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defined by the formula v (x) = x : f . If v1 ,v2 aut[T, I ] by the ab ove formula, then v2 v1 Y0 = h-1 (((h0 0 This formula establishes an isomorphism aut[T, I ] (W with the injection h-1 and a bijection of the group v 0

and if f1 ,f2 G corresp ond to v1 and v2 Y0 ) : f1 ) : f2 ) = h-1 ((h0 Y0 ) : (f1 f2 )). 0 : G). The image of the set W is the set Y0 f.

Theorem 7.2. Any action of the group G on a set has an imprint in the class of topological spaces and in the class of graphs. The proof is similar to the proofs of Theorems 6.3 and 6.2 and uses the construction in the proof of Theorem 7.1. Definition 7.3. By an action W : G (inv = U ) of some group G on a given set W with an invariant system of subsets U (V U V W ) we mean a family Q of self-bijections such that q Q & V U q : W W & q V U and Q is closed with resp ect to sup erp osition and a group isomorphic to G with resp ect to sup erp osition, i.e., there is a bijection P : Q G for which q, q Q P (q (q )) = P (q )P (q ) . In other words, an action of a group (on a set) with an invariant system of subsets is an action of the group on the set such that every self-bijection takes any subset in a system into a subset in the same system. Definition 7.4. By the induced induction I/A on a subset A of an inductor space [T, I ] we mean the set of inductor pairs {[x, B ]I/A | x A, B = C A, [x, C ]I I } . Definition 7.5. An inductor s imprint ) of an action of a group G of the action with an injection H inductors for the induced induction pace [T, I ] on a set W and the im on HW . In is said to b e an inductor imprint with invariant system of subsets U ages of the subsets in U are a gen this case, we write aut[T, I ]/HW (an I-imprint, an if it is an imprint erating system of W : G(inv = U ) .

Lemma 7.1. An action of a group on a set with an invariant system of subsets W : G(inv = U ) is an action W : G(inv = tr(U )), where tr(U ) = {A B | A, B U, A B = } . Proof. Any bijection takes a union of sets into the union of their images and any intersection of sets into the intersection of their images. Therefore, tr(U ) is invariant together with U with resp ect to any system of bijections. Lemma 7.2. Any action of a group G on a set W with an invariant system of subsets U admits an inductor imprint. Proof. Let us use the construction in the proof of Theorem 7.1. The image of HW in this Iimprint coincides with the layer Y0 . This layer is equipp ed with the loop induction. Therefore, this is not a desired imprint in general. Let us add another layer with a nongroup index YW and with the corresp onding bijection hW : YW W for T = Yc |c G {0; W } ; T = Yc |c G {0} . Let us complete the graph J with the arrows (w, x), w YW , x Y0 , hW (w) = h0 (x), where (7.1) w = w(x) = h-1 (h0 (x)). W -1 Here the induced loop induction on the layer Y0 is preserved. Write H = hW . On the layer YW , define the I-pairs of the form [w, V ]I , where V H U and w V . Let us complete them by the inductors of closure by the axiom of transitive union. By construction, the family of inductors of this induction coincides with tr(U ). By Lemma 7.1, this system of inductors is also invariant with resp ect to the action of the group. In this case, the self-bijections of the layer YW corresp ond under the automorphisms of [T, I ] (by the mapping (7.1)) to the bijections of the layer Y0 in the automorphisms of [T ,I ] given in Theorem 7.1. No automorphism can take a p oint of the layer Y0 to a p oint of the layer YW and conversely, b ecause the inductors of the layer YW are disjoint from the layers Yg , g G. If the group G is nontrivial, then the layer YW is not isomorphic to the layers Yg , g G, and they cannot b e mapp ed into one another under automorphisms. Therefore, the automorphism group of the space this constructed is isomorphic to the automorphism group of the space in Theorem 7.1, and hence isomorphic to G. Using the mapping H as an injection H : W T , we obtain the desired imprint. Finally, if a homeomorphism of YW and one of the layers Yg , g G, exists, then the group G is trivial, g = e, b ecause these layers are rigid. In this
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case, one should delete the layer YW from the construction and set H = h-1 . This trivially solves e the problem of imprint for the identity action. Theorem 7.3. Every subgroup G of the homeomorphism group of a topological space (X | ) admits an inductor imprint with an invariant system of open sets. These imprints can be constructed in the class of topological spaces. Proof. The existence of an imprint follows from Lemma 7.2, W = X ; U = ; H = h-1 ; W H = h-1 . One cannot pass to an imprint in the class of graphs in general by using an argument 0 similar to that in Theorem 6.2 b ecause the induction on the layer YW in this construction is a top ological space and its reduction to a graph is imp ossible in general; however, the passage to the class of top ological spaces is similar to Theorem 6.3. To this end, one must extend the family of op en sets obtained in Theorem 6.3 by the family of sets of the form UH = {uV = H V H V | V }, where H V YW , H V Y0 , together with the sets generated by the ab ove sets by the axioms of arbitrary union and finite intersection. Since a top ology is closed with resp ect to these op erations, it follows that there are no new inductors on the layer YW under the canonical mapping of the top ology. On the layer Y0 , under the passage from the loop induction to the discrete top ology, according to Theorem 6.3, all inductors of the form [x, v ]I , v Y0 , where x v , necessarily occur, but these do not influence the automorphism group b ecause they are invariant with resp ect to any bijection. The new induced inductors of the form H V b elong to this class of subsets and do not change the induced induction [T, I ]/Y0 . Therefore, the set of inductors [T, I ]/Y0 YW thus obtained is invariant with resp ect to the action of automorphisms representing the group G. The induced induction is preserved on the other layers. Therefore, if the layer YW is not homeomorphic to any layer of the form Yg , g G, then the prop erty of inductor imprint of the action X : G(inv ) is preserved. The case in which the layer YW is homeomorphic to one of the layers Yg , g G, is treated as in the proof of Lemma 7.2. 8. SPACES WITH CONIC INDUCTION Definition 8.1. By a conic space Rc[n] we mean a space Rn with an induction for which the generating system of inductors of a p oint y are cones representable in a system of Cartesian coordinates x1 ,... ,xn in the form 0 (x1 - y1 )2 - (x2 - y2 )2 - ··· - (xn - yn )2 , 0 y1 - x1 H, H > 0. The assumption |y1 - x1 | H leads to the so-called biconic space Rb[n], the condition 0 y1 - x1 to the so-called ful l conic space Rfc[n], and, if no conditions on y1 - x1 are imp osed, then the space is referred to as a ful l biconic space Rfb[n]. The corresp onding automorphism groups are denoted, according to the structure of inductors, by aut Rc[n], etc. A conic space can b e regarded as a vector space equipp ed with a system of spherical cones obtained as all p ossible parallel shifts of a chosen cone. In biconic spaces, the cones are two-sided with resp ect to the vertex. In nonfull spaces, along with full cones, cones of b ounded height are also considered. The parameter H defines this condition for a sp ecific cone in the generating set. The transitive union can produce diverse b ounds of nonfull cones. The conic inductor spaces do not b elong to classes of graphs or top ological spaces. Remark 8.1. Using the axiom of transitive union, one can readily show that aut Rc[n] = aut Rfc[n] and aut Rb[n] = aut Rfb[n]. Therefore, it suffices to study the automorphism groups for the classes of full spaces only. Distinguishing of the ab ove four classes of spaces of conic typ e is related to applications to mathematical modeling. The use of b ounded cones (with the parameter H ) defines a top ology on the time axis in models of mathematical physics [8, 12]. However, this is inessential for the purp oses of the present pap er. In what follows, we consider only full conic and biconic spaces. We are interested in their automorphism groups. As was proved in [8], the affine automorphism group of a conic space coincides with the canonical action of the Lorentz group (on the Minkowski space of corresp onding dimension) extended by all parallel translations, rotations, and reflections in the hyp erplane of the spherical section of the cone, and also by uniform dilations (multiplications of vectors by a p ositive numb er). Below we refer to this action as an affine extension of the (action of the) Lorentz group and denote it by ALor[n] and denote the abstract group by itself by GAL[n].
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Below we show that, for the dimensions 1 and 2, the full automorphism group is significantly larger than its affine subgroup, whereas the full automorphism group coincides with the affine subgroup b eginning with the dimension 3. Denote by the sup erp osition of two actions on a set, by в the direct product of groups, by the direct product of actions on the direct product of the corresp onding sets (in the linear space this is reduced to the direct sum of the subspaces), and by Gaut[T, I ] an abstract group isomorphic to the automorphism group of the space. Theorem 8.1. The group aut Rc[1] is formed by al l positively monotone bijections of R1 , the group aut Rc[2] is ((aut Rc[1] : Z1 ) (aut Rc[1] : Z2 )) h2 (2), where Z1 and Z2 stand for two generators of the two-dimensional cone, the action hn (i), i n, on Rn is the inversion (the multiplication by -1) of the xi axis, and, in particular, h2 (2) makes a permutation of the generators of the cone on R2 , the group Gaut Rc[2] is Gaut Rc[1] в Gaut Rc[1] в S2 , where S2 = Ghn (i) stands for the group of order two, aut Rc[n] = ALor[n] for n 3, and Gaut Rc[n] = GAL[n] for n 3. Theorem aut Rc[1] Rh respect to the Gaut Rb[2] = Gaut Rb[n] = 8.2. The group aut Rb[1] is formed by al l monotone bijections of R1 , aut (1), GRb[1] = GRc[1] в S2 , where the action of Rh (n) is the reflection of origin (the multiplication of vectors by -1), aut Rb[2] = aut Rc[2] h2 (1) Gaut Rc[1] в Gaut Rc[1] в S2 в S2 , aut Rb[n] = ALor[n] Rh (n) for n GAL[n] в S2 for n 3. Rb[1] = Rn with h2 (2), 3, and

Proof. The actions of the groups aut Rb[n] and aut Rc[n] differ only by the direct additional multiplication by the inversion of the axis of the cone, which corresp onds to the coordinate x1 (this is conditioned by the corresp onding symmetry of the generating system of biconic induction). Therefore, Theorem 8.2 immediately follows from Theorem 8.1. The relations for the abstract groups immediately follow from relations for the actions of automorphism groups. For this reason, it suffices to carry out the proof of the relations for the automorphisms of the conic induction. Case n = 1. In this case, the conic space is the "directed line:" a neighb orhood of a p oint is the left half-line in which this p oint is the right end. The continuous mappings in this induction are left continuous functions. An automorphism which is a continuous self-bijection, can b e only an arbitrary strictly monotone continuous real function which is unb ounded in all directions. Case n = 2. In this case, the cone (the inductor of a p oint) is an angle with a vertex at the p oint. The generators of the cone are the sides of the angle. The corresp onding generators of all cones (the inductors) are parallel. Under an automorphism, any generator of any cone passes to a generator of the image of the cone. The mapping can take a generator either to the corresp onding generator or to the opp osite one. Let us pass to the system of coordinates whose axes are parallel to the generators with the origin at some p oint (0, 0) and with directions e and t. On the lines of these vectors, the induction is induced by the directed lines. If U is an automorphism, then either U (x, y ) = U (xe + yt) = U (0, 0) + V (x)e + W (y )t, where V and W are automorphisms of the directed lines or U (x, y ) = U (xe + yt) = U (0, 0) + V (x)t + W (y )e, where V and W are mutual homeomorphisms of the lines. With regard to the ab ove set of automorphisms of a directed line, this corresp onds to the assertion of the theorem. Case n = 3. To prove the assertion of the theorem, it suffices to show that all automorphisms are affine, i.e., that the class of lines is preserved under the action (this is really sufficient b ecause the group of affine automorphisms of Rc[n] coincides with the extended action of the Lorentz group [8]), or, in other words, that every automorphism takes any line to a line. Introduce some notions. An I-cone is a cone which is the union of all inductors of a single p oint in Rc[3] (the inductor of a full conic induction). A C-line is a line continuing a generator of an I-cone. A C-plane is a plane tangent to some I-cone. A T-line is a line partially b elonging to the interior of an I-cone. A T-plane is a plane formed by lines parallel to a T-line. A P-plane is a secant plane for an I-cone. A P-line is a line b elonging to a P-plane. Corresp ondingly, we generally sp eak of C-objects, T-objects, and P-objects. Lemma 8.1. Any automorphism of a conic space Rc[n] is continuous in the topology of Rn . Proof. Supp ose that a sequence of p oints r (1),r (2),... converges to a p oint r. In this case, one can construct a sequence of emb edded I-cones (K (i)|i = 1,... ) with vertices at some p oints g(1), g(2), ... , and with heights H (1), H (2), ... , where limi g(i) = r and limi H (i) = 0 and all
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points r (j ), the p oint r. emb edded s u(r (j )), j

j i, b elong to the interior of the cone K (i). The only common p oint of all cones is Applying an arbitrary automorphism u to the I-cones K (1), K (2), ... , we obtain an ystem of I-cones (u K (i)|i = 1,... ) with a single common p oint u(r ). Here the images i, are placed inside the I-cone u K (i). This implies that limi u(r (i)) = u(r ).

Lemma 8.2. Every P-line is the intersection of two C-planes. Proof. Consider a P-line L. Choose a p oint r on L and place the vertex of the I-cone K having no other intersection p oints with L there. Draw two distinct tangent planes to the cone that contain L (recall that the dimension of the space is not less than three). Since every plane tangent to an I-cone is a C-plane, this will prove the lemma. Let us prove that these planes always exist for a cone and for a line passing through the vertex from outside. Consider the plane V of the circular cross-section of the cone K. Two cases are p ossible: the line L can b e either parallel or not parallel to V. If the line L is parallel to the plane, then it is orthogonal to the axis of the cone. In this case, there are two generators of the cone K , which are orthogonal to the line L. The planes spanned by these generators and the line L are the desired tangent planes. If the line is not parallel to the plane, then they have an intersection p oint, which we denote by s. Let the circle Z b e the section of the cone K by the plane V. Draw two tangents from the p oint s to the circle Z in the plane V. Denote the p oints of tangency by a and b. Each of the lines sa and sb is tangent to the cone K in one of these p oints. The lines ra and rb are generators of the cone. Therefore, each of the planes of the triangles asr and bsr contains a generator and a tangent of the cone K, and these lines intersect. Hence, these are tangent planes to the cone, and they meet along the line sr = L. Lemma 8.3. Any automorphism takes any C-object to a C-object and any P-object to a P-object of the same type. Any T-line is taken into a continuous line intersecting the interior of the I-cone (we do not claim here that this image is a line ). Proof. By the definition of an automorphism, the image and the preimage of any I-cone is an I-cone. Since any bijection is strictly monotone with resp ect to the emb edding of subsets, the b oundary (the surface) of any I-cone is taken to the surface of the image. Every C-line is a line of tangency of two I-cones one of which is placed inside the other (tangency along a generator). This is an intersection of two surfaces of I-cones, which passes to a similar tangency, i.e., the image of any C-line is a C-line again. In this case, every C-plane is taken to a continuous set consisting of disjoint C-lines. Choose an arbitrary p oint r on an arbitrary C-plane B and draw a C-line L through r along which the plane B is tangent to the cone K . Each line of this kind of the C-plane is a tangent line of infinitely many I-cones whose vertices b elong to L. Consider the set QL of all I-cones of this kind. Their union KL = QL is a half-space b ounded by the plane B. Under our automorphism, this union of cones is taken to a similar union, and the b oundary of the half-space is taken to the b oundary of the image. Hence, the image of any C-plane is a C-plane. Thus, any automorphism takes any C-ob ject to a C-ob ject of the same typ e. By Lemma 8.2, the image of a P-line passing through the vertex of an I-cone K is the intersection of the images of two C-planes, and, as was proved ab ove, this intersection is a P-line b ecause it can b e represented as the intersection of two C-planes external with resp ect to the I-cone given by the image of K . Since all lines on a P-plane are P-lines, it follows that any automorphism takes all lines on a P-plane into lines. Any three P-lines whose intersections define a triangle are taken by any automorphism to a similar pattern. By the continuity (Lemma 8.1) and the bijective prop erty of any automorphism, the corresp onding triangle defines a plane which is the image of the original plane. All lines in the image are P-lines, and hence the image is a P-plane. The interior of an I-cone passes under any automorphism to the interior of the target I-cone. Therefore, it follows from Lemma 8.1 that any T-line is taken to a continuous curve passing through the interior of the target I-cone. Lemma 8.4. The automorphisms are affine on the C-objects and the P-objects. Proof. For any P-plane, this was proved in the proof of Lemma 8.3. Since every P-line b elongs to some P-plane, it follows that the automorphism is affine on the line. On any C-plane tangent to an I-cone (and hence not contained in the I-cone), all lines are either C-lines or P-lines. Each of these lines is taken by any automorphism to a line. Thus, the automorphism is affine on any C-plane. Since every C-line is contained to some C-plane, it follows that the mapping is affine on any C-line.
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Lemma 8.5. Any T-object is taken to a T-object of the same type under any automorphism u of a conic space. Proof. Consider an arbitrary T-plane P . By definition, there is an I-cone K restricted with resp ect to height by a circular section Z , and the intersection P K consists of two generatrices L1 and L2 and the vertex A. The intersection L3 = P Z is a P P-line. By Lemma 8.4, uL1 and uL2 are generatrices of the I-cone uK which meet at the vertex uA of uK , and the image of the P -line uL3 is a line and b elongs to the plane uZ. Since the entire plane P can b e represented as the union of C -lines parallel to L1 and intersecting L2 and L3 , it follows that uP is a T-plane. Thus, the image of an arbitrary T-plane is a T-plane. Every T-line is the intersection of two T-planes and meets the interior of any I-cone whose vertex b elongs to the line. The prop erty to enter the interior of any cone of this kind is an invariant of any automorphism. The intersection of two T-planes is taken to the intersection of their image planes. Hence, the image of an arbitrary T-line is a T-line. End of the pro of of Theorem 8.1. It follows from Lemmas 8.4 and 8.5 that, if the dimension of the space is equal to 3, then all automorphisms preserve the class of lines. This means that all the automorphisms are affine transformations of the vector space. Let n > 3. Note that the proof for n = 3 did not use any mapping of the conic space onto itself. Moreover, we have proved that a homeomorphism of any three-dimensional conic space is an affine mapping. Supp ose that the desired assertion is proved for n = m 3. Consider the case n = m + 1 and proceed by induction. Choose a basis: Rn = L{e0 ,e1 ,... ,em }, where e0 is the axis of the I-cone. One can write out the algebraic sum Rn = L{e0 ,e1 ,... ,em-1 } L{e0 ,e1 ,... ,em-2 ,em }. Each of the summands is an m-dimensional conic space. Under any automorphism (or isomorphism) u, by the induction assumption, the images of these subspaces are subspaces of the same typ e, and the mappings are affine. The dimension of the space is preserved b ecause the mapping u is bijective. If a line l Rn is given, then we can always choose e1 and e2 in such a way that l L{e0 ,e1 ,e2 }, and hence l L{e0 ,e1 ,... ,em-1 } (this is achieved, for instance, by a successive orthogonalization of the triple of vectors e0 , v , w - v, where w and v are any two distinct p oints on the line l). Therefore, the image of ul is a line. This completes the induction, and thus proves the theorem. REFERENCES
1. D. Koning, Theorie der end lichen und unend lichen Graphen (Leipzig, 1936). 2. R. Frucht, "Herstellung von Graphen mit vorgegebener abstrakten Gruppe," Compos. Math. 6, 239250 (1938). 3. R. Frucht, "Graphs of Degree Three witha Given Abstract Group," Canad. J. Math. 1, 365378 (1949). 4. G. Sabidussi, "Graphs with Given Group and Given Graph-Theoretical Properties," Canad. J. Math. 9, 515525 (1957). 5. G. Sabidussi, "On the Minimum Order of Graphs with Given Automorphism Group," Monatsh. Math. 63, 693696 (1959). 6. H. Izbicki, "Unendliche Graphen endlichen Grades mit vorgegebener Eigenschaften," Monatsh. Math. 63, 298301 (1959). 7. A. V. Koganov, "Inductor Spaces and Processes," Dokl. Akad. Nauk 324 (5), 953958 (1992) [Soviet Phys. Dokl. 37 (6), 275278 (1992)]. 8. A. V. Koganov, "Processes and Automorphisms on Inductor Spaces," Russ. J. Math. Phys. 4 (3), 315339 (1996). 9. A. V. Koganov, "Inductor Spaces as a Tool of Modeling," in Problems in Cybernetics (Algebra, Hypergeometry, Probability, Modeling ) (Ed. by V. B. Betelin) (RAS, Moscow, 1999), pp. 119181. 10. A. V. Koganov, "Automorphisms of Conical Inductor Spaces," in Problems in Cybernetics (Algebra, Hypergeometry, Probability, Modeling ) (Ed. by V. B. Betelin) (RAS, Moscow, 1999), pp. 182189. 11. A. V. Koganov, "Intrinsic Images of the Action of Groups on Inductor Spaces," in Mathematics. Computer. Education. Col lection of Scientific Works, No. 8, Part 2 (Progress-Traditsiya, Moscow, 2001), pp. 489495. 12. A. V. Koganov, "Local Operators and Differentiation on Inductor Spaces," in Mathematics. Computer. Education. Col lection of Scientific Works, No. 9, Part 2 (R&C-Dynamics, MoscowIzhevsk, 2002), pp. 373384.

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No. 1

2008