Our aim is to calculate some graphs associated with two of the larger sporadicsimple groups, Fi₂₄ and the Baby Monster. Firstly we calculate the point line collinearity graph for a maximal 2-local geometry of Fi₂₄. If T is such a geometry, then the point line collinearity graph G will be the graph whose vertices are the points in T, with any two vertices joined by an edge if and only if they are incident with a common line. We found that the graph has diameter 5 and we give its collapsed adjacency matrix. We also calculate part of the commuting involution graph, C, for the class 2C of the Baby Monster, whose vertex set is the conjugacy class 2C, with any two elements joined by an edge if and only if they commute. We have managed to place all vertices inside C whose product with a fixed vertex t does not have 2 power order, with all evidence pointing towards C having diameter 3.

In this thesis, we will see a way to use representation theory and the theory of linear algebraic groups to characterize certain family of finite groups. In Chapter 1, we see the history of preceding work. In particular, J. G. Thompson’s classification of minimal finite simple nonsolvable groups and characterization of solvable groups will be given. In Chapter 2, we will describe some background knowledge underlying this project and notation that will be widely used in this thesis. In Chapter 3, the main theorem originally conjectured by Guralnick and Tiep will be stated together with the base theorem which is a reduced version of main theorem to the case where we have a quasisimple group. Main theorem explains a way to characterize the finite groups with a composition factor of order divisible by two distinct primes p and q as the finite groups containing nontrivial 2-element x, p-element y, q-element z such that xyz = 1. In this thesis we more focus on the proof of showing a finite group G with a composition factor of order divisible by two distinct prime p and q contains nontrivial 2-element x, p-element y, q-element z such that xyz = 1. In Chapter 4, we will prove a set of lemmas and proposition which will be used as key tools in the proof of the base theorem. In Chapters 5 to 7, we will establish the base theorem in the cases where a quasisimple group G has its simple quotient isomorphic to alternating groups or sporadic groups (Chapter 5), classical groups (Chapter 6), and exceptional groups (Chapter 7). In Chapter 8, we show that any finite group G admitting nontrivial 2-element x, p- element y, q-element z such that xyz = 1 for two distinct odd primes p and q admits a composition factor of order divisible by pq. Also, we show that the question if a finite group G with a composition factor of order divisible by two distinct prime p and q contains nontrivial 2-element x, p-element y, q-element z such that xyz = 1 can be reduced to the base theorem.

No description available.

In any group G, we may extend the definition of the conjugacy class of an element to the conjugacy class of a k-tuple, for a positive integer k. When k = 2, we are forming the conjugacy classes of ordered pairs, when k = 3, we are forming the conjugacy classes of ordered triples, etc. In this report we explore a generalized question which Professor B. Doug Park has posed (for k = 2). For an arbitrary k, is it true that: (G has finitely many k-conjugacy classes) implies (G is finite)? Supposing to the contrary that there exists an infinite group G which has finitely many k-conjugacy classes for all k = 1, 2, 3, ..., we present some preliminary analysis of the properties that G must have. We then investigate known classes of groups having some of these properties: universal locally finite groups, existentially closed groups, and Engel groups.

In any group G, we may extend the definition of the conjugacy class of an element to the conjugacy class of a k-tuple, for a positive integer k. When k = 2, we are forming the conjugacy classes of ordered pairs, when k = 3, we are forming the conjugacy classes of ordered triples, etc. In this report we explore a generalized question which Professor B. Doug Park has posed (for k = 2). For an arbitrary k, is it true that: (G has finitely many k-conjugacy classes) implies (G is finite)? Supposing to the contrary that there exists an infinite group G which has finitely many k-conjugacy classes for all k = 1, 2, 3, ..., we present some preliminary analysis of the properties that G must have. We then investigate known classes of groups having some of these properties: universal locally finite groups, existentially closed groups, and Engel groups.

In this thesis, we mainly address two contrasting topics in the area of saturated fusion systems. The first concerns the notion of a centraliser of a subsystem E of a fusion system F, and we give new proofs of the existence of such an object in the case where E is normal in F. The second concerns the development of the theory of `trees of fusion systems', an analogue for fusion systems of Bass-Serre theory for finite groups. A major theorem finds conditions on a tree of fusion systems for there to exist a saturated completion, and this is applied to construct and classify certain fusion systems over p-groups with an abelian subgroup of index p. Results which do not fall into either of the above categories include a new proof of Thompson's normal p-complement Theorem for saturated fusion systems and characterisations of certain quotients of fusion systems which possess a normal subgroup.

The foundation of the work of this thesis is based around the involutory progenitor and the finite homomorphic images found therein. This process is developed by Robert T. Curtis and he defines it as 2^{*n} :N {pi w | pi in N, w} where 2^{*n} denotes a free product of n copies of the cyclic group of order 2 generated by involutions. We repeat this process with different control groups and a different array of possible relations to discover interesting groups, such as sporadic, linear, or unitary groups, to name a few. Predominantly this work was produced from transitive groups in 6,10,12, and 18 letters. Which led to identify some appealing groups for this project, such as Janko group J1, Symplectic groups S(4,3) and S(6,2), Mathieu group M12 and some linear groups such as PGL2(7) and L2(11) . With this information, we performed double coset enumeration on some of our findings, M12 over L_2(11) and L_2(31) over D15. We will also prove their isomorphism types with the help of the Jordan-Holder theorem, which aids us in defining the make up of the group. Some examples that we will encounter are the extensions of L_2(31)(center) 2 and A5:2^2.

La première partie est consacrée à l'étude d'attaques génériques sur des schémas de Feistel dissymétriques. Ces attaques sont en fait des distingueurs qui calculent sur une partie des clairs-chiffrés le nombre de paires vérifiant un système d'égalités et de non-égalités sur un groupe fini. La recherche de ce type d'attaques a été automatisée et améliorée, notamment en tenant compte de goulots d'étranglement. Plus généralement, des travaux sur ce type de systèmes, que l'on désigne par les termes << théorie du miroir >> sont exposés dans cette partie. En particulier, on décrit le problème de la somme de deux bijections sur un groupe fini.La deuxième partie décrit un des candidats à la compétition SHA-3 : la fonction de hachage CRUNCH. Cette fonction reprend un schéma de Feistel dissymétrique et utilise la somme de deux bijections. De plus, un nouveau mode d'enchaînement a été utilisé.Dans la dernière partie on traite de problème d'authentification à divulgation nulle de connaissance. D'abord avec les polynômes à plusieurs variables, puis avec un problème difficile lié aux groupes symétriques. Une illustration est donnée avec le groupe du Rubik's Cube.Enfin une méthode originale pour tenter de trouver une solution aux équations de Brent est donnée en annexe. / The first part is dedicated to the study of generic attacks in unbalanced Feistel schemes. All these attacks are distinguishers that counts how many number of couples (plain text, cipher text) verify a system of equalities and non-equalities on a finite groupe. With the help of algorithms we have found all the possible attacks, and some attacks with a neck bottle have been rejected automatically. More generally, we describe some works about the "mirror theory" that deals about that kind of systems. We specially describe the problem of the sum of two bijections in a finite group.The second part describes one of the candidate of the SHA-3 competition : the hash function called CRUNCH. This function includes the sum of two bijections, and each bijection is an unbalanced Feistel Scheme. A new chaining process for long messages is given.In the last part we deal with zero-knowledge authentication problems. The first protocol is based on multivariate polynomials. The second is linked to a difficult problem in symmetric groups. We take the example of the Rubik's cube group.Finally, we reveal some works on Brent equations. We build an algorithm that may find one solution.

viii, 107 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / This dissertation consists of two related parts. In the first portion we use the tracial Rokhlin property for actions of a finite group G on stably finite simple unital C *-algebras containing enough projections. The main results of this part of the dissertation are as follows. Let A be a stably finite simple unital C *-algebra and suppose a is an action of a finite group G with the tracial Rokhlin property. Suppose A has real rank zero, stable rank one, and suppose the order on projections over A is determined by traces. Then the crossed product algebra C * ( G, A, à à ±) also has these three properties. In the second portion of the dissertation we introduce an analogue of the tracial Rokhlin property for C *-algebras which may not have any nontrivial projections called the projection free tracial Rokhlin property . Using this we show that under certain conditions if A is an infinite dimensional simple unital C *-algebra with stable rank one and à à ± is an action of a finite group G with the projection free tracial Rokhlin property, then C * ( G, A, à à ±) also has stable rank one. / Adviser: Phillips, N. Christopher

In 1896, Frobenius began the study of character theory while factoring the group determinant. Later in 1963, Brauer pointed out that the relationship between characters and their groups was still not fully understood. He published a series of questions that he felt would be important to resolve. In response to these questions, Johnson, Mattarei, and Sehgal developed the idea of a weak Cayley table map between groups. The set of all weak Cayley table maps from one group to itself also has a group structure, which we will call the weak Cayley table group. We will examine the weak Cayley table group of AGL(1; p) and the dicyclic groups, a nd a normal subgroup of the weak Cayley table group for a special case with Camina pairs and Semi-Direct products with a normal Hall-π subgroup, and look at some nontrivial weak Cayley table elements for certain p-groups. We also define a relative weak Cayley table and a relative weak Cayley table map. We will examine the relationship between relative weak Cayley table maps and weak Cayley table maps, automorphisms and anti-automorphisms, characters and spherical functions.