Automorphism group

It is shown that the creative r.e. subspaces fall into infinitely many distinct elementary classes. The techniques also extend to give some new results about orbits of creative subspaces and subfields in L ∗ ( V ∞ ) {L^*}({V_\infty }) and L ∗ ( F ∞ ) {L^*}({F_\infty }) respectively. Finally within each of these new elementary classes we construct infinitely many further orbits in the automorphism group of L ( V ∞ ) L({V_\infty }) .

We investigated the benefit of exploiting the symmetries of graphs for partitioning. We represent the model to be simulated by a weighted graph. Graph symmetries are studied in the theory of permutation groups and can be calculated in polynomial time with the nauty algorithm . We designed an algorithm to extract useful symmetries from the automorphism group, which can be used to create partitions derived from the graph's structure. Our approach is focused on composite graphs, for which identical subgraphs reoccur in the graph. If these identical subgraphs can be mapped onto each other by symmetries, the subgraphs are replaced by equivalent multivertices, resulting in a 'natural' aggregation of vertices. This approach is applied to parallel simulation of a detailed IP-switch with a conservative synchronous algorithm. The experimental results show that even for good partitions, global and temporal load imbalances are inevitable.

The Petra Šparl Award was established in 2017 to recognise in each even-numbered year the best paper published in the previous five years by a young woman mathematician in one of the two journals Ars Mathematica Contemporanea (AMC) and The Art of Discrete and Applied Mathematics (ADAM). It was named after Dr Petra Šparl, a talented woman mathematician who died mid-career in 2016 after a battle with cancer. The first award was made in 2018 to Dr Monika Pilśniak (AGH University, Poland) for a paper she published in AMC on the distinguishing index of graphs. Nominations for the 2020 award were invited in 2019, and all cases were considered by a committee (consisting of the three of us, listed below) appointed by the Editors-in-Chief of AMC and ADAM. As judges we were impressed by the large number of papers in these journals over the five years 2015-2019 having a woman author or co-author, with many of these being women in the early stages of their career. With helpful commentaries from co-authors and/or nominators, we drew up a long list of candidates for the 2020 award, sought reports from referees on those, and also considered the papers themselves, before making a decision. In fact it seemed very appropriate to us to celebrate these awards in a year in which Slovenia was to be hosting the European Congress of Mathematicians, by recommending two awards. Sadly the ECM is being postponed to 2021, but we are proceeding with an announcement of the 2020 awards. The two winners of the Petra Šparl Award for 2020 are as follows:

was a talented woman mathematician with a promising future who worked in graph theory and combinatorics, but died mid-career in 2016 after a battle with cancer. In her memory, the Petra Šparl Award was established recently to recognise in each even-numbered year the best paper published in the previous five years by a young woman mathematician in one of the two journals Ars Mathematica Contemporanea (AMC) and The Art of Discrete and Applied Mathematics (ADAM). Nominations for the inaugural award were invited in AMC in 2017, and cases were considered by a committee (consisting of the three of us) appointed by Dragan Marušič and Tomaž Pisanski as editors of AMC and ADAM. As judges we were impressed by the large number of papers in AMC over the five years 2013-2017 having a woman author or co-author: almost 60 in total, with well over half of those being women in the early stages of their career. With helpful commentaries from co-authors (in some cases) we drew up a long list of candidates for the 2018 award, sought reports from referees on those, and also considered the papers themselves, before making a decision, which was unanimous. The winner of the Petra Šparl

A graph is said to be 1 2 -transitive if its automorphism group acts transitively on vertices and edges but not on arcs. For each n 11, a 1 2 -transitive graph of valency 4 and girth 6, with the automorphism group isomorphic to A n _Z 2 , is given.

A graph is said to be one-regular if its automorphism group acts regularly on the set of its arcs. A construction of an infinite family of infinite one-regular graphs of valency 4 is given. These graphs are Cayley graphs of almost abelian groups and hence of polynomial growth.

The problem of lifting graph automorphisms along covering projections is considered in a purely combinatorial setting. Because of certain natural applications and greater generality, graphs are allowed to have semiedges. This requires careful reexamination of the whole subject, which leads to simpli cation and generalization of several known results. For example, Cayley graphs, including those with involutory or repeated generators, are characterized as regular coverings over one-vertex graphs. The ordinary and the permutation voltage constructions are uni ed into the concept of a voltage space, constituting the crucial tool for combinatorialization of the lifting problem. Particular attention is paid to the structure of lifted groups, with focus on split extensions having a nice geometrical and combinatorial description. Some applications of these results to regular maps on surfaces are given.

An n-bicirculant is a graph having an automorphism with two orbits of length n and no other orbits. Symmetry properties of p-bicirculants, p a prime, are extensively studied. In particular, the actions of their automorphism groups are described in detail in terms of certain algebraic representation of such graphs.

A graph X is said to be distance-balanced if for any edge uv of X, the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. A graph X is said to be strongly distance-balanced if for any edge uv of X and any integer k, the number of vertices at distance k from u and at distance k + 1 from v is equal to the number of vertices at distance k + 1 from u and at distance k from v. Obviously, being distance-balanced is metrically a weaker condition than being strongly distance-balanced. In this paper, a connection between symmetry properties of graphs and the metric property of being (strongly) distance-balanced is explored. In particular, it is proved that every vertex-transitive graph is strongly distance-balanced. A graph is said to be semisymmetric if its automorphism group acts transitively on its edge set, but does not act transitively on its vertex set. An infinite family of semisymmetric graphs, which are not distance-balanced, is constructed. Finally, we give a complete classification of strongly distance-balanced graphs for the following infinite families of generalized Petersen graphs: GP(n, 2), GP(5k +1, k), GP(3k ± 3, k), and GP(2k + 2, k).

The main aim of these lectures is to study the connection between symplectic symmetries of K3 surfaces and the Mathieu group M24, and its Enriques analogy, that is, a conjectural connection between semi-symplectic symmetries of Enriques surfaces and another Mathieu group M12. Algebraic varieties are the subject of study of algebraic geometers. The place of K3 and Enriques surfaces among them can be depicted in the following way:

Let G be a group. The orbits of the natural action of Aut(G) on G are called "automorphism orbits" of G, and the number of automorphism orbits of G is denoted by ω(G). We prove that if G is a soluble group with finite rank such that ω(G) < ∞, then G contains a torsion-free characteristic nilpotent subgroup K such that G = K ⋊ H, where H is a finite group. Moreover, we classify the mixed order soluble groups of finite rank such that ω(G) = 3.

Denote by ω(G) the number of orbits of the action of Aut(G) on the finite group G. We prove that if G is a finite nonsolvable group in which ω(G) 5, then G is isomorphic to one of the groups A 5 , A 6 , P SL(2, 7) or P SL(2, 8). We also consider the case when ω(G) = 6 and show that if G is a nonsolvable finite group with ω(G) = 6, then either G ≃ P SL(3, 4) or there exists a characteristic elementary abelian 2-subgroup N of G such that G/N ≃ A 5 .

Let G be a group. The orbits of the natural action of Aut(G) on G are called "automorphism orbits" of G, and the number of automorphism orbits of G is denoted by ω(G). In this paper the finite non-solvable groups G with ω(G) ≤ 6 are classified -this solves a problem posed by Markus Stroppel -and it is proved that there are infinitely many finite non-solvable groups G with ω(G) = 7. Moreover it is proved that for a given number n there are only finitely many finite groups G without non-trivial abelian normal subgroups and such that ω(G) ≤ n, generalizing a result of Kohl. MG was supported by FEMAT-Fundação de Estudos em Ciências Matemáticas Proc. 037/2016.

Let G be a group. The orbits of the natural action of Aut(G) on G are called "automorphism orbits" of G, and the number of automorphism orbits of G is denoted by ω(G). We prove that if G is a soluble group with finite rank such that ω(G) < ∞, then G contains a torsion-free characteristic nilpotent subgroup K such that G = K ⋊ H, where H is a finite group. Moreover, we classify the mixed order soluble groups of finite rank such that ω(G) = 3.

Let G be a group. The orbits of the natural action of Aut ⁡ ( G ) {\operatorname{Aut}(G)} on G are called “automorphism orbits” of G, and the number of automorphism orbits of G is denoted by ω ⁢ ( G ) {\omega(G)} . In this paper the finite non-solvable groups G with ω ⁢ ( G ) ≤ 6 {\omega(G)\leq 6} are classified – this solves a problem posed by Markus Stroppel – and it is proved that there are infinitely many finite non-solvable groups G with ω ⁢ ( G ) = 7 {\omega(G)=7} . Moreover, it is proved that for a given number n there are only finitely many finite groups G without non-trivial abelian normal subgroups and such that ω ⁢ ( G ) ≤ n {\omega(G)\leq n} , generalizing a result of Kohl.

Denote by ω(G) the number of orbits of the action of Aut(G) on the finite group G. We prove that if G is a finite nonsolvable group in which ω(G) 5, then G is isomorphic to one of the groups A 5 , A 6 , P SL(2, 7) or P SL(2, 8). We also consider the case when ω(G) = 6 and show that if G is a nonsolvable finite group with ω(G) = 6, then either G ≃ P SL(3, 4) or there exists a characteristic elementary abelian 2-subgroup N of G such that G/N ≃ A 5 .

Richardson varieties play an important role in intersection theory and in the geometric interpretation of the Littlewood-Richardson Rule for flag varieties. We discuss three natural generalizations of Richardson varieties which we call projection varieties, intersection varieties, and rank varieties. In many ways, these varieties are more fundamental than Richardson varieties and are more easily amenable to inductive geometric constructions. In this paper, we study the singularities of each type of generalization. Like Richardson varieties, projection varieties are normal with rational singularities. We also study in detail the singular loci of projection varieties in Type A Grassmannians. We use Kleiman's Transversality Theorem to determine the singular locus of any intersection variety in terms of the singular loci of Schubert varieties. This is a generalization of a criterion for any Richardson variety to be smooth in terms of the nonvanishing of certain cohomology classes which has been known by some experts in the field, but we don't believe has been published previously. Contents 1. Introduction 1 2. The singularities of intersection varieties and Richardson varieties 4 3. Projection Varieties 10 4. Singularities of Grassmannian projection varieties 13 References 25

The component-wise or Schur product C ∗C of two linear error correcting codes C and C over certain finite field is the linear code spanned by all component-wise products of a codeword in C with a codeword in C. When C = C, we call the product the square of C and denote it C. Motivated by several applications of squares of linear codes in the area of cryptography, in this paper we study squares of so-called matrixproduct codes, a general construction that allows to obtain new longer codes from several “constituent” codes. We show that in many cases we can relate the square of a matrix-product code to the squares and products of their constituent codes, which allow us to give bounds or even determine its minimum distance. We consider the well-known (u, u+v)-construction, or Plotkin sum (which is a special case of a matrix-product code) and determine which parameters we can obtain when the constituent codes are certain cyclic codes. In addition, we use the same techniques to study the ...

The covering number of a nontrivial finite group $G$, denoted $\sigma(G)$, is the smallest number of proper subgroups of $G$ whose set-theoretic union equals $G$. In this article, we focus on a dual problem to that of covering numbers of groups, which involves maximal subgroups of finite groups. For a nontrivial finite group $G$, we define the intersection number of $G$, denoted $\iota(G)$, to be the minimum number of maximal subgroups whose intersection equals the Frattini subgroup of $G$. We elucidate some basic properties of this invariant, and give an exact formula for $\iota(G)$ when $G$ is a nontrivial finite nilpotent group. In addition, we determine the intersection numbers of a few infinite families of non-nilpotent groups. We conclude by discussing a generalization of the intersection number of a nontrivial finite group and pose some open questions about these invariants.

We prove that the automorphism group of the braid group on four strands acts faithfully and geometrically on a CAT(0) 2-complex. This implies that the automorphism group of the free group of rank two acts faithfully and geometrically on a CAT(0) 2-complex, in contrast to the situation for rank three and above.

We consider complex K3 surfaces with a non-symplectic group acting trivially on the algebraic cycles. Vorontsov and Kondō classified those K3 surfaces with transcendental lattice of minimal rank. The purpose of this note is to study the Galois representations associated to these K3 surfaces. The rank of the transcendental lattices is even and varies from 2 to 20, excluding 8 and 14. We show that these K3 surfaces are dominated by Fermat surfaces, and hence they are all of CM type. We will establish the modularity of the Galois representations associated to them. Also we discuss mirror symmetry for these K3 surfaces in the sense of Dolgachev, and show that a mirror K3 surface exists with one exception.

In this survey the structure of one of the powerful group presentations, introduced by Alain Bretto is studied which is called G-graph. This survey contains some sections of properties, examples, characterization and groups automorphism of G-graphs. There are also some well-known graphs that have the structure of G-graphs. Moreover, in this survey we compare G-graph with the Cayley graph especially to examine the Hamiltonian property of such graphs for a given group. Furthermore by using the construction of a hypergraph, the spectra of the G-graph is reported. Finally, in the last section, the structure of the G-graph is considered as a network and connectivity property is studied.

Let Ω be a m-set, where m > 1, is an integer. The Hamming graph H(n, m), has Ω n as its vertex-set, with two vertices are adjacent if and only if they differ in exactly one coordinate. In this paper, we provide a proof on the automorphism group of the Hamming graph H(n, m), by using elementary facts of group theory and graph theory.

Let Ω be a m-set, where m > 1, is an integer. The Hamming graph H(n, m), has Ω n as its vertex-set, with two vertices are adjacent if and only if they differ in exactly one coordinate. In this paper, we provide a proof on the automorphism group of the Hamming graph H(n, m), by using elementary facts of group theory and graph theory.

Let $\Omega$ be a $m$-set, where $m&gt;1$, is an integer. The Hamming graph $H(n,m)$, has $\Omega ^{n}$ as its vertex-set, with two vertices are adjacent if and only if they differ in exactly one coordinate. In this paper, we provide a proof on the automorphism group of the Hamming graph $H(n,m)$, by using elementary facts of group theory and graph theory.

We prove that the only primes which may divide the order of the automorphism group of a putative binary self-dual doubly-even [120, 60, 24] code are 2, 3, 5, 7, 19, 23 and 29. Furthermore we prove that automorphisms of prime order p ≥ 5 have a unique cycle structure.

We prove that the only primes which may divide the order of the automorphism group of a putative binary self-dual doubly-even [120, 60, 24] code are 2, 3, 5, 7, 19, 23 and 29. Furthermore we prove that automorphisms of prime order p ≥ 5 have a unique cycle structure.

Let V be a simple vertex operator algebra which admits the continuous, faithful action of a compact Lie group G of automorphisms. We establish a Schur-Weyl type duality between the unitary, irreducible modules for G and the irreducible modules for V G which are contained in V where V G is the space of G-invariants of V. We also prove a concomitant Galois correspondence between vertex operator subalgebras of V which contain V G and closed Lie subgroups of G in the case that G is abelian. These results extend those of [DM1] and [DM2].

We prove that the automorphism group of P(ω)/fin remains simple if ℵ 2 Cohen reals are added to a model of ZFC + CH. 1 1 This paper is based on a part of author's Dissertation written under the supervision of Professor Sabine Koppelberg whom the author would like to thank for her encouragement during the preparation of the thesis. Thanks are also due to the referee for some valuable suggestions on improvement of the formulation.

An abstract polytope of rank n is said to be chiral if its automorphism group has precisely two orbits on the flags, such that adjacent flags belong to distinct orbits. The present paper describes a general method for deriving new finite chiral polytopes from old finite chiral polytopes of the same rank. In particular, the technique is used to construct many new examples in ranks 3, 4 and 5.

A (face-)primer hypermap is a regular oriented hypermap with no regular proper quotients with the same number of hyperfaces. Primer hypermaps are then regular hypermaps whose automorphism groups induce faithful actions on their hyperfaces. In this paper we classify the primer hypermaps with a prime number of hyperfaces. This classification generalises an earlier dual classification by Du, Kwak and Nedela of regular oriented maps with a prime number of vertices and simple underlying graph.

A hypermap is (hypervertex-) bipartite if its hypervertices can be 2-coloured in such a way that "neighbouring" hypervertices have different colours. It is bipartiteuniform if within each of the sets of hypervertices of the same colour, hyperedges and hyperfaces, all the elements have the same valency. The flags of a bipartite hypermap are naturally 2-coloured by assigning the colour of its adjacent hypervertices. A hypermap is bipartite-regular if the automorphism group acts transitively on each set of coloured flags. If the automorphism group acts transitively on the set of all flags, the hypermap is regular. In this paper we classify the bipartite-uniform hypermaps on the sphere (up to duality). Two constructions of bipartite-uniform hypermaps are given. All bipartite-uniform spherical hypermaps are shown to be constructed in this way. As a by-product we show that every bipartite-uniform hypermap H on the sphere is bipartite-regular. We also compute their irregularity group and index, and also their closure cover H ∆ and covering core H ∆ .

In this paper we classify the reflexible and chiral regular oriented maps with p faces of valency n, and then we compute the asymptotic behaviour of the reflexible to chiral ratio of the regular oriented maps with p faces. The limit depends on p and for certain primes p we show that the limit can be 1, greater than 1 and less than 1. In contrast, the reflexible to chiral ratio of regular polyhedra (which are regular maps) with Suzuki automorphism groups, computed by Hubard and Leemans [13], has produced a nill asymptotic ratio.

A (face-)primer hypermap is a regular oriented hypermap with no regular proper quotients with the same number of hyperfaces. Primer hypermaps are then regular hypermaps whose automorphism groups induce faithful actions on their hyperfaces. In this paper we classify the primer hypermaps with a prime number of hyperfaces. This classification generalises an earlier dual classification by Du, Kwak and Nedela of regular oriented maps with a prime number of vertices and simple underlying graph.

A subgroup G of automorphisms of a graph X is said to be 1 2 -arc-transitive if it is vertex-and edge-but not arc-transitive. The graph X is said to be 1 2 -arc-transitive if Aut X is 1 2 -arc-transitive. The interplay of two different concepts, maps and hypermaps on one side and 1 2 -arc-transitive group actions on graphs on the other, is investigated. The correspondence between regular maps and 1 2 -arctransitive group actions on graphs of valency 4 given via the well known concept of medial graphs (European J. Combin. 19 (1998) 345) is generalised. Any orientably regular hypermap H gives rise to a uniquely determined medial map whose underlying graph Y admits a 1 2 -arc-transitive group action of the automorphism group G of the original hypermap H. Moreover, the vertex stabiliser of the action of G on Y is cyclic. On the other hand, given graph X and G ≤ Aut X acting 1 2 -arc-transitively with a cyclic vertex stabiliser, we can construct an orientably regular hypermap H with G being the orientation preserving automorphism group. In particularly, if the graph X is 1 2 -arc-transitive, the corresponding hypermap is necessarily chiral, that is, not isomorphic to its mirror image. Note that the associated 1 2 -arc-transitive group action on the medial graph induced by a map always has a stabiliser of order two, while when it is induced by a (pure) hypermap the stabiliser can be cyclic of arbitrarily large order. Hence moving from maps to hypermaps increases our chance of getting different types of 1 2 -arc-transitive group action. Indeed, in last section we have applied general results to construct 1 2 -arc-transitive graphs with cycle stabilisers of arbitrarily large orders.

In this paper, we observe the possibility that the group \(S_{n}\times S_{m}\) acts as a flag-transitive automorphism group of a block design with point set \(\{1,\ldots ,n\}\times \{1,\ldots ,m\},4\leq n\leq m\leq 70\). We prove the equivalence of that problem to the existence of an appropriately defined smaller flag-transitive incidence structure. By developing and applying several algorithms for the construction of the latter structure, we manage to solve the existence problem for the desired designs with \(nm\) points in the given range. In the vast majority of the cases with confirmed existence, we obtain all possible structures up to isomorphism.

In this paper we consider the possibility that groups S n wr S 2 , 4 ≤ n ≤ 63, act flagtransitively on block designs with n 2 points. We rule out n ∈ {7, 15, 19, 23, 31, 35, 43, 47, 59} and confirm the required action for other n in the given range through performing the construction of base blocks of the desired designs. Our construction makes use of flagtransitive incidence structures and weakly flag-transitive incidence structures.

Using the list of 2607 so far constructed (96,20,4) difference sets as a source, we checked the related symmetric designs upon isomorphism and analyzed their full automorphism groups. New (96, and (96,19,2,4) regular partial difference sets are constructed, together with the corresponding strongly regular graphs.

We present the results of a research which aims to determine, up to isomorphism and complementation, all primitive block designs with the projective line Fq ∪ {∞} as the set of points and PSL (2, q) as an automorphism group. The obtained designs are classified by the type of a block stabilizer. The results are complete, except for the designs with block stabilizers in the fifth Aschbacher's class. In particular, the problem is solved if q is a prime. We include formulas for the number of such designs with q = p 2 α 3 β , α, β nonnegative integers. 2010 Mathematics Subject Classification. 05B05.

We discuss the notion of criticality of semilinear differential equations and systems, its relations to scaling transformations and the Noether approach to Pokhozhaev's identities. For this purpose we propose a definition for criticality based on the S. Lie symmetry theory. We show that this definition is compatible with the well-known notion of critical exponent by considering various examples. We also review some related recent papers.

For a subgroup L of the symmetric group S ℓ , we determine the minimal base size of GL d (q) ≀ L acting on V d (q) ℓ as an imprimitive linear group. This is achieved by computing the number of orbits of GL d (q) on spanning m-tuples, which turns out to be the number of d-dimensional subspaces of Vm(q). We then use these results to prove that for certain families of subgroups L, the affine groups whose stabilisers are large subgroups of GL d (q) ≀ L satisfy a conjecture of Pyber concerning bases.

This thesis is concerned with the following two areas in the theory of finite permutation groups: A. transitive groups of degree p, where p -4q+1 and p,q are prime numbers, B. automorphism groups of 2-graphs and some related algebras. These two subjects will be referred to as Problem A and Problem B? we introduce them separately. Chapter 3 of this thesis consists of a proof of this theorem and some results needed for this proof are given in Chapter 1. In Chapter 2 we prove some general facts about 2-transitive and 3-transitive groups; most of these involve deductions about the way a group G acts on a set n. from information about the permutation characters *S ,^ of G associated with Jlw , JT. 1 respectively (where fl0"" and Jlf are the sets of ordered and unordered pairs of distinct elements ofjfL). The most important fact is: PROPOSITION 2.2. Suppose that G is generously 2-transitive and 2-priraitive of degree n on fl , and let k be the rank of G^ on _n_\{o<} (where o£ e .Q_). Then ||l|( -\\-q\\ ^ k 2+k-2; and if equality holds then either k = 2 or k = 3 and n is even. This fact is used in the proof of Theorem 3.1-The results in Chapter 2 on 3-transitive groups are used in Chapter 4 f where further study of transitive insoluble groups of degree p = 4q+1 is carried out. 'and go on till you come to the end: then stop. '"