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Isomorphic Cayley Graphs on Non-Isomorphic Groups) UNIVERZA NA PRIMORSKEM FAKULTETA ZA MATEMATIKO, NARAVOSLOVJE IN INFORMACIJSKE TEHNOLOGIJE Matematiˇcneznanosti Studijskiˇ program 2. stopnje Mary Agnes SERVATIUS Izomorfni Cayleyevi grafi nad neizomorfnimi grupami (Isomorphic Cayley Graphs on Non-Isomorphic Groups) Magistrsko delo (Master's Thesis) Mentor: izr. prof. dr. Istv´anKov´acs Koper, 2014 Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 ii Kljuˇcnadokumentacijska informacija Ime in PRIIMEK: Mary Agnes SERVATIUS Naslov magistrske naloge: Izomorfni Cayleyevi grafi nad neizomorfnimi grupami Kraj: Koper Leto: 2014 Steviloˇ listov: 56 Steviloˇ slik: 27 Steviloˇ referenc: 18 Mentor: izr. prof. dr. Istv´anKov´acs UDK: Kljuˇcnebesede: Cayleyjev graf, cikliˇcengraf, isomorfnost grafov, grupa avtomorfizmov grafov, leksikografiˇcenprodukt grafov, konˇcnaabelova grupa, permutacijska grupa, venˇcniprodukt permutacijskih grup, loˇcnotranzitiven graf. Math. Subj. Class. (2010): 05C25 20B25 20B25 05C60 Izvleˇcek V magistrskem delu obravnavamo Cayleyeve grafe in grupe. Poseben poudarek je namenjen problemu Cayleyevih izomorfizmov. Poglavje 1 vsebuje uvod v glavni izrek, ki poda potrebni in zadostni pogoj za izomorfnost dveh Cayleyevih digrafov X1 = cay(G1;S1) in X2 = cay(G2;S2), kjer sta G1 in G2 neizomorfni abelski 2- grupi, digrafa X1 in X2 pa imata regularno cikliˇcnogrupo avtomorfizmov. Omenjeni rezultat je razˇsiritevrezultata Morrisove (glej J. Graph Theory 3 (1999), 345{362) v zvezi s p-grupami Gi, kjer je p liho praˇstevilo. Poglavje 2 vsebuje osnovne definicije in potrebno predpripravo za kasnejˇsapoglavja. V 3. poglavju so obravnavani izbrani primeri. V 4. poglavju je podan dokaz glavnega izreka. Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 iii Key words documentation Name and SURNAME: Mary Agnes SERVATIUS Title of Masters degree: Isomorphic Cayley Graphs on Non-isomorphic Groups Place: Koper Year: 2014 Number of pages: 56 Number of figures: 27 Number of references: 18 Mentor: Assoc. Prof. Istv´anKov´acs,PhD UDK: Key words: Cayley graph, cyclic graph, isomorphic graphs, automorphism groups of graphs, lexicographic products of graphs, elementary abelian groups, permutation groups, wreath product, arc transitive graph Math. Subj. Class. (2010): 05C25 20B25 20B25 05C60 Abstract In this thesis we explore Cayley graphs and groups, in particular, we are concerned with the Cayley isomorphism problem. Chapter 1 contains a high level introduction to the main theorem, in which a nec- essary and sufficient condition is given for two Cayley digraphs X1 = Cay(G1;S1) and X2 = Cay(G2;S2) to be isomorphic, where G1 and G2 are non-isomorphic abelian 2-groups, and the digraphs X1 and X2 have a regular cyclic group of au- tomorphisms. This result extends that of Morris (see J. Graph Theory 3 (1999), 345{362) concerning p-groups Gi, where p is an odd prime. Chapter 2 introduces some preliminaries, and Chapter 3 provides a selection of examples. Chapter 4 contains a proof of the main theorem. Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 iv Contents 1 Introduction 1 2 Preliminaries 4 2.1 Graphs . .4 2.1.1 Basic definitions . .4 2.2 Groups . .6 2.2.1 Basic definitions . .6 2.2.2 Automorphism groups . .8 2.2.3 Permutation groups and group actions . 11 2.2.4 Imprimitivity block systems . 13 2.2.5 Wreath products of groups . 14 2.3 Cayley graphs . 17 2.3.1 Arc transitive Cayley digraphs on Cyclic groups . 19 2.3.2 CI-Graphs . 19 3 Non-isomorphic groups with isomorphic Cayley graphs. 22 3.1 Infinite graphs and free groups . 23 3.2 Graphs of the Platonic solids . 27 4 Proof of the Main Theorem 34 4.1 W-subgroups of abelian groups . 34 4.2 Proof of Theorem 4.1.4 . 37 5 Conclusions 48 Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 v Bibliography 49 Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 vi Acknowledgements Thanks are due to Dragan Marusiˇcfor this opportunity, to my mentor Istv´anKov´acs for his patience, and to my father, Herman Servatius, who taught me everything I know. Thank you Papa, for letting me peek at the vast explored and unexplored territories of mathematics, for going on random walks with me on objects in space, and most of all for distracting me in my most difficult times with your apparently endless supply of mathematical treats. It does not escape my notice that the time, energy and money spent on this project vastly outweigh its value. Many years ago, when I was much too young and naive to be suspicious of a grand promise, I signed a contract which effectively forced me to work for years at a job to which I was completely unqualified for less than nothing. My anger over this situation is the only thing which shields me from my grief and humiliation. I am not sure if I will ever completely recover. Work on this thesis was generously supported by grant TI508-7529165 from the Trowbridge Institute, Massachusetts, USA. My maiden name is Mary Agnes Franziska Sophie Servatius, and my married name is Mary Milaniˇc. The name on the cover of this thesis does not reflect the legal status of my name in the countries of which I am a citizen, although it is being used to refer to me. Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 1 Chapter 1 Introduction A necessary and sufficient condition is given for two Cayley digraphs X1 = Cay(G1;S1) and X2 = Cay(G2;S2) to be isomorphic, where the groups Gi are nonisomorphic abelian 2-groups, and the digraphs Xi have a regular cyclic group of automorphisms. Our result extends that of Morris (see J. Graph Theory 3 (1999), 345{362) concern- ing p-groups Gi, where p is an odd prime. Let G be a finite group with identity element e.A Cayley digraph Cay(G; S) on G generated by a connection set S ⊆ G; e2 = S; is a digraph with the vertex set G and the arc set f(x; y) j x−1y 2 Sg. In this paper we consider the problem of finding efficient necessary and sufficient conditions for two Cayley digraphs Cay(G1;S1) and Cay(G2;S2) to be isomorphic, such that the groups Gi are nonisomorphic. Here we restrict to the special case that Gi are p-groups. For an overview on the general problem we refer to the survey paper by Li [12]. 2 The simplest case is that the groups Gi have order p , and this has been charac- n terized by Joseph in [10]. Joseph's result was generalized to jGij = p by Morris in [16] as follows. Recall first that, for two digraphs X1 and X2, the wreath product X1 o X2 is the digraph with vertex set V (X1) × V (X2), and an arc from (x1; x2) to (y1; y2) if and only if (x1; y1) is an arc of X1, or x1 = y1 and (x2; y2) is an arc of X2. We remark that the wreath product Aut(X1) o Aut(X2) acts as a group of automorphisms of X1 o X2, explaining also the term wreath product. Here we follow the convention that Aut(X1) o Aut(X2) has active group Aut(X1) and passive group Aut(X2); i.e., Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 2 jX1j the order j Aut(X1) o Aut(X2)j = j Aut(X1)j · j Aut(X2)j . Second, the set of all abelian groups of a fixed prime-power order pn is partially n ordered by the relation ≤po; where for two abelian groups G and H of order p , G ≤po H if and only if there is chain H1 < H2 < ··· < Hm = H of subgroups of H such that 1. H1;H2=H1;:::;Hm=Hm−1 are all cyclic, and ∼ 2. G = H1 × H2=H1 × · · · × Hm=Hm−1: The main result of Morris is the following (see [16, Theorem 1.1]). Theorem 1.1.1. Let X = Cay(G; S) be a Cayley digraph on an abelian group G of order pn, where p is an odd prime. Then the following are equivalent: (1) The digraph X is isomorphic to a Cayley digraph on both Zpn and H, where H is an abelian group with jHj = pn. (2) There exists a chain of subgroups G1 < ··· < Gm−1 in G such that (i) G1, G2=G1; : : : ; G=Gm−1 are cyclic groups, (ii) G1 × G2=G1 × · · · × G=Gm−1 ≤po H, (iii) for each 1 ≤ i ≤ m − 1, S n Gi is a union of Gi-cosets. (3) There exist Cayley digraphs X1;:::;Xm on cyclic p-groups H1;:::;Hm such ∼ that H1 × · · · × Hm ≤po H and X = Xm o · · · o X1. Moreover, Theorem 1.1.1 with the assumption G = H was shown to be true for any (not necessarily abelian) group G of order pn (see [16, Theorem 6.1]), as well as when p = 2 and H is an elementary abelian group of order 2n (see [16, Theorem 5.1]). Our goal in this thesis is to extend Theorem 1.1.1 to the case when p = 2, that is, to show the following theorem. Servatius, M. Isomorphic Cayley graphs on non-isomorphic groups Univerza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologije, 2014 3 Theorem 1.1.2. Let X = Cay(G; S) be a Cayley digraph on an abelian group G of order 2n.
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