Groups II

For a permutation group G acting on a set V, a subset I of G is said to be an intersecting set if for every pair of elements g,h∈I there exists v∈V such that g(v)=h(v). The intersection densityρ(G) of a transitive permutation group G is the maximum value of the quotient |I|/|Gv| where Gv is a stabilizer of a point v∈V and I runs over all intersecting sets in G. If Gv is the largest intersecting set in G then G is said to have the Erdős–Ko–Rado (EKR)-property, and moreover, G has the strict-EKR-property if every intersecting set of maximum size in G is a coset of a point stabilizer. Intersecting sets in G coincide with independent sets in the so-called derangement graphΓG, defined as the Cayley graph on G with connection set consisting of all derangements, that is, fixed-point free elements of G. In this paper a conjecture regarding the existence of transitive permutation groups whose derangement graphs are complete multipartite graphs, posed by Meagher, Razafimahatratra and Spiga in Meagher et al. (2021) is proved. The proof uses direct product of groups. Questions regarding maximum intersecting sets in direct and wreath products of groups and the (strict)-EKR-property of these group products are also investigated. In addition, some errors appearing in the literature on this topic are corrected.

The main object of this paper is the determination of all the possible groups whose group of isomorphisms is either the symmetric group of order 6 or the synimetric group of order 24. We shall also determine the infinite system of groups whose group of cogredient isomorphisms is the former of these two symmetric groups. It will be proved that this system includes one and only one group (which is not the direct product of an abelian and a non-abelian group) for every power of 2. It is well known that every simple isomorphism of a group G with itself may be obtained by transforming G by means of operators that transform it into itself.t In what follows we shall generally employ this method of making G simply isomorphic with itself. In a few cases it will be convenient to employ two special methods, which 'we proceed to explain. The first of these two methods may be employed when G contains a subgroup H' which is composed entirely of operators which are selfconjugate under G and which is also simply isomorphic to a quotient group of G with respect to a selfconjugate subgroup which includes H'. In this case we may evidently multiply all of the operators of each one of the various divisions of G with respect to this quotient group by the corresponding operator of H' and thus obtain a simple isomorphisin of G with itself.-To illustrate this method we may employ the direct product G12 of' the symmetric group of order 6 anid an operator s1 of order two. If 'we multiply each of the six operators of G12 which are not contained in its cyclical subgroup of order 6 by s1 we obtain a simple isomorphism of G12 with itself. It is evident that this isomorphism corresponds to the selfconjugate operator of order two in the group of isomorphisms of G12 t It is important to observe that any operator t1 of the group of isomorphisms of G which is obtailned in this manner is selfconjugate under this group of isomorphisms whenever H' is composed of characteristic operators

Introduction. In the appendix of Hilton's Finite Groups (1908), page 233, the question whether non-abelian group can have an abelian group of isomorphisms occurs among a few interesting questions still awaiting solution. A non-abelian group of order 64 whose group of isomorphisms is abelian and of order 128 was later constructed by G. A. Miller.t No other discussian of the problem appears in the literature of Mathematics. In fact, an exhaustive analysis of the properties of non-abelian group whose group of isomorphisms is abelian seems to involve some difficulty, for the reason that little is known concerning isomorphisms of non-abelian group. The nature of an abelian group whose group of isomorphisms is abelian was determined by G. A. Miller,+ who proved that necessary and sufficient condition that an operation of the group of isomorphisms of an abelian group be invariant under this group is that it should transform every operation of this abelian group into the same power of itself. From this it is obvious that the only abelian groups whose groups of isomorphisms are abelian are the cyclic groups. In what follows we shall develop few necessary conditions which must be satisfied by non-abelian group whose group of isomorphisms is abelian. 1. The fundamental theorem. Wle assume non-abelian group G which is restricted solely by the hypothesis that its group of isomorphisms I is abelian. Since the central quotient group of G is simply isomorphic with the group of inner isomorphisms of G, it is clear that every commutator of G is invariant under G. The group G is accordingly the direct product of its Sylow subgroups.? Since every Sylow subgroup of G must correspond to itself in any isomorphism of G with itself, we may confine ourselves to the case where G is of order pm. For convenience we shall introduce the following notation: the symbol G shall consistently denote non-abelian group whose group of isomorphisms I

The workshop Permutation groups organised by Robert Guralnick (Southern California), Cheryl Praeger (Western Australia), Jan Saxl (Cambridge) and Katrin Tent (Bielefeld) was held August 5th–11th, 2007. The focus was recent developments in permutation group theory and their influence on, and from, group representation theory, algebraic graph theory and other areas of geometry and combinatorics, algebraic geometry, model theory and infinite symmetrical structures. It was well attended with 50 participants. Especially valuable was the program to support young researchers which gave the opportunity for a considerable number of young mathematicians to participate in the workshop. They benefited from the ‘Oberwolfach experience’ and also enlivened and contributed enormously to the success of the meeting. The workshop had a friendly and vibrant atmosphere, aided by the excellent facilities of the Institute. All participants appreciated the beautiful lecture by John G. Thompson entitled ‘The divisor matrix and {\rm SL}(2,\Z) ’. (In fact, this meeting was the first Oberwolfach workshop in over twenty years that Thompson had attended.) There were 26 other talks, and the program featured in each of the first five sessions at least one talk by a student or young researcher at the beginning of their career. This facilitated interaction between younger and more established researchers. The number of talks was restricted to give plenty of time for discussion and cooperation among the participants. Other communications were presented as posters. Highlights of the workshop addressed both fundamental permutation group theory, and also significant outcomes from applying permutation group theory and methods in combinatorics, model theory and other areas. Some topics arose in more than one context, forming new connections. A common feature of many contributions was application of the finite simple group classification, or new results on the structure of simple groups. As examples, we mention two highlights reported by early career participants. The question of base size of permutation actions is of importance in computational group theory as well as in the study of the graph isomorphism problem. Recent research has thrown much light on the base sizes of actions of almost simple groups in particular. Burness reported on the very recent proof of a conjecture of Cameron and Kantor dating from 1993 concerning non-standard permutation actions of almost simple groups: these are all primitive actions apart from actions of alternating and symmetric groups on subsets or partitions, and of classical groups on subspaces or pairs of complementary subspaces. In all the latter families the minimum base size is unbounded as the group order increases. The somewhat surprising conjecture, now proved, is that the minimum base size of non-standard permutation groups is at most 7, with the unique example where this bound is attained being the Mathieu group M_{24} in its natural action on 24 points. Expander graphs play an important role in computer science and combinatorics, and, for example, are significant in modelling and analysing communication networks. In particular, their diameters grow only logarithmically with the numbers of vertices. This property on diameters was proved in 1989 by Babai, Kantor and Lubotzky to hold for suitable Cayley graphs of simple groups: specifically for each finite nonabelian simple group G , they constructed a valency 7 Cayley graph for G with diameter at most a constant times \log(|G|) . Although their graphs were not expanders, they conjectured that there exists an absolute constant k such that all infinite families of simple groups give rise to families of valency k Cayley graphs that are expanders. Kassabov lectured on amazing progress towards proving this conjecture – it is now proved for all simple groups apart from the Suzuki groups. It was a very happy and successful workshop. Indeed, several of the most distinguished participants commented that this was one of the best Oberwolfach meetings they had attended, with plenty of time for collaboration, and outstanding talks. Extended abstracts of the talks mentioned, as well as all the others, are given below (in the order in which they were presented).

A subgroup H is called c -normal in a group G if there exists a normal subgroup N of G such that HN = G and H ∩ N ≤ H G , where H G ≕ Core( H ) is the maximal normal subgroup of G which is contained in H . We obtain the c -normal subgroups in symmetric and dihedral groups. Also we find the number of c -normal subgroups of order 2 in symmetric groups. We conclude by giving a program in GAP for finding c -normal subgroups. AMS Classification : 20D25. Keywords : c -normal, symmetric, dihedral. Introduction The relationship between the properties of maximal subgroups of a finite group G and the structure of G has been studied extensively. The normality of subgroups in a finite group plays an important role in the study of finite groups. It is well known that a finite group G is nilpotent if and only if every maximal subgroup of G is normal in G . In Wang introduced the concept of c -normality of a finite group. He used the c -normality of a maximal subgroup to give some conditions for the solvability and supersolvability of a finite group. For example, he showed that G is solvable if and only if M is c -normal in G for every maximal subgroup M of G . In this paper, we obtain the c -normal subgroups in symmetric and dihedral groups, and also we find the number of c -normal subgroups of order 2 in symmetric groups.

Conditional diagnosability of multiprocessor systems based on Cayley graphs generated by transpositions

Equivalence classes of matchings and lattice-square designs

In this paper we apply Polya's Theorem to the problem of enumerating Cayley graphs on permutation groups up to isomorphisms induced by conjugacy in the symmetric group. We report the results of a search of all three-regular Cayley graphs on permutation groups of degree at most nine for small diameter graphs. We explore several methods of constructing covering graphs of these Cayley graphs. Examples of large graphs with small diameter are obtained.

Subgroup majorization

The process of imbedding a group in a larger group of some prescribed type has been one of the most useful tools in the investigation of properties of groups. The three principal types of representation of groups, each with its particular field of usefulness, are the following: 1. Permutation groups. 2. Monomial groups. 3. Linear or matrix representations of groups. These three types of representation correspond to an imbedding of the group in the following groups: 1. The symmetric group. 2. The complete monomial group. 3. The full linear group. The symmetric group and the full linear group have both been exhaustively investigated and many of their principal properties are known. A similar study does not seem to exist for the complete monomial group. Such a general theory seems particularly desirable in view of the numerous recent investigations on finite groups in which the monomial representations are used in one form or another to obtain deep-lying theorems on the properties of such groups. The present paper is an attempt to fill this lacuna. In this paper the monomial group or symmetry is taken in the most general sense(') where one considers all permutations of a certain finite number of variables, each variable being multiplied also by some element of a fixed arbitrary group H. In the first chapter the simplest properties such as transformation, normal form, centralizer, etc., are discussed. Some of the auxiliary theorems appear to have independent interest. One finds that the symmetry contains a normal subgroup, the basis group, consisting of all those elements which do not permute the variables. The symmetry splits over the basis group with a group isomorphic to the symmetric group as one representative group. A complete solution of the problem of finding all representative groups in this splitting of the symmetry is given. This result is of interest since it gives a general idea of the solution of the splitting problem in a fairly complicated case. In the second chapter all normal subgroups of the symmetry are deter-

The main aim of this chapter is to prepare students for their first course in abstract algebra. We begin by introducing binary operations and discuss the meaning of a “well-defined” binary operation. Algebraic structures are defined and then we move on to groups, subgroups, and normal subgroups. Proof strategies are presented that deal with these latter two concepts. We also investigate permutation groups and the symmetric group. The chapter also introduces rings and then ends on the topics of quotient algebras, quotient groups, and quotient rings.

We present polynomial-time algorithms for computation in quotient groups G/K of a permutation group G. In effect, these solve, for quotient groups, the problems that are known to be in polynomial-time for permutation groups. Since it is not computationally feasible to represent G/K itself as a permutation group, the methodology for the quotient-group versions of such problems frequently differ markedly from the procedures that have been observed for the K = 1 subcases. Whereas the algorithms for permutation groups may have rested on elementary notions, procedures underlying the extension to quotient groups often utilize deep knowledge of the structure of the group. In some instances, we present algorithms for problems that were not previously known to be in polynomial time, even for permutation groups themselves (K = 1). These problems apparently required access to quotients.

On the theory of groups with extremal layers

Let H be a closed subgroup of a regular abelian paratopological group G. The group reflexion G ( of G is the group G endowed with the strongest group topology, weaker that the original topology of G. We show that the quotient G/H is Hausdorff (and regular) if H is closed (and locally compact) in G ( . On the other hand, we construct an example of a regular abelian paratopological group G containing a closed discrete subgroup H such that the quotient G/H is Hausdorff but not regular. In this paper we study the properties of the quotients of paratopological groups by their normal subgroups. By a paratopological group G we understand a group G endowed with a topology � making the group operation continuous, see (ST). If, in addition, the operation of taking inverse is continuous, then the paratopological group (G;�) is a topological group. A standard example of a paratopological group failing to be a topological group is the Sorgefrey line L, that is the real line R endowed with the Sorgefrey topology (generated by the base consisting of half-intervals (a;b), a < b). Let (G;�) be a paratopological group and HG be a closed normal subgroup of G. Then the quotient group G=H endowed with the quotient topology is a paratopological group, see (Ra). Like in the case of topological groups, the quotient homomorphism � : G ! G=H is open. If the subgroup HG is compact, then the quotient G=H is Hausdorff (and regular) provided so is the group G, see (Ra). The compactness of H in this result cannot be replaced by the local compactness as the following simple example shows. Example 1. The subgroup H = f(−x;x) : x 2 Qg is closed and discrete in the square G = L 2 of the Sorgenfrey line L. Nonetheless, the quotient group G=H fails to be Hausdorff: for any irrational x the coset (−x;x) + H cannot be separated from zero (0;0) + H. A necessary and sufficient condition for the quotientG=H to be Hausdorff is the closedness of H in the topology of group reflexion G ( of G. By the group reflexion G ( = (G;� ( ) of a paratopological group (G;�) we under- stand the group G endowed with the strongest topology � ( � � turning G into a topological group. This topology admits a categorial description: � ( is a unique topology on G such that � (G;� ( ) is a topological group; � the identity homomorphism id : (G;�) ! (G;� ( ) is continuous;

LetH be a hyperbolic normal subgroup of infinite index in a hyperbolic group G. It follows from work of Rips and Sela [16] (see below), that H has to be a free product of free groups and surface groups if it is torsion-free. From [14], the quotient group Q is hyperbolic and contains a free cyclic subgroup. This gives rise to a hyperbolic automorphism [2] of H . By iterating this automorphism, and scaling the Cayley graph of H , we get a sequence of actions of H on δi-hyperbolic metric spaces, where δi → 0 as i → ∞. From this, one can extract a subsequence converging to a small isometric action on a 0-hyperbolic metric space, i.e. an R-tree. By the JSJ splitting of Rips and Sela [16], [17], the outer automorphism group of H is generated by internal automorphisms. One notes further, that a hyperbolic automorphism cannot preserve any splitting over cyclic subgroups and that the limiting action is in fact free. Hence, by a theorem of Rips [16], H has to be a free product of free groups and surface groups if it is torsion-free. Thus the collection of normal subgroups possible is limited. However, the class of groups G can still be fairly large. Examples can be found in [3], [5] and [13]. For the purposes of this paper we choose a finite generating set of G that contains a finite generating set of H . Let ΓG and ΓH be the Cayley graphs of G, H with respect to these generating sets. There is a continuous proper embedding i of ΓH into ΓG. Every hyperbolic group admits a compactification of its Cayley graph by adjoining the Gromov boundary consisting of