There are different ways to associate to a finite group a certain graph. An interesting question is to analyse the relations between the structure of the group, given in group-theoretical terms, and the structure of the graph, given in the language of graph theory. This survey paper presents some contributions to this research line

Ballester-Bolinches, Adolfo

Cossey, John

Esteban Romero, Ramón

There are different ways to associate to a finite group a certain graph. An interesting question is to analyse therelations between the structure of the group, given in group-theoretical terms, and the structure of the graph, given in the language of gra

Cossey, Peter (John)

Esteban-Romero, R

The Australian National University

Note di Matematica ISSN 1123-2536, e-ISSN 1590-0932Note Mat. 33 (2013) no. 1, 89–94. doi:10.1285/i15900932v33n1p89Graphs and Classes of Finite GroupsA. Ballester-BolinchesDepartament d’Àlgebra, Universitat de València, Dr. Moliner, 50, 46100 Burjassot,València, Spainadolfo.ballester@uv.esJohn CosseyDepartment of Mathematics, Mathematical Sciences Institute, Australian NationalUniversity, Canberra, ACT 0200, Australiajohn.cossey@anu.edu.auR. Esteban-RomeroDepartment de Matemàtica Aplicada, Universitat Politècnica de València, Camı́ de Vera s/n,46022 València, Spainresteban@mat.upv.esAbstract. There are different ways to associate to a finite group a certain graph. Aninteresting question is to analyse the relations between the structure of the group, given ingroup-theoretical terms, and the structure of the graph, given in the language of graph theory.This survey paper presents some contributions to this research line.Keywords: finite groups, classes of groups, graphsMSC 2000 classification: primary 20D10, secondary 05C25All groups in this paper are finite. We will consider only graphs which areundirected, simple (that is, with no parallel edges), and without loops. Thesegraphs will be characterised by the set of vertices and the adjacency relationbetween the vertices. Only basic concepts about graphs will be needed forthis paper. They can be found in any book about graph theory or discretemathematics, for example [12].Given a group G, there are many ways to associate a graph with G bytaking families of elements or subgroups as vertices and letting two vertices bejoined by an edge if and only if they satisfy a certain relation. We may askabout characterising group structural properties by means of the properties ofthe associated graph. In recent years, there has been considerable interest inthis line of research (see [1, 2, 3, 5, 7, 10]).If X is a class of groups, Delizia, Moravec, and Nicotera [10] associate agraph ΓX(G) with a group G by taking the nontrivial elements of G as verticesand letting a, b ∈ G\{1} be joined by an edge if 〈a, b〉 ∈ X. If we chooseX = A, where A is the class of all abelian groups, then the graph is just thecommuting graph of G and so we may think of this graph as a generalisation ofhttp://siba-ese.unisalento.it/ c© 2013 Università del Salentothe commuting graph. This graph has been used to study simple groups sincethe paper of Stellmacher [18].For an arbitrary class, the graph considered by Delizia, Moravec, and Nico-tera is too general to give much information and so we restrict the classes weconsider. We suppose that our class X is subgroup closed and contains A.Delizia, Moravec, and Nicotera restrict their attention to groups which areX-transitive. A group G is said to be X-transitive, or an XT -group, if whenever〈a, b〉 ∈ X and 〈b, c〉 ∈ X, then 〈a, c〉 ∈ X (a, b, c ∈ G). Note that the A-transitive groups are just the CT -groups, or groups in which the centraliser ofany non-identity element is an abelian subgroup. CT -groups are of historicalimportance as an early example of the type of classification that would beused in the FeitThompson theorem and the classification of simple groups andwere classified by several mathematicians (Weisner [20], Suzuki [19], Wu [23]).Characterisations of XT -groups are known for several classes of groups (see [11]for a survey).Let G be an XT -group. If C is a connected component of ΓX(G), then Cis a complete graph, C ∪ {1} is a subgroup of G and {C ∪ {1} : C a connectedcomponent of ΓX(G)} is a partition of G ([17] p.145). As a consequence of theclassification of groups with a partition (see for example [17, Theorems 3.5.10and 3.5.1]) we have the following result.Theorem 1. Let G be an XT -group.(1) The connected components of ΓX(G) form a normal partition of G\{1},that is, conjugates of connected components are again connected compo-nents.(2) Either ΓX(G) is connected or G is one of the following groups: a Frobeniusgroup, PSL(2, 2h), Sz(2h) (a Suzuki group, h = 2k + 1 > 1).Note that a group in the partition has the property that every 2-generatorsubgroup is an X-group. In general this will not ensure that the group is anX-group. For instance, N2, the class of (finite) nilpotent groups of class at most2, is an example, since the free group of exponent 3 on 3 generators has class3, but every 2-generator subgroup has class at most 2 (see [16]). A class X ofgroups has been called 2-recognisable (2-erkennbar) by Brandl [9] if a group Gis in X if and only if every 2-generator subgroup of G is in X.Observe that G is an XT -group and ΓX(G) is connected, then every 2-generator subgroup of G is an X-group and so if X is 2-recognisable then G ∈ X.Many classes of groups are known to be 2-recognisable; abelian, nilpotent,supersoluble and soluble among them. Some of the classes of supersolublegroups that have been extensively investigated in recent years are 2-recognisable.90We consider here the following classes: the class T of soluble T -groups (solu-ble groups in which normality is transitive), the class PT of soluble PT -groups(soluble groups in which permutability is transitive) and the class PST of sol-uble PST -groups (soluble groups in which Sylow permutability is transitive).That each of these classes is subgroup closed, it is contained in the class of allsupersoluble groups and contains all abelian groups is well known (a descriptionof these classes and their properties can be found in [6]).Lemma 2. The classes T, PT, and PST are all 2-recognisable.We now define D to be the class of groups G with all Sylow subgroups cyclic(these are just the groups with a cyclic normal subgroup whose quotient is cyclicand whose order and index are coprime, see [24, Theorem V.3.11]). We thendefine FX to be the class of Frobenius groups with the property that the kernelK is an X-group, G/K ∈ D and for each prime p | |G/K| and each prime q | |K|,p does not divide q − 1.Theorem 3. Let X be a 2-recognisable class of soluble groups containingA ∪D. Then the class of all XT -groups is contained in X ∪ FX.Theorem 4. Let X be a 2-recognisable class of soluble groups containingA ∪D and G be an XT -group.(1) G ∈ X if and only if ΓX(G) is connected.(2) G 6∈ X if and only if G is a Frobenius group and K\{1} is a connectedcomponent of ΓX(G) (where K is the kernel of G).Bearing in mind the above results, it is natural to ask for the smallest 2-recognisable class of groups containing A ∪D.Proposition 5. Let T0 be the class of T-groups G with G/G′ cyclic or Gabelian. Then T0 is the smallest 2-recognisable class of groups containing A∪D.The following variation of the commuting graph gives a characterisation forthe groups in which all subgroups are permutable. We will call it the graphof permutability of cyclic subgroups (see [4]). Given a group G, consider thegraph in which the vertices are the cyclic subgroups of G and in which everytwo vertices are adjacent when they permute. A group has all subgroups per-mutable if and only if the graph of permutability of cyclic subgroups is complete.A related graph whose vertices are the non-normal subgroups was studied byBianchi, Gillio, and Verardi (see [7, 8, 13]).The prime graph of a group has also attracted the attention of many re-searchers. The vertices of this graph are the prime numbers dividing the orderof the group G and two different vertices p and q are connected if and onlyif G has an element of order pq. For instance, in the cyclic group of order 6,91this graph is complete, but in the symmetric group of degree 3, this graph con-sists of two isolated vertices. The first references known to the authors of thisgraph correspond to Gruenberg and Kegel, in an unpublished manuscript, andto Williams, who studied the number of connected components of the primegraph of a finite group (see [14, 21, 22]). Abe and Iiyori studied in [2] a general-isation of the prime graph in the following way: given a group G, they constructthe graph ΓG whose vertices are the prime numbers dividing the order of G andin which given two different vertices p and q, they are adjacent if and only if Gpossesses a soluble group of order divisible by pq. Abe and Iiyori [2] proved:Theorem 6. If G is a non-abelian simple group, then ΓG is connected, butnot complete.Herzog, Longobardi, and Maj [15] have considered the graph whose verticesare the non-trivial conjugacy classes of a group G and in which two non-trivialconjugacy classes C and D of G are adjacent if and only if there exists c ∈ Cand d ∈ D such that cd = dc. They show that if G is a soluble group, thenthis graph has at most two connected components, each of diameter at most 15.They also study the structure of the groups for which there exist no edgesbetween non-central conjugacy classes and the relation between this graph andthe prime graph.In [5] (see also [4]), the authors define for a group G a graph Γ(G) whosevertices are the conjugacy classes of cyclic subgroups of G and in which twovertices ClG(〈x〉) and ClG(〈y〉) are adjacent if and only if we can find an elementg ∈ G such that 〈x〉 permutes with 〈yg〉. The main result of [5] is:Theorem 7. A group G is a soluble PT-group if and only if the graph Γ(G)is complete.AcknowledgementsThe first author and third author have been supported by the research grantMTM2010-19938-C03-01 from MICINN (Spain).References[1] A. Abdollahi, S. Akbari, and H. R. Maimani. Non-commuting graph of a group. J.Algebra, 298(2):468–492, 2006.[2] S. Abe and N. Iiyori. A generalization of prime graphs of finite groups. Hokkaido Math.J., 29(2):391–407, 2000.[3] A. Ballester-Bolinches and J. Cossey. Graphs, partitions and classes of groups. Monatsh.Math., 166:309–318, 2012.92[4] A. Ballester-Bolinches, J. Cossey, and R. Esteban-Romero. A characterization via graphsof the soluble groups in which permutability is transitive. Algebra Discrete Math., 4:10–17,2009.[5] A. Ballester-Bolinches, J. Cossey, and R. Esteban-Romero. On a graph related to per-mutability in finite groups. Ann. Mat. Pura Appl., 189(4):567–570, 2010.[6] A. Ballester-Bolinches, R. Esteban-Romero, and M. Asaad. Products of finite groups,volume 53 of de Gruyter Expositions in Mathematics. Walter de Gruyter, Berlin, 2010.[7] M Bianchi, A. Gillio, and L. Verardi. Subgroup-permutability and affine planes. Geome-triae Dedicata, 85:147–155, 2001.[8] M. Bianchi, A. Gillio Berta Mauri, and L. Verardi. Finite groups and subgroup-permutability. Ann. Mat. Pura Appl., IV. Ser., 169:251–268, 1995.[9] R. Brandl. Zur Theorie der untergruppenabgeschlossenen Formationen: Endlichen Va-rietäten. J. Algebra, 73:1–22, 1981.[10] C. Delizia, P. Moravec, and C. Nicotera. Finite groups in which some property of two-generator subgroups is transitive. Bull. Austral. Math. Soc., 75:313–320, 2007.[11] C. Delizia, P. Moravec, and C. Nicotera. Transitivity of propeties of 2-generator sub-groups. In M. Bianchi, P. Longobardi, M. Maj, and C. M. Scoppola, editors, IschiaGroup Theory 2008. Proceedings of the Conference. Naples, Italy, 1–4 April 2008, pages68–78, Singapore, 2009. World Scientific.[12] R. Diestel. Graph theory, volume 173 of Graduate Texts in Mathematics. Springer-Verlag,Berlin, third edition, 2005.[13] A. Gillio Berta Mauri and L. Verardi. On finite groups with a reducible permutability-graph. Ann. Mat. Pura Appl., IV. Ser., 171:275–291, 1996.[14] K. W. Gruenberg. Free abelianised extensions of finite groups. In C. T. C. Wall, editor,Homological group theory. Proc. Symp., Durham 1977, volume 36 of Lond. Math. Soc.Lect. Note Ser., pages 71–104, Cambridge, 1979. London Mathematical Society, Cam-bridge Univ. Press.[15] M. Herzog, P. Longobardi, and M. Maj. On a commuting graph on conjugacy classes ofgroups. Comm. Algebra, 37(10):3369–3387, 2009.[16] H. Neumann. Varieties of Groups. Springer Verlag, New York, 1967.[17] R. Schmidt. Subgroup lattices of groups, volume 14 of De Gruyter Expositions in Mathe-matics. Walter de Gruyter, Berlin, 1994.[18] B. Stellmacher. Einfache Gruppen, die von einer Konjugiertenklasse vno Elementen derOrdnung drei erzeugt werden. J. Algebra, 30:320–354, 1974.[19] M. Suzuki. The nonexistence of a certain type of simple groups of odd order. Proc. Amer.Math. Soc., 8:686–695, 1957.[20] L. Weisner. Groups in which the normaliser of every element except identity is abelian.Bull. Amer. Math. Soc., 31:413–416, 1925.[21] J. S. Williams. The prime graph components of finite groups. In B. Cooperstein andG. Mason, editors, The Santa Cruz Conference on Finite Groups (Proc. Symp. PureMath., June 25–July 20, 1979, volume 37 of Proc. Symp. Pure Math., pages 195–196,Providence, RI, USA, 1980. American Mathematical Society (AMS), American Mathe-matical Society.[22] J. S. Williams. Prime graph components of finite groups. J. Algebra, 69:487–513, 1981.93[23] Y.-F. Wu. Groups in which commutativity is a transitive relation. J. Algebra, 207:165–181, 1998.[24] H. J. Zassenhaus. The theory of groups. Chelsea Publishing Company, New York, secondedition, 1958.94

English

Graphs and classes of finite groups

Ballester-Bolinches, A.

Esteban-Romero, R.

Università del Salento: ESE - Salento University Publishing

Note di Matematica ISSN 1123-2536, e-ISSN 1590-0932Note Mat. 33 (2013) no. 1, 89–94. doi:10.1285/i15900932v33n1p89Graphs and Classes of Finite GroupsA. Ballester-BolinchesDepartament d’A`lgebra, Universitat de Vale`ncia, Dr. Moliner, 50, 46100 Burjassot,Vale`ncia, Spainadolfo.ballester@uv.esJohn CosseyDepartment of Mathematics, Mathematical Sciences Institute, Australian NationalUniversity, Canberra, ACT 0200, Australiajohn.cossey@anu.edu.auR. Esteban-RomeroDepartment de Matema`tica Aplicada, Universitat Polite`cnica de Vale`ncia, Camı´ de Vera s/n,46022 Vale`ncia, Spainresteban@mat.upv.esAbstract. There are different ways to associate to a finite group a certain graph. Aninteresting question is to analyse the relations between the structure of the group, given ingroup-theoretical terms, and the structure of the graph, given in the language of graph theory.This survey paper presents some contributions to this research line.Keywords: finite groups, classes of groups, graphsMSC 2000 classification: primary 20D10, secondary 05C25All groups in this paper are finite. We will consider only graphs which areundirected, simple (that is, with no parallel edges), and without loops. Thesegraphs will be characterised by the set of vertices and the adjacency relationbetween the vertices. Only basic concepts about graphs will be needed forthis paper. They can be found in any book about graph theory or discretemathematics, for example [12].Given a group G, there are many ways to associate a graph with G bytaking families of elements or subgroups as vertices and letting two vertices bejoined by an edge if and only if they satisfy a certain relation. We may askabout characterising group structural properties by means of the properties ofthe associated graph. In recent years, there has been considerable interest inthis line of research (see [1, 2, 3, 5, 7, 10]).If X is a class of groups, Delizia, Moravec, and Nicotera [10] associate agraph ΓX(G) with a group G by taking the nontrivial elements of G as verticesand letting a, b ∈ G\{1} be joined by an edge if 〈a, b〉 ∈ X. If we chooseX = A, where A is the class of all abelian groups, then the graph is just thecommuting graph of G and so we may think of this graph as a generalisation ofhttp://siba-ese.unisalento.it/ c© 2013 Universita` del Salentothe commuting graph. This graph has been used to study simple groups sincethe paper of Stellmacher [18].For an arbitrary class, the graph considered by Delizia, Moravec, and Nico-tera is too general to give much information and so we restrict the classes weconsider. We suppose that our class X is subgroup closed and contains A.Delizia, Moravec, and Nicotera restrict their attention to groups which areX-transitive. A group G is said to be X-transitive, or an XT -group, if whenever〈a, b〉 ∈ X and 〈b, c〉 ∈ X, then 〈a, c〉 ∈ X (a, b, c ∈ G). Note that the A-transitive groups are just the CT -groups, or groups in which the centraliser ofany non-identity element is an abelian subgroup. CT -groups are of historicalimportance as an early example of the type of classification that would beused in the FeitThompson theorem and the classification of simple groups andwere classified by several mathematicians (Weisner [20], Suzuki [19], Wu [23]).Characterisations of XT -groups are known for several classes of groups (see [11]for a survey).Let G be an XT -group. If C is a connected component of ΓX(G), then Cis a complete graph, C ∪ {1} is a subgroup of G and {C ∪ {1} : C a connectedcomponent of ΓX(G)} is a partition of G ([17] p.145). As a consequence of theclassification of groups with a partition (see for example [17, Theorems 3.5.10and 3.5.1]) we have the following result.Theorem 1. Let G be an XT -group.(1) The connected components of ΓX(G) form a normal partition of G\{1},that is, conjugates of connected components are again connected compo-nents.(2) Either ΓX(G) is connected or G is one of the following groups: a Frobeniusgroup, PSL(2, 2h), Sz(2h) (a Suzuki group, h = 2k + 1 > 1).Note that a group in the partition has the property that every 2-generatorsubgroup is an X-group. In general this will not ensure that the group is anX-group. For instance, N2, the class of (finite) nilpotent groups of class at most2, is an example, since the free group of exponent 3 on 3 generators has class3, but every 2-generator subgroup has class at most 2 (see [16]). A class X ofgroups has been called 2-recognisable (2-erkennbar) by Brandl [9] if a group Gis in X if and only if every 2-generator subgroup of G is in X.Observe that G is an XT -group and ΓX(G) is connected, then every 2-generator subgroup of G is an X-group and so if X is 2-recognisable then G ∈ X.Many classes of groups are known to be 2-recognisable; abelian, nilpotent,supersoluble and soluble among them. Some of the classes of supersolublegroups that have been extensively investigated in recent years are 2-recognisable.90We consider here the following classes: the class T of soluble T -groups (solu-ble groups in which normality is transitive), the class PT of soluble PT -groups(soluble groups in which permutability is transitive) and the class PST of sol-uble PST -groups (soluble groups in which Sylow permutability is transitive).That each of these classes is subgroup closed, it is contained in the class of allsupersoluble groups and contains all abelian groups is well known (a descriptionof these classes and their properties can be found in [6]).Lemma 2. The classes T, PT, and PST are all 2-recognisable.We now define D to be the class of groups G with all Sylow subgroups cyclic(these are just the groups with a cyclic normal subgroup whose quotient is cyclicand whose order and index are coprime, see [24, Theorem V.3.11]). We thendefine FX to be the class of Frobenius groups with the property that the kernelK is an X-group, G/K ∈ D and for each prime p | |G/K| and each prime q | |K|,p does not divide q − 1.Theorem 3. Let X be a 2-recognisable class of soluble groups containingA ∪D. Then the class of all XT -groups is contained in X ∪ FX.Theorem 4. Let X be a 2-recognisable class of soluble groups containingA ∪D and G be an XT -group.(1) G ∈ X if and only if ΓX(G) is connected.(2) G 6∈ X if and only if G is a Frobenius group and K\{1} is a connectedcomponent of ΓX(G) (where K is the kernel of G).Bearing in mind the above results, it is natural to ask for the smallest 2-recognisable class of groups containing A ∪D.Proposition 5. Let T0 be the class of T-groups G with G/G′ cyclic or Gabelian. Then T0 is the smallest 2-recognisable class of groups containing A∪D.The following variation of the commuting graph gives a characterisation forthe groups in which all subgroups are permutable. We will call it the graphof permutability of cyclic subgroups (see [4]). Given a group G, consider thegraph in which the vertices are the cyclic subgroups of G and in which everytwo vertices are adjacent when they permute. A group has all subgroups per-mutable if and only if the graph of permutability of cyclic subgroups is complete.A related graph whose vertices are the non-normal subgroups was studied byBianchi, Gillio, and Verardi (see [7, 8, 13]).The prime graph of a group has also attracted the attention of many re-searchers. The vertices of this graph are the prime numbers dividing the orderof the group G and two different vertices p and q are connected if and onlyif G has an element of order pq. For instance, in the cyclic group of order 6,91this graph is complete, but in the symmetric group of degree 3, this graph con-sists of two isolated vertices. The first references known to the authors of thisgraph correspond to Gruenberg and Kegel, in an unpublished manuscript, andto Williams, who studied the number of connected components of the primegraph of a finite group (see [14, 21, 22]). Abe and Iiyori studied in [2] a general-isation of the prime graph in the following way: given a group G, they constructthe graph ΓG whose vertices are the prime numbers dividing the order of G andin which given two different vertices p and q, they are adjacent if and only if Gpossesses a soluble group of order divisible by pq. Abe and Iiyori [2] proved:Theorem 6. If G is a non-abelian simple group, then ΓG is connected, butnot complete.Herzog, Longobardi, and Maj [15] have considered the graph whose verticesare the non-trivial conjugacy classes of a group G and in which two non-trivialconjugacy classes C and D of G are adjacent if and only if there exists c ∈ Cand d ∈ D such that cd = dc. They show that if G is a soluble group, thenthis graph has at most two connected components, each of diameter at most 15.They also study the structure of the groups for which there exist no edgesbetween non-central conjugacy classes and the relation between this graph andthe prime graph.In [5] (see also [4]), the authors define for a group G a graph Γ(G) whosevertices are the conjugacy classes of cyclic subgroups of G and in which twovertices ClG(〈x〉) and ClG(〈y〉) are adjacent if and only if we can find an elementg ∈ G such that 〈x〉 permutes with 〈yg〉. The main result of [5] is:Theorem 7. A group G is a soluble PT-group if and only if the graph Γ(G)is complete.AcknowledgementsThe first author and third author have been supported by the research grantMTM2010-19938-C03-01 from MICINN (Spain).References[1] A. Abdollahi, S. Akbari, and H. R. Maimani. Non-commuting graph of a group. J.Algebra, 298(2):468–492, 2006.[2] S. Abe and N. Iiyori. A generalization of prime graphs of finite groups. Hokkaido Math.J., 29(2):391–407, 2000.[3] A. Ballester-Bolinches and J. Cossey. Graphs, partitions and classes of groups. Monatsh.Math., 166:309–318, 2012.92[4] A. Ballester-Bolinches, J. Cossey, and R. Esteban-Romero. A characterization via graphsof the soluble groups in which permutability is transitive. Algebra Discrete Math., 4:10–17,2009.[5] A. Ballester-Bolinches, J. Cossey, and R. Esteban-Romero. On a graph related to per-mutability in finite groups. Ann. Mat. Pura Appl., 189(4):567–570, 2010.[6] A. Ballester-Bolinches, R. Esteban-Romero, and M. Asaad. Products of finite groups,volume 53 of de Gruyter Expositions in Mathematics. Walter de Gruyter, Berlin, 2010.[7] M Bianchi, A. Gillio, and L. Verardi. Subgroup-permutability and affine planes. Geome-triae Dedicata, 85:147–155, 2001.[8] M. Bianchi, A. Gillio Berta Mauri, and L. Verardi. Finite groups and subgroup-permutability. Ann. Mat. Pura Appl., IV. Ser., 169:251–268, 1995.[9] R. Brandl. Zur Theorie der untergruppenabgeschlossenen Formationen: Endlichen Va-rieta¨ten. J. Algebra, 73:1–22, 1981.[10] C. Delizia, P. Moravec, and C. Nicotera. Finite groups in which some property of two-generator subgroups is transitive. Bull. Austral. Math. Soc., 75:313–320, 2007.[11] C. Delizia, P. Moravec, and C. Nicotera. Transitivity of propeties of 2-generator sub-groups. In M. Bianchi, P. Longobardi, M. Maj, and C. M. Scoppola, editors, IschiaGroup Theory 2008. Proceedings of the Conference. Naples, Italy, 1–4 April 2008, pages68–78, Singapore, 2009. World Scientific.[12] R. Diestel. Graph theory, volume 173 of Graduate Texts in Mathematics. Springer-Verlag,Berlin, third edition, 2005.[13] A. Gillio Berta Mauri and L. Verardi. On finite groups with a reducible permutability-graph. Ann. Mat. Pura Appl., IV. Ser., 171:275–291, 1996.[14] K. W. Gruenberg. Free abelianised extensions of finite groups. In C. T. C. Wall, editor,Homological group theory. Proc. Symp., Durham 1977, volume 36 of Lond. Math. Soc.Lect. Note Ser., pages 71–104, Cambridge, 1979. London Mathematical Society, Cam-bridge Univ. Press.[15] M. Herzog, P. Longobardi, and M. Maj. On a commuting graph on conjugacy classes ofgroups. Comm. Algebra, 37(10):3369–3387, 2009.[16] H. Neumann. Varieties of Groups. Springer Verlag, New York, 1967.[17] R. Schmidt. Subgroup lattices of groups, volume 14 of De Gruyter Expositions in Mathe-matics. Walter de Gruyter, Berlin, 1994.[18] B. Stellmacher. Einfache Gruppen, die von einer Konjugiertenklasse vno Elementen derOrdnung drei erzeugt werden. J. Algebra, 30:320–354, 1974.[19] M. Suzuki. The nonexistence of a certain type of simple groups of odd order. Proc. Amer.Math. Soc., 8:686–695, 1957.[20] L. Weisner. Groups in which the normaliser of every element except identity is abelian.Bull. Amer. Math. Soc., 31:413–416, 1925.[21] J. S. Williams. The prime graph components of finite groups. In B. Cooperstein andG. Mason, editors, The Santa Cruz Conference on Finite Groups (Proc. Symp. PureMath., June 25–July 20, 1979, volume 37 of Proc. Symp. Pure Math., pages 195–196,Providence, RI, USA, 1980. American Mathematical Society (AMS), American Mathe-matical Society.[22] J. S. Williams. Prime graph components of finite groups. J. Algebra, 69:487–513, 1981.93[23] Y.-F. Wu. Groups in which commutativity is a transitive relation. J. Algebra, 207:165–181, 1998.[24] H. J. Zassenhaus. The theory of groups. Chelsea Publishing Company, New York, secondedition, 1958.94

Graphs and Classes of Finite Groups

A. Ballester-Bolinches

John Cossey

R. Esteban-romero

CiteSeerX

ESE - Salento University Publishing

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

Note di Matematica ISSN 1123-2536, e-ISSN 1590-0932Note Mat. 00 (2015) no. 0, 1–6. doi:10.1285/i15900932v00n0p1Graphs and Classes of Finite GroupsA. Ballester-BolinchesDepartament d’Àlgebra, Universitat de València, Dr. Moliner, 50, 46100 Burjassot, València,Spainadolfo.ballester@uv.esJohn CosseyDepartment of Mathematics, Mathematical Sciences Institute, Australian NationalUniversity, Canberra, ACT 0200, Australiajohn.cossey@anu.edu.auR. Esteban-RomeroDepartment de Matemàtica Aplicada, Universitat Politècnica de València, Camí de Vera s/n,46022 València, Spainresteban@mat.upv.esReceived: . . . . . . ; accepted: . . . . . .Abstract. There are different ways to associate to a finite group a certain graph. Aninteresting question is to analyse the relations between the structure of the group, given ingroup-theoretical terms, and the structure of the graph, given in the language of graph theory.This survey paper presents some contributions to this research line.Keywords: finite groups, classes of groups, graphsMSC 2000 classification: primary 20D10, secondary 05C25All groups in this paper are finite. We will consider only graphs which areundirected, simple (that is, with no parallel edges), and without loops. Thesegraphs will be characterised by the set of vertices and the adjacency relationbetween the vertices. Only basic concepts about graphs will be needed forthis paper. They can be found in any book about graph theory or discretemathematics, for example [12].Given a group G, there are many ways to associate a graph with G bytaking families of elements or subgroups as vertices and letting two vertices bejoined by an edge if and only if they satisfy a certain relation. We may askabout characterising group structural properties by means of the properties ofthe associated graph. In recent years, there has been considerable interest inthis line of research (see [1, 2, 3, 5, 7, 10]).If X is a class of groups, Delizia, Moravec, and Nicotera [10] associate a graphΓX(G) with a group G by taking the nontrivial elements of G as vertices andletting a, b ∈ G\{1} be joined by an edge if 〈a, b〉 ∈ X. If we choose X = A, whereA is the class of all abelian groups, then the graph is just the commuting graphhttp://siba-ese.unisalento.it/ c© 2015 Università del Salentoof G and so we may think of this graph as a generalisation of the commutinggraph. This graph has been used to study simple groups since the paper ofStellmacher [18].For an arbitrary class, the graph considered by Delizia, Moravec, and Nico-tera is too general to give much information and so we restrict the classes weconsider. We suppose that our class X is subgroup closed and contains A.Delizia, Moravec, and Nicotera restrict their attention to groups which areX-transitive. A group G is said to be X-transitive, or an XT -group, if whenever〈a, b〉 ∈ X and 〈b, c〉 ∈ X, then 〈a, c〉 ∈ X (a, b, c ∈ G). Note that the A-transitive groups are just the CT -groups, or groups in which the centraliser ofany non-identity element is an abelian subgroup. CT -groups are of historicalimportance as an early example of the type of classification that would be usedin the Feit–Thompson theorem and the classification of simple groups and wereclassified by several mathematicians (Weisner [20], Suzuki [19], Wu [23]). Char-acterisations of XT -groups are known for several classes of groups (see [11] fora survey).Let G be an XT -group. If C is a connected component of ΓX(G), then Cis a complete graph, C ∪ {1} is a subgroup of G and {C ∪ {1} : C a connectedcomponent of ΓX(G)} is a partition of G ([17] p.145). As a consequence of theclassification of groups with a partition (see for example [17, Theorems 3.5.10and 3.5.1]) we have the following result.Theorem 1. Let G be an XT -group.(1) The connected components of ΓX(G) form a normal partition of G\{1},that is, conjugates of connected components are again connected compon-ents.(2) Either ΓX(G) is connected or G is one of the following groups: a Frobeniusgroup, PSL(2, 2h), Sz(2h) (a Suzuki group, h = 2k + 1 > 1).Note that a group in the partition has the property that every 2-generatorsubgroup is an X-group. In general this will not ensure that the group is anX-group. For instance, N2, the class of (finite) nilpotent groups of class at most2, is an example, since the free group of exponent 3 on 3 generators has class3, but every 2-generator subgroup has class at most 2 (see [16]). A class X ofgroups has been called 2-recognisable (2-erkennbar) by Brandl [9] if a group Gis in X if and only if every 2-generator subgroup of G is in X.Observe that G is an XT -group and ΓX(G) is connected, then every 2-generator subgroup of G is an X-group and so if X is 2-recognisable then G ∈ X.Many classes of groups are known to be 2-recognisable; abelian, nilpotent,supersoluble and soluble among them. Some of the classes of supersoluble groups2that have been extensively investigated in recent years are 2-recognisable. Weconsider here the following classes: the class T of soluble T -groups (solublegroups in which normality is transitive), the class PT of soluble PT -groups (sol-uble groups in which permutability is transitive) and the class PST of solublePST -groups (soluble groups in which Sylow permutability is transitive). Thateach of these classes is subgroup closed, it is contained in the class of all super-soluble groups and contains all abelian groups is well known (a description ofthese classes and their properties can be found in [6]).Lemma 2. The classes T, PT, and PST are all 2-recognisable.We now define D to be the class of groups G with all Sylow subgroups cyclic(these are just the groups with a cyclic normal subgroup whose quotient is cyclicand whose order and index are coprime, see Zassenhaus [24, Theorem V.3.11]).We then define FX to be the class of Frobenius groups with the property thatthe kernel K is an X-group, G/K ∈ D and for each prime p | |G/K| and eachprime q | |K|, p does not divide q − 1.Theorem 3. Let X be a 2-recognisable class of soluble groups containingA ∪D. Then the class of all XT -groups is contained in X ∪ FX.Theorem 4. Let X be a 2-recognisable class of soluble groups containingA ∪D and G be an XT -group.(1) G ∈ X if and only if ΓX(G) is connected.(2) G 6∈ X if and only if G is a Frobenius group and K\{1} is a connectedcomponent of ΓX(G) (where K is the kernel of G).Bearing in mind the above results, it is natural to ask for the smallest 2-recognisable class of groups containing A ∪D.Proposition 5. Let T0 be the class of T-groups G with G/G′ cyclic or Gabelian. Then T0 is the smallest 2-recognisable class of groups containing A∪D.The following variation of the commuting graph gives a characterisation forthe groups in which all subgroups are permutable. We will call it the graph ofpermutability of cyclic subgroups (see [4]). Given a group G, consider the graphin which the vertices are the cyclic subgroups ofG and in which every two verticesare adjacent when they permute. A group has all subgroups permutable if andonly if the graph of permutability of cyclic subgroups is complete. A relatedgraph whose vertices are the non-normal subgroups was studied by Bianchi,Gillio, and Verardi (see [7, 8, 13]).The prime graph of a group has also attracted the attention of many re-searchers. The vertices of this graph are the prime numbers dividing the orderof the group G and two different vertices p and q are connected if and only if3G has an element of order pq. For instance, in the cyclic group of order 6, thisgraph is complete, but in the symmetric group of degree 3, this graph consistsof two isolated vertices. The first references known to the authors of this graphcorrespond to Gruenberg and Kegel, in an unpublished manuscript, and to Wil-liams, who studied the number of connected components of the prime graph ofa finite group (see [14, 21, 22]). Abe and Iiyori studied in [2] a generalisationof the prime graph in the following way: given a group G, they construct thegraph ΓG whose vertices are the prime numbers dividing the order of G and inwhich given two different vertices p and q, they are adjacent if and only if Gpossesses a soluble group of order divisible by pq. Abe and Iiyori [2] proved:Theorem 6. If G is a non-abelian simple group, then ΓG is connected, butnot complete.Herzog, Longobardi, and Maj [15] have considered the graph whose verticesare the non-trivial conjugacy classes of a group G and in which two non-trivialconjugacy classes C and D of G are adjacent if and only if there exists c ∈ Cand d ∈ D such that cd = dc. They show that if G is a soluble group, then thisgraph has at most two connected components, each of diameter at most 15. Theyalso study the structure of the groups for which there exist no edges betweennon-central conjugacy classes and the relation between this graph and the primegraph.In [5] (see also [4]), the authors define for a group G a graph Γ(G) whosevertices are the conjugacy classes of cyclic subgroups of G and in which twovertices ClG(〈x〉) and ClG(〈y〉) are adjacent if and only if we can find an elementg ∈ G such that 〈x〉 permutes with 〈yg〉. The main result of [5] is:Theorem 7. A group G is a soluble PT-group if and only if the graph Γ(G)is complete.AcknowledgementsThe first author and third author have been supported by the research grantMTM2010-19938-C03-01 from MICINN (Spain).References[1] A. Abdollahi, S. Akbari, and H. R. Maimani. Non-commuting graph of a group. J.Algebra, 298(2):468–492, 2006.[2] S. Abe and N. Iiyori. A generalization of prime graphs of finite groups. Hokkaido Math.J., 29(2):391–407, 2000.[3] A. Ballester-Bolinches and J. Cossey. Graphs, partitions and classes of groups. Monatsh.Math., 166:309–318, 2012.4[4] A. Ballester-Bolinches, J. Cossey, and R. Esteban-Romero. A characterization via graphsof the soluble groups in which permutability is transitive. Algebra Discrete Math., 4:10–17,2009.[5] A. Ballester-Bolinches, J. Cossey, and R. Esteban-Romero. On a graph related to per-mutability in finite groups. Ann. Mat. Pura Appl., 189(4):567–570, 2010.[6] A. Ballester-Bolinches, R. Esteban-Romero, and M. Asaad. Products of finite groups,volume 53 of de Gruyter Expositions in Mathematics. Walter de Gruyter, Berlin, 2010.[7] M Bianchi, A. Gillio, and L. Verardi. Subgroup-permutability and affine planes. Geo-metriae Dedicata, 85:147–155, 2001.[8] M. Bianchi, A. Gillio Berta Mauri, and L. Verardi. Finite groups and subgroup-permutability. Ann. Mat. Pura Appl., IV. Ser., 169:251–268, 1995.[9] R. Brandl. Zur Theorie der untergruppenabgeschlossenen Formationen: Endlichen Vari-etäten. J. Algebra, 73:1–22, 1981.[10] C. Delizia, P. Moravec, and C. Nicotera. Finite groups in which some property of two-generator subgroups is transitive. Bull. Austral. Math. Soc., 75:313–320, 2007.[11] C. Delizia, P. Moravec, and C. Nicotera. Transitivity of propeties of 2-generator sub-groups. In M. Bianchi, P. Longobardi, M. Maj, and C. M. Scoppola, editors, IschiaGroup Theory 2008. Proceedings of the Conference. Naples, Italy, 1–4 April 2008, pages68–78, Singapore, 2009. World Scientific.[12] R. Diestel. Graph theory, volume 173 of Graduate Texts in Mathematics. Springer-Verlag,Berlin, third edition, 2005.[13] A. Gillio Berta Mauri and L. Verardi. On finite groups with a reducible permutability-graph. Ann. Mat. Pura Appl., IV. Ser., 171:275–291, 1996.[14] K. W. Gruenberg. Free abelianised extensions of finite groups. In C. T. C. Wall, editor,Homological group theory. Proc. Symp., Durham 1977, volume 36 of Lond. Math. Soc.Lect. Note Ser., pages 71–104, Cambridge, 1979. London Mathematical Society, Cam-bridge Univ. Press.[15] M. Herzog, P. Longobardi, and M. Maj. On a commuting graph on conjugacy classes ofgroups. Comm. Algebra, 37(10):3369–3387, 2009.[16] H. Neumann. Varieties of Groups. Springer Verlag, New York, 1967.[17] R. Schmidt. Subgroup lattices of groups, volume 14 of De Gruyter Expositions in Math-ematics. Walter de Gruyter, Berlin, 1994.[18] B. Stellmacher. Einfache Gruppen, die von einer Konjugiertenklasse vno Elementen derOrdnung drei erzeugt werden. J. Algebra, 30:320–354, 1974.[19] M. Suzuki. The nonexistence of a certain type of simple groups of odd order. Proc. Amer.Math. Soc., 8:686–695, 1957.[20] L. Weisner. Groups in which the normaliser of every element except identity is abelian.Bull. Amer. Math. Soc., 31:413–416, 1925.[21] J. S. Williams. The prime graph components of finite groups. In B. Cooperstein andG. Mason, editors, The Santa Cruz Conference on Finite Groups (Proc. Symp. PureMath., June 25–July 20, 1979, volume 37 of Proc. Symp. Pure Math., pages 195–196,Providence, RI, USA, 1980. American Mathematical Society (AMS), American Mathem-atical Society.[22] J. S. Williams. Prime graph components of finite groups. J. Algebra, 69:487–513, 1981.5[23] Y.-F. Wu. Groups in which commutativity is a transitive relation. J. Algebra, 207:165–181, 1998.[24] H. J. Zassenhaus. The theory of groups. Chelsea Publishing Company, New York, secondedition, 1958.6

[EN] There are different ways to associate to a finite group a certain graph. An interesting question is to analyse the relations between the structure of the group, given in group-theoretical terms, and the structure of the graph, given in the language of graph theory. This survey paper presents some contributions to this research line.The first author and third author have been supported by the research grant MTM2010-19938-C03-01 from MICINN (Spain).Ballester-Bolinches, A.; Cossey, J.; Esteban Romero, R. (2013). Graphs and classes of finite groups. NOTE DI MATEMATICA. 33(1):89-94. https://doi.org/10.1285/i15900932v33n1p89899433

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[EN] There are different ways to associate to a finite group a certain graph. An interesting question is to analyse the relations between the structure of the group, given in group-theoretical terms, and the structure of the graph, given in the language of graph theory. This survey paper presents some contributions to this research line.The first author and third author have been supported by the research grant MTM2010-19938-C03-01 from MICINN (Spain).Ballester-Bolinches, A.; Cossey, J.; Esteban Romero, R. (2013). Graphs and classes of finite groups. NOTE DI MATEMATICA. 33(1):89-94. doi:10.1285/i15900932v33n1p89S899433

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