EMBEDDING PERMUTATION GROUPS INTO WREATH PRODUCTS IN PRODUCT ACTION

Deterministic polynomial factoring over finite fields: A uniform approach via [formula omitted]-schemes

First we define and study the exponentiation of a cellular algebra by a permutation group which is analogous to the corresponding operation (the wreath product in primitive action) in permutation group theory. Necessary and sufficient conditions for the resulting cellular algebra to be primitive and Schurian are given. This enables us to construct infinite series of primitive non-Schurian algebras. Also we define and study for cellular algebras the notion of a base which is similar to that for permutation groups. We present an upper bound for the size of an irredundant base of a primitive cellular algebra in terms of the parameters of its standard representation. This produces new upper bounds for the order of the automorphism group of such an algebra and in particular for the order of a primitive permutation group. Finally we generalize to 2-closed primitive algebras some classical theorems for primitive groups and show that the hypothesis for a primitive algebra to be 2-closed is essential.

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).

An asymmetric coloring of a graph is a coloring of its vertices that is not preserved by any non-identity automorphism of the graph. The motion of a graph is the minimal degree of its automorphism group, i.e., the minimum number of elements that are moved (not fixed) by any non-identity automorphism. We confirm Tom Tucker's “Infinite Motion Conjecture” that connected locally finite graphs with infinite motion admit an asymmetric 2-coloring. We infer this from the more general result that the inverse limit of an infinite sequence of finite permutation groups with disjoint domains, viewed as a permutation group on the union of those domains, admits an asymmetric 2-coloring. The proof is based on the study of the interaction between epimorphisms of finite permutation groups and the structure of the setwise stabilizers of subsets of their domains. We note connections of the subject to computational group theory, asymptotic group theory, highly regular structures, and the Graph Isomorphism problem, and list a number of open problems.

The purpose of this thesis is to examine the narrative of images and the form of visual art in Hayao Miyazaki's “Ponyo on the Cliff” from the viewpoint of affect. For the multilateral interpretation of the worldview implied by the text, an attempt to explore the movement and shape difference of objects from the aspect of affect is necessary. The approach through the 'group theory' that generalizes the connection between these affective imagination and semantic motifs expands the meaning recognition of growth stories. As a result of examining the transformation of geometric shapes in animation pictures as a symmetrical group and exploring the correspondence between narrative connection and narrative as a permutation group, the subject matter became clearer and richer. This article has the significance of considering the narration and morphological symbols of animation in terms of the convergence of art and learning, and presenting a methodology to analyze the morphological properties as affect as a group theory.

Direct product and uniqueness of automorphism groups of graphs

This book is a collection of articles, some introductory, some extended surveys, and some containing previously unpublished research, on a range of topics linking infinite permutation group theory and model theory. Topics covered include: oligomorphic permutation groups and omega-categorical structures; totally categorical structures and covers; automorphism groups of recursively saturated structures; Jordan groups; Hrushovski's constructions of pseudoplanes; permutation groups of finite Morley rank; applications of permutation group theory to models of set theory without the axiom of choice. There are introductory chapters by the editors on general model theory and permutation theory, recursively saturated structures, and on groups of finite Morley rank. The book is almost self-contained, and should be useful to both a beginning postgraduate student meeting the subject for the first time, and to an active researcher from either of the two main fields looking for an overview of the subject.

If V is a (possibly infinite) set, G a permutation group on $${V, v\in V}$$ , and Ω is an orbit of the stabiliser G v , let $${G_v^{\Omega}}$$ denote the permutation group induced by the action of G v on Ω, and let N be the normaliser of G in Sym(V). In this article, we discuss a relationship between the structures of G v and $${G_v^\Omega}$$ . If G is primitive and G v is finite, then by a theorem of Betten et al. (J Group Theory 6:415–420, 2003) we can conclude that every composition factor of the group G v is also a composition factor of the group $${G_v^{\Omega(v)}}$$ . In this paper we generalize this result to possibly imprimitive permutation groups G with infinite vertex-stabilisers, subject to certain restrictions that can be expressed in terms of the natural permutation topology on Sym(V). In particular, we show the following: If $${\Omega=u^{G_v}}$$ is a suborbit of a transitive closed subgroup G of Sym(V) with a normalizing overgroup N ≤ N Sym(V)(G) such that the N-orbital $${\{(v^g,u^g) \mid u\in \Omega, g\in N\}}$$ is locally finite and strongly connected (when viewed as a digraph on V), then every closed simple section of G v is also a section of $${G_v^\Omega}$$ . To demonstrate that the topological assumptions on G and the simple sections of G v cannot be omitted in this statement, we give an example of a group G acting arc-transitively on an infinite cubic tree, such that the vertex-stabiliser G v is isomorphic to the modular group $${{\rm PSL}(2,\mathbb{Z}) \cong C_2*C_3}$$ , which is known to have infinitely many finite simple groups among its sections.

Association schemes and permutation groups

A foundational area of mathematics, group theory has broad applications in many scientific domains. An integrated review of its use in cryptography and chemistry is presented in this work. Group theory simplifies structural predictions in chemistry by making it easier to analyze molecular symmetries, molecular orbitals, and spectroscopy. Group theory in cryptography provides the security underpinnings of contemporary encryption techniques, especially through discrete logarithm issues, homomorphisms, and permutation groups. We examine group-theoretical isomorphisms, homomorphism stability, and representations using both recent and classic literature, emphasizing their dual applicability in the information and biological sciences. Opportunities for interdisciplinary advances, such as improved chemical modeling and post-quantum cryptography protocols, are suggested by the synergy between these domains.

We prove two theorems in the ergodic theory of infinite permutation groups. First, generalizing a theorem of Nessonov for the infinite symmetric group, we show that every non-singular action of a non-archimedean, Roelcke precompact, Polish group on a measure space $(\Omega , \mu )$ admits an invariant $\sigma $-finite measure equivalent to $\mu $. Second, we prove the following de Finetti-type theorem: if $G \curvearrowright M$ is a primitive permutation group with no algebraicity verifying an additional uniformity assumption, which is automatically satisfied if $G$ is Roelcke precompact, then any $G$-invariant, ergodic probability measure on $Z^{M}$, where $Z$ is a Polish space, is a product measure.

A permutation group H acting on a set X is said to be graphical if there is a graph G such that Γ(G), the automorphism group of G, is identical to H. Characterisation of graphical permutation groups seems to be difficult. Kagno and Chao have shown that the group generated by a single m-cycle is not graphical. Here we study the group generated by a permutation such that it consists of disjoint cycles whose lengths are multiples of the length of one of its cycles. Our results are obtained by constructing certain graphs which we call generalised permutation graphs. We also study graphical cycle permutation groups of order pm where p is a prime.

A proof of a conjecture of Sabidussi on graphs idempotent under the lexicographic product

We describe the integration of permutation group algorithms with proof planning. We consider eight basic questions arising in computational permutation group theory, for which our code provides both answers and a set of certificates enabling a user, or an intelligent software system, to provide a full proof of correctness of the answer. To guarantee correctness we use proof planning techniques, which construct proofs in a human-oriented reasoning style. This gives the human mathematician the necessary insight into the computed solution, as well as making it feasible to check the solution for relatively large groups.

A rack of order $n$ is a binary operation $\vartriangleright$ on a set $X$ of cardinality $n$, such that right multiplication is an automorphism. More precisely, $(X,\vartriangleright)$ is a rack provided that the map $x\mapsto x\vartriangleright y$ is a bijection for all $y\in X$, and $(x\vartriangleright y)\vartriangleright z=(x\vartriangleright z)\vartriangleright (y\vartriangleright z)$ for all $x,y,z\in X$.The paper provides upper and lower bounds of the form $2^{cn^2}$ on the number of isomorphism classes of racks of order $n$. Similar results on the number of isomorphism classes of quandles and kei are obtained. The results of the paper are established by first showing how an arbitrary rack is related to its operator group (the permutation group on $X$ generated by the maps $x\mapsto x\vartriangleright y$ for $y\in Y$), and then applying some of the theory of permutation groups. The relationship between a rack and its operator group extends results of Joyce and of Ryder; this relationship might be of independent interest.