According to Theorem 2. Therefore, G is solvable and by Theorem 2. Finally, using similar argument as in Theorem 2. We have: Theorem 2. References  J. Gorenstein, Finite Groups, New York, James and M. Robinson, A Course in the Theory of Groups, 2nd ed. Shahryari and M. Shahabi, Subgroups which are the union of two conjugacy classes, Bull. Iranian Math. Shahabi, Subgroups which are the union of three conjugate classes, J. Algebra , , This makes interesting the study of factorized groups whose subgroup factors are connected by certain permutability properties.
Baer see  proved that if a group G is the product of two normal supersoluble subgroups, then G is supersoluble if and only if the commutator subgroup of G is nilpotent. Further studies have been done by several authors within the framework of formation theory. More precisely, Maier  proves that the above property on totally permutable products is not only true for the class of all supersoluble groups but also for all saturated formations F containing U, the class of supersoluble groups.
Furthermore, it turns out that the residual, projectors and normalizers associated to these kind of formations behave well in totally permutable products see  and . Mutually permutable products are studied in  and . The results contained in those papers should be seen in the light of the fact that structural information about permutable products is notoriously hard.
They have a good behaviour only in some special cases, for instance when the conmutator subgroup is nilpotent. In the last years some weaker versions of the concepts of totally and mutually permutable products have been studied. Permutability of each factor only with some concrete families of subgroups of the other one is analyzed. In this context, our main objective in  and  is to obtain relevant properties of m-permutable products. It is clear that totally permutable products and mutually permutable products are m-permutable.
However, there exists m-permutable products which are not mutually permutable. Notice that in this case, the groups in question are soluble. Gr be the pairwise m-permutable product of the subgroups G1 , G2 ,. On the other hand, it is well-known that formations composed of nilpotent groups are subgroup closed [13, IV, 1. The answer to this question is negative in general as the following example shows. Let F be the class of soluble groups whose Carter subgroups are 2-groups and take for G the symmetric group of degree 3. By [13, IV, 1. Since the Carter subgroups of G are 2-groups it follows that the formation generated by G is contained in F.
Clearly A is not in F. Gr be the pairwise permutable product of the subgroups G1 , G2 ,. Assume that F is a saturated formation such that G belongs to F. Then each subgroup Gi belongs to F. Combining Theorems 1 and 2, we have the following result. The celebrated theorem of Kegel and Wielandt states the solubility of every group which is the product of two nilpotent subgroups.
The group P SL 2, 7 is a non-soluble group which is the product of a nilpotent group and a supersoluble group. In the following we present some theorems of Kegel-Wielandt type for products connected by certain permutability properties. If H is supersoluble, K is nilpotent and K permutes with every maximal subgroup of H, then G is soluble.
If H and K are supersoluble, then G is soluble. The following theorem analyzes the case K nilpotent and it is an application of the above results. Assume that H is supersoluble and K is nilpotent. If K permutes with every Sylow subgroup of H, then G is supersoluble. Remarks a Permutability of H with every maximal subgroup of K is essential in order to get supersolubility in Theorem 5.
Let G be the symmetric group of degree four. However, G is not supersoluble. Notice that H does not permute with the maximal subgroups of K. Moreover, G is the m-permutable product of H and K. However G is not a supersoluble group. Notice that K does not permute with every Sylow subgroup of H.
References  M. Asaad and A. Ballester-Bolinches and M. Pedraza-Aguilera and M. Soc, Vol. Algebra, Vol. Algebra, Vol , , Edinburgh Math. Maier concerning formations. Ballester-Bolinches Xiuyun Guo and M. Group Theory, Vol. Ballester-Bolinches, John Cossey and M. Accepted in Comm. Andrews Lecture Notes Series , Vol. I, 68 Carocca and R. Maier, Theorems of Kegel-Wielandt type, Proc. I, Doerk and T. Maier, A completeness property of certain formations, Bull. Although theories of formations and Fitting classes are quite independent generalizations of the classical theory of Sylow and Hall, many of their results have been motivated by the good behaviour of the Fitting formation of nilpotent groups as a class of groups.
This paper is a survey article containing a detailed account of recent works concerning these classes of nilpotent type. Let us start by looking at the class of nilpotent groups. Therefore, the natural question arising from the above fact is the following: Which are the subgroup-closed saturated formations F for which the set of all F-subnormal subgroups is a lattice in every group G?
We recall here the concept of F-subnormal subgroup. In fact, these classes were also studied by Lockett  from this point of view. He proved that they are dominant Fitting classes and obtained the exact description of the associated injectors.
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So, in this framework, it is natural to ask about the behaviour of the F-radical and the Finjectors with respect to the F-subnormal subgroups of a group. This idea provides also interesting characterizations of the lattice formations in terms of Fitting type properties : Theorem 3 Let F be an sn -closed saturated formation containing N. Then the following statements are pairwise equivalent: 1 F is a lattice formation.
Lattice formations have been also involved recently in the study of F-normality associated to saturated formations. Originally the notion of F-normality was restricted to maximal subgroups. Moreover, in this case, lattice formations turn out to be again the subgroup-closed saturated formations for which the set of all FDnormal subgroups is a lattice i! Besides, the analogous results to Theorem 1 and Theorem 2 remain valid for F-Dnormal subgroups in place of F-subnormal ones.
In this respect, it is well-known that the Fitting subgroup of a group which is the product of two nilpotent subgroups is a factorized group see . The next result which appears in  shows that this property also characterizes the aforesaid subgroup-closed Fitting formations. Theorem 4 Let F be a subgroup-closed Fitting formation containing N. Then the following statements are equivalent: 1 F is a lattice-formation.
First of all, notice that, in view of property 4 of Theorem 3, it is natural to wonder which are the subgroup-closed saturated formations F closed under taking products of F-subnormal subgroups. Of course, lattice formations satisfy this property, but so does the class of p-nilpotent groups, for every prime p. Therefore it is of interest to know the structure of F-critical groups with respect to some class F. In the Kourovka Notebook  Shemetkov proposed the following question: Which are the saturated formations F satisfying that every F-critical group is either a Schmidt group or a cyclic group of prime order?
We say that a saturated formation F has the Shemetkov property if every F-critical group is either a Schmidt group or a cyclic group of prime order. The groups in this extension can be caracterized through an analogue to the Frobenius p-nilpotence criterion. More concretely we have: Theorem 5 Let F be a subgroup-closed saturated formation containing N. The equivalence between 1 , 3 and 4 has been obtained in . Moreover, some authors of the Gomel School have also studied related problems see ,  and . We are going to see how this family of saturated formations appears again in the context of factorized groups.
So the following question can be formulated: Which are the subnormal subgroup closed saturated formations containing N which are closed under triply factorized groups? A particular case of this question is when C is an F-subnormal subgroup of G. The answer to both questions leads to the family of saturated formations of nilpotent type into consideration see . Theorem 6 Let F be an sn -closed saturated formation containing N.
We remark that Vasilev also studied related problems in  and . For instance, the factorizer of a normal subgroup of a factorized group always has this form. So, the following characterization of the above formations is also of interest . Theorem 7 Let F be an sn -closed saturated formation containing N.
Extensions of p-nilpotent groups In this section we meet extensions of the class of nilpotent groups through the existence of normal complements. Theorem 8 Let F be a subgroup-closed saturated formation containing N. Clearly, the saturated formations described in this theorem are extensions of the formations of all nilpotent and p-nilpotent groups p a prime. Again, this family of classes appears when considering some interesting questions in the framework of factorized and trifactorized groups. In particular, we consider an extension of the properties studied in Theorem 6 and 7.
More concretely, in  it is proved: Theorem 9 Let F be an sn -closed saturated formation containing N. Now we want to draw attention to the fact that they are also Fitting classes. From this point of view, one would hope them to be dominant, like the class of nilpotent groups. However, if F is a class as above, F is not in general a dominant Fitting class even if we consider classes as in section 4. So the next step is to look for characterization theorems. In  we provide a characterization for these classes to be dominant. In fact, to prove this theorem, the exact description of the F-maximal subgroups of a group G containing the F-radical GF is obtained for these classes.
This construction rests strongly on a well-known result of Lockett  about the permutability of normally embedded subgroups of a group into which a given Hall system of the group reduces. It is interesting to point out that classical constructions of nilpotent injectors, Lockett injectors, as well as p-nilpotent injectors p a prime , appear as particular cases of the above construction. References  B.
Amberg, S. Franciosi and F. Ballester-Bolinches, A note on saturated formations, Arch. Ballester-Bolinches, K. Doerk and M. Algebra , Ballester-Bolinches, A. Pedraza-Aguilera, Finite trifactorized groups and formations, J. Algebra, , A 64 , Algebra , — Shemetkov on critical groups, J. Kegel, Zur Struktur mehrfach faktorisierter endlicher Gruppen, Math. Kamornikov , On two problems of L. Shemtekov, Siberian Math. Kamornikov, Permutability of subgroups and F-subnormality, Siberian Math. Thesis, University of Warwick, Shemetkov and A.
Nauk BSSR 34 , no. Press, Minsk, Vasilev, S. Kamornikov and V. For instance, subnormality of each subgroup, normality of all Sylow subgroups, centrality of every chief factor and normality of all maximal subgroups. If U is a proper subgroup of a group G then m U is the least upper bound of the types of all properly ascending chains from U to G. Let G be the split extension of a quasicyclic 2-group by its involution.
Then G is a locally nilpotent group, in fact it is hypercentral, but it is not a B-group. Then Gi is nilpotent of class i. The corresponding statement holds for locally nilpotent groups and is of great importance. Moreover in any group G there is a unique maximal normal locally nilpotent subgroup called the Hirsch-Plotkin radical containing all normal locally nilpotent subgroups of G see[16, However in this case it plays an important role as it inherits the properties of the B-radical. This result is also true for the Hirsch-Plotkin radical of a periodic locally soluble group see [10, 1.
Such groups have been widely studied over the past 50 years by several authors see[1, Ch. Some authors have been interested in the study of factorised groups whose subgroup factors are connected by certain permutability properties. In fact, the following question can be formulated. Some weaker versions of these permutable products due to Carocca  and to Beidleman, Heineken, Galoppo and Manfredino  have been fruitful in studying the structure of a product of two soluble groups. Now we obtain a result of this kind, using the derived lengths of the subgroups A and B instead of their classes, under some restrictions on the permutability of their derived series.
Let us denote by dG the derived length of a soluble group G. Ballester-Bolinches and S. Preprint  A. Preprint  J. Beidleman, A. Galoppo, H. Heineken and M. Pure Appl. Berlin, Cambridge Phil. Palermo, serie II, 23 , In this paper we will calculate these quantities for a p-group of class 2 with cyclic centre, where p is either an odd prime or 2. Thus every permutation matrix over C is a quasi-permutation matrix. See [BGHS]. Now I would like to state a problem from Prof.
Brian Hartley Calculating p G , c G and q G Lemma 2. Let Z G be cyclic. Proof : See [HB1], Lemma 4. Then G contains a generalized quaternion section. Proof : See [F] Theorem 2. In other words G only can have Q8 as a section. Lemma 2. Proof : See [HB1], Corollary 3. Proof : This follows from [I], Corollary Proof : See [HB1], Theorem 4. Then p G is the smallest index of a subgroup with trivial core that is, containing no non-trivial normal subgroup. Proof : See [HB1], Corollary 2. Proof : By Lemma 2. Hence by [R], 5. Therefore C is a maximal abelian subgroup of G. In this case G contains Q8 as a section.
Also let G have no Q8 section. Proof : This follows from Lemmas 2. It is easy to prove the results given in the following table. Proof : See [DH], Theorem Proof : See Theorem 2. Proof : This follows from [HB1], Corollaries 3. Then a if G is a central product of D1 ,. So the result follows. The rest of the proof is the same as part a. Theorem 3. By Lemmas 2.
References [HB1] H. Burns, B. Goldsmith, B. Hartley, R. Doerk, T. Hawkes, Finite soluble groups, de Gruyter, Berlin, Robinson, A course in the theory of groups Springer, New York, Then we study their structure to obtain new methods for constructing additional families. Automorphism groups of Riemann surfaces with this maximal number of automorphisms are called Hurwitz groups. The article by Conder  contains a nice survey of known results about Hurwitz groups. Corresponding problems concerning Klein surfaces have also received a good deal of attention and we present a summary of known results here.
A Klein surface is the orbit space of a Riemann surface under the action of a symmetry, that is, an anticonformal automorphism of order two. Throughout the paper, the letter p will be used exclusively to denote the algebraic genus of a Klein surface. A Klein surface is endowed with a dianalytic structure, see , and it may be non-orientable and with non empty boundary. The notion of a Klein surface is a generalization of that of a Riemann surface in the sense that Riemann surfaces may be seen as orientable Klein surfaces with empty boundary.
All groups of automorphisms of bordered Klein surfaces arise in this way. An epimorphism whose kernel is a bordered surface NEC group is called a bordered smooth epimorphism. However, the genus itself does not determine topologically a Klein surface, since two more data are required, namely, its orientability and the number of its boundary components.
It turns out that some of these genera have more than one topological type with maximal symmetry. The same happens in higher genus, see , and in fact, there exists no bound independent of p for the number of topological types with maximal symmetry within a single genus, . Other results concerning the orientability of a Klein surface with maximal symmetry have been obtained in . Thus any result concerning Klein surfaces can be stated in terms of real algebraic curves and, in fact, automorphisms of Klein surfaces correspond to real birational transformations of real algebraic curves.
The boundary of the surface is homeomorphic to the set of real points of the curve, so the number of empty period cycles of the surface NEC group uniformizing the surface coincides with the number of ovals of the real curve. There is also an important correspondence between bordered Klein surfaces with maximal symmetry and regular maps, see , ,  and .
Further G is isomorphic to the automorphism group of the map M, and the number of boundary components of X is equal to the number of vertices of M. Let q be an odd prime. Each group has order 12n2 m and index 6m. Each group has order 36n2 m and index 6m. We illustrate it with the following examples. Another interesting example where this method is applied deals with the family of 2, 3, r; s -groups, which has been extensively studied, see  and the references given there.
This allows one to study the relationship between them from a group theoretic point of view. References  N. Alling, N. Bujalance, F. Cirre, P. Bujalance, J. Etayo, J. Gamboa, G. Gromadzki, Automorphism groups of compact bordered Klein surfaces. Lecture Notes in Math. Burnside, Theory of groups of finite order 2nd ed. Conder, Generators for alternating and symmetric groups. Conder, More on generators for alternating and symmetric groups. Oxford Ser. Conder, Hurwitz groups: a brief survey. Coxeter, The abstract groups Gm,n,p , Trans. Coxeter, W. Moser, Generators and relations for discrete groups.
Etayo, Klein surfaces with maximal symmetry and their groups of automorphisms. Etayo, C. Algebra, 42, , 1 , 29— Greenleaf, C. May, Bordered Klein surfaces with maximal symmetry. Hall, Automorphisms and coverings of Klein surfaces, Ph. Howie, R. Thomas, Proving certain groups infinite. Geometric group theory, Vol. Lecture Note Ser. Press, Cambridge, Macbeath, On a theorem of Hurwitz, Proc. Glasgow Math. Macbeath, The classification of non-euclidean plane crystallographic groups, Canad.
May, Automorphisms of compact Klein surfaces with boundary. May, Large automorphism groups of compact Klein surfaces with boundary I. May, Cyclic automorphism groups of compact bordered Klein surfaces. Houston J. May, The species of bordered Klein surfaces with maximal symmetry of low genus. Singerman, Automorphisms of compact non-orientable Riemann surfaces. Singerman, Orientable and non-orientable Klein surfaces with maximal symmetry. Singerman, PSL 2, q as an image of the extended modular group with applications to group actions on surfaces.
Sinkov, On generating the simple group LF 2, 2N by two operators of periods two and three. Wilson, Riemann surfaces over regular maps. Box , Safat , Kuwait Dedicated to R. Khazal Abstract The paper classifies those locally finite groups having a proper nontrivial subgroup which is comparable with any other element of the subgroup lattice.
The description of groups G with L G a chain is well-known. In a chain, every element is comparable with the others. Such a subgroup H will be called a breaking point for the lattice L G. For the sake of convenience, we shall call these groups BP-groups. Of course, BP-groups cannot be decomposed as nontrivial direct products. These simple considerations are valuable in what follows and we shall use them without any further reference. This focuses the discussion on nonabelian BP-groups.
These quasi finite examples show that BP-groups need not be soluble, nor locally finite; the class of BP-groups is thus large enough to warrant a more serious investigation. We shall restrict ourselves here to the particular case of locally finite BP-groups; the cyclic p-groups of order at least p2 and the generalized quaternion groups are examples of finite BP-groups.
The main result of this note shows that the examples described above exhaust all locally finite BP-groups: Theorem 1. The notation is standard and the proofs are elementary. In particular, G has a unique subgroup of order p. We prove next that H is finite. Thus Z G is a breaking point for L G.
The maximality of Z G follows from 1.
Thus A being not normal in G and having finite index are contradictory. This completes the proof of the Lemma. Proof Let G be a locally finite BP-group. This follows from the lemma and from Satz 8. If G is infinite, then any two nontrivial elements of G generate a finite subgroup T of G which has just one minimal subgroup. For p odd, this subgroup T is cyclic. We reached the stage where all locally finite BP-groups were classified, except those which are infinite nonabelian 2-groups. From now on, G will denote a locally finite infinite nonabelian BP-group which is a 2-group.
Since G is nonabelian, there exists a pair of non commuting elements in G which generate a nonabelian subgroup K of G, which is a generalized quaternion 2-group. These groups are discussed in , p. This concludes the proof of the Theorem. In Satz 3. It is also shown there for such infinite nonabelian p-groups of odd order that Z G is the largest breaking point of L G. A number of interesting questions remain still open: a Are there infinite BP-groups which are not locally finite and with non modular lattice L G?
Such groups would necessarily have infinite proper abelian subgroups of infinite index. Prof A. This hints that a complete classification of BP-groups is far from being an easy task. Wehrfritz, Locally finite groups, North Holland German  T. Romane vol A 1 One of the key strengths of group theory comes from the use of groups to measure and analyse the symmetries of objects, whether these be physical objects in 2 or 3 dimensions , or more purely mathematical objects such as roots of polynomials or vectors or indeed other groups.
This is now bearing unexpected fruit in areas such as structural chemistry with the study of fullerenes for example , and interconnection networks where Cayley graphs and other graphs constructed from groups often have ideal properties for communication systems. The aim of this paper and the associated short course of lectures given at the Groups St Andrews conference in Oxford is to describe a number of instances of symmetry groups of mathematical objects where the order of the group is as large as possible with respect to the genus, size or type of the object.
Please note that this paper makes no claims to be a comprehensive survey of the theme of maximum symmetry, or even of each of the topics dealt with. In Section 2, we describe some of the tools that have proved useful to the author and others in this area, including the low index subgroups process and Schreier coset graphs. Section 3 deals with automorphism groups of compact Riemann surfaces, with particular reference to Hurwitz groups. Section 4 concerns regular maps, on both orientable and non-orientable surfaces, and goes on to describe some recent work on the maximum number of automorphisms of a closed non-orientable surface of given genus.
Finally, Section 6 describes two instances where unexpected results have arisen, and Section 7 lists a number of open problems. A surface will be taken as a closed 2-manifold without boundary. As such, graphs in this context are simple, with undirected edges, no loops and no multiple edges. A multigraph is a generalisation of a graph, in which multiple edges are allowed between any pair of distinct vertices. Finally, a map is a 2-cell embedding of a connected graph or multigraph into a surface so that the connected components of the complementary space obtained by removing the graph or multigraph from the surface are all homeomorphic to open disks, called faces.
Excellent descriptions of these may be found in the book by Charles Sims . This involves a backtrack search through a tree, with nodes at level k in the tree corresponding to pseudo subgroups generated by k elements. Tests are built in to avoid generating the same subgroup more than once by rejecting sub-trees and also to avoid conjugates of subgroups found earlier in the search tree isomorph rejection.
The algorithm stops when the whole search tree has been traversed. Example 2. Numerous examples exist in the literature for example  and in documentation for Magma and GAP. This can be particularly helpful in a search for small concrete examples of such factor groups, which can then be used as building blocks for larger examples, as will be seen later. Another important observation to make about the low index subgroups algorithm is that distinct sub-trees can be processed independently. This provides a basis for distributed processing or parallelisation, of either the basic algorithm or special adaptations.
One such adaptation involves pursuing only selected branches of the search tree: for example those which correspond to subgroups avoiding a given set of elements and their conjugates. This in turn enables a search up to much higher index within given computing resources. The reduction in computing time can be spectacular: Example 2. This provided equivalent computing power to a medium-sized supercomputer, at a fraction of the cost!
Applications will be described in Sections 4. This graph provides a diagrammatic representation of the action of G on cosets of H by right multiplication. In fact these things are essentially interchangeable: the coset table, the coset graph, and permutations induced by the group generators. See  for many examples. A number of observations are worth making about Schreier coset graphs. It follows that a Schreier generating-set for H in G corresponds to the set of edges of the coset graph not used in a spanning tree. Schreier coset graphs have other theoretical applications, for example to the following theorem which provides a necessary condition for transitivity of a group generated by a set of permutations due independently to Ree and Singerman : If G is the group generated by permutations x1 , x2 ,.
Coset graphs can also have important more practical applications. This is related to abelianisation of the ReidemeisterSchreier process, but will not be pursued in detail here. This and the other two possibilities which are similar are illustrated in Figure 2. Figure 2.
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Composition of coset graphs for the 2, 3, 7 triangle group Interesting things can happen when coset graphs are linked together in this way. For example, the 2, 3, 7 triangle group has permutation representations of degrees 14, 64 and 22 in which the groups generated by the permutations are isomorphic to PSL2 13 , A64 and A22 respectively.
The corresponding coset graphs can be 70 CONDER composed to create a transitive permutation representation of degree , in which the permutations generate the Hall-Janko simple group J2 , of order The cycle structure of the commutator [x, y] in the latter representation is 13 A Riemann surface is a connected 2-manifold endowed with a complex analytic structure called an atlas that allows local coordinatisation — somewhat analogous to a book of maps of the planet Earth.
As usual, automorphisms form a group under composition, known as the automorphism group of X and denoted by Aut X. The parameters mi are the orders of branch points. This leads to: Theorem 3. It follows that every Hurwitz group has a nonabelian simple quotient, and hence it is natural to look among the nonabelian simple groups for examples of Hurwitz groups. Several other simple groups and families of simple groups have been shown to be Hurwitz using character-theoretic techniques.
Good knowledge of the maximal subgroup structure and local analysis can often be used to account for subgroups generated by these pairs, and hence to determine whether or not G itself is so generated. Particular attention has been paid to the sporadic simple groups in this context. It is also the orientation-preserving subgroup of the group of automorphisms of a regular map on a surface of the same genus; see Section 4. Similar methods are applied in . Next, we note that there are several ways of constructing larger Hurwitz groups from given examples.
Finally, we note some non-existence results. The map is orientable or non-orientable according to whether the underlying surface on which the graph is embedded is orientable or non-orientable. Each dart is made up of two blades, one corresponding to each face containing the edge e except in degenerate situations where an edge lies in just one face, but these will not concern us much here. An automorphism of a map M is a permutation of its blades, preserving the properties of incidence, and as usual these form a group under composition, called the automorphism group of the map, and denoted by Aut M.
Now if there exist automorphisms R and S with the property that R cyclically permutes the consecutive edges of some face f in single steps around f , and S cyclically permutes the consecutive edges incident to some vertex v of f in single steps around v , then following Steve Wilson  we may call M a rotary map. Under more currently accepted terminology, M is also called a regular map in the sense of Brahana, who generated early interest  in such objects in the s. In this case, again by connectedness, Aut M acts transitively on vertices, on edges, and on faces of the map M , and it follows that M is combinatorially regular, with all its faces bordered by the same number of edges, say p, and all its vertices having the same degree, say q.
When M is rotary, R and S may be chosen by replacing one of them by its inverse if necessary so that the automorphism RS interchanges the vertex v with one of its neighbours along an edge e on the border of f , interchanging f with the other face containing e in the process. On the other hand, if no such automorphism a exists, then the rotary map M is called chiral , and its automorphism group is generated by the rotations R and S. Chiral maps are necessarily orientable.
Further details and some historical background may be found in [29, 38, 60]. Indeed: Theorem 4. Suppose G is any group generated by three involutions a, b and c such that ac has order 2, and ab and bc have orders greater than 2, say p and q respectively. As an illustration, we have the following well-known construction for regular maps on orientable surfaces of every possible genus: Example 4.
Also let K be a cyclic group of arbitrary order n, generated say by z. Thus G is a 2, p, qm -generated group, having the original 2, p, q -generated group H as a quotient. For increasing n, we obtain a family of such groups, with orders in arithmetic progression. This construction given in  has numerous applications. The complete genus spectrum of non-orientable regular maps is not known. Apart from genus 2 and 3 which are somewhat trivial exceptions , it is known that no such maps exist on non-orientable surfaces of genus 18, 24, 27, 39 or 48 by unpublished work of Antonio Breda and Steve Wilson.
Also recently Wilson and the author have shown there is no such map of genus This may be contrasted with orientable regular maps, which are known to exist for all possible genera see Example 4. We will return to this matter later. By Theorem 4. Such a test can easily be built into a post-processing phase of the normal subgroups adaptation of the low index subgroups process, if desired.
The details and results for all three types of regular map of small genus may be found in . It is also interesting to ask for a lower bound on the maximum number of conformal automorphisms of a compact orientable surface of given genus. To explain this further, some more background material is needed. In some cases more advanced methods are required. For example, often Sylow theory shows G has a cyclic normal subgroup K of order q, with CG K of low index, and then by the Schur-Zassenhaus theorem CG K has a quotient of order q, while the Reidemeister-Schreier process shows that all subgroups of low index are generated by elements of order coprime to q, making this impossible.
See [28, 53] for further details. Under composition, the symmetries of a graph X form a group called the automorphism group of X, and denoted by Aut X. Finite graphs with maximum symmetry are very easy to classify: the largest possible number of automorphisms of a graph on n vertices is n! These examples, however, are rather uninteresting, and graphs of more frequent attention are those which lie in between these two extremes but have an automorphism group which acts transitively on vertices, edges, arcs, or directed walks of a given length. If Aut X has a single orbit on vertices, then the graph X is said to be vertextransitive.
Similarly if Aut X is transitive on the edges of X, or on arcs directed edges of X, then X is edge-transitive or arc-transitive respectively. Figure 4. Also the background theory can be taken much further, to produce some very strong conditions on maximum symmetry. In Tutte proved the following remarkable theorem by local analysis: Theorem 5.
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This may be contrasted starkly with the 4-valent case, where generalisations of the graph in Figure 6 show that the stabiliser of a vertex can be arbitrarily large. Figure 6. If X is 2-arc-transitive, then the stabiliser of a vertex v is doubly-transitive on the neighbourhood X v of v.
More recently, using the normal subgroups adaptation of the low index subgroups algorithm described in Section 2. In particular, this extends the Foster census  of such graphs of order up to compiled largely by hand by by R. It can now be observed that in any transitive permutation representation of G5 , all orbits of H have lengths dividing 48, and decompose into orbits of A which are linked together by cycles of the permutations induced by h and a. This observation makes it easy to construct multitudes of transitive permutation representations of G5 of arbitrarily large degree, in a similar way to representations of the 2, 3, 7 triangle group, and hence multitudes of 5-arc-transitive cubic graphs.
The smallest such example is the one associated with G2 3 , on vertices, and larger examples can be constructed as covers of given examples under certain conditions see [30, 57]. The proof is based on a careful selection of permutation representations of the generic group R4,7 as building blocks for constructing transitive permutation representations of arbitrarily large degree, as in . In particular, H has order The generator b takes the role of the arc-reversing involution a in the construction described in Section 5.
Each block in turn was made up of representations of smaller degree, linked together by multiple transpositions of b. Taking k copies of the block A and l copies of B, we may link these together into a chain to produce a transitive permutation representation of R4,7 on n points. If the order in which the blocks are linked is chosen carefully, then the permutation induced by bh will have a single cycle of length , and lengths of all other cycles will be relatively prime to In particular, this produces an unexpectedly succinct presentation for the group SL3 ZZ.
Note: only groups of Lie type remain to be considered. Problem 2: What is the complete genus spectrum of non-orientable regular maps? Problem 3: Is it true that for every positive integer g there exists a regular map on an orientable surface of genus g such that the underlying graph is simple?
Note: the underlying graphs of the families of examples customarily used to show orientable maps exist for all possible genera have multiple edges. Note: considerable progress has been made on this by Cheryl Praeger. Conder, R. Wilson and A. Woldar, The symmetric genus of sporadic groups, Proc. Martin, Cusps, triangle groups and hyperbolic 3-folds, J.
Series A 55 , — Everitt, Regular maps on non-orientable surfaces, Geometriae Dedicata 56 , — Conder, Asymmetric combinatorially-regular maps, J. Algebraic Combinatorics 5 , — Walker, Vertex-transitive graphs with arbitrarily large vertexstabilizers, J. Algebraic Combinatorics 8 , 29— Theory Ser. B 81 , — Conder, Hurwitz groups with given centre, preprint. Conder, C. Maclachlan, S.
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Combinatorics 10 , — Encyclopedia of Mathematics and its Applications, vol. Tutte, A family of cubical graphs, Proc. Tutte, On the symmetry of cubic graphs, Canad. Wilson, The Monster is a Hurwitz group, preprint. Normality and subnormality are the most elementary ones. From the study of conjugacy classes of subgroups the property of pronormality arises. This fact motivated the concept of dual pronormality. This property emerges both as a weaker condition than normality and as a dual concept to pronormality. Dual pronormal subgroups are close to N -injectors, for the class N of nilpotent groups, and, in this study, groups containing several relevant classes of dual pronormal subgroups were also taken into consideration.
This development shows how far dual pronormality is from normality and subnormality and how dual pronormality can provide additional information. In particular, groups in which Carter subgroups are dual pronormal have been studied. This leads to groups in which Carter subgroups and N -injectors coincide. In our context the case when these subgroups are dual pronormal has also been studied. This investigation is taken further in the context of Fitting classes by means of the extension to F-dual pronormality, for a Fitting class F, in ,  and . This property appears in a natural way from the observation that the Fitting subgroup of a group is the radical for the class of nilpotent groups.
Fischer F-subgroups, and in particular F-injectors, associated to a Fitting class F, in the soluble universe, are F-dual pronormal subgroups. This allows Fischer F-subgroups of soluble groups, for a Fitting class F, to be characterized as F-dual pronormal F-maximal subgroups. Later on the study of Fitting classes whose injectors enjoy other embedding properties, such as normal embedding or permutability, has been of interest see . To be more exact, initially a Fitting class F of soluble groups was proved to be normal if, and only if, every F-dual pronormal subgroup has respectively one of the following embedding properties: normality, subnormality, pronormality, normal embedding.
Notice that normality and A-normality coincide in this case. Normality fails in this development, even in the soluble case. As mentioned, the extension of normality that works in this case is A-normality. Hawkes in , p. Our aim here is to provide an up-to-date account of the achievements regarding dual pronormal and F-dual pronormal subgroups, for a Fitting class F. We refer to for unexplained notation and concepts.
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Definition 2. Assume that either G is soluble or F is subgroup-closed. Then GF normalizes H. This property does not characterize F-dual pronormal subgroups. A cyclic subgroup of order 4 of Sym 4 , the symmetric group of degree 4, is an example for this. In soluble groups, Fischer F-subgroups and, in particular, F-injectors, associated to a Fitting class F, are F-dual pronormal subgroups.
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