A group G is called locally finite if every finitely generated subgroup of G is finite. In this thesis we study the centralizers of subgroups in simple locally finite groups. Hartley proved that in a linear simple locally finite group, the fixed point of every semisimple automorphism contains infinitely many elements of distinct prime orders. In the first part of this thesis, centralizers of finite abelian subgroups of linear simple locally finite groups are studied and the following result is proved: If G is a linear simple locally finite group and A is a finite d-abelian subgroup consisting of semisimple elements of G, then C_G(A) has an infinite abelian subgroup isomorphic to the direct product of cyclic groups of order p_i for infinitely many distinct primes pi. Hartley asked the following question: Let G be a non-linear simple locally finite group and F be any subgroup of G. Is CG(F) necessarily infinite? In the second part of this thesis, the following problem is studied: Determine the nonlinear simple locally finite groups G and their finite subgroups F such that C_G(F) contains an infinite abelian subgroup which is isomorphic to the direct product of cyclic groups of order pi for infinitely many distinct primes p_i. We prove the following: Let G be a non-linear simple locally finite group with a split Kegel cover K and F be any finite subgroup consisting of K-semisimple elements of G. Then the centralizer C_G(F) contains an infinite abelian subgroup isomorphic to the direct product of cyclic groups of order p_i for infinitely many distinct primes p_i.

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.

The aim of this thesis is to generate original symmetric presentations for finite non-abelian simple groups. We will discuss many permutation progenitors, including but not limited to 2*14 : D28, 2∗9 : 3•(32), 3∗9 : 3•(32), 2∗21 : (7X3) : 2 as well as monomial progenitors, including 7∗5 :m A5, 3∗5 :m S5. We have included their homomorphic images which include the Mathieu group M12, 2•J2, 2XS(4, 5), as well as, many PGL′s, PSL′s and alternating groups. We will give proofs of the isomorphism types of each progenitor, either by hand using double coset enumeration or computer based using MAGMA. We have also constructed Cayley graphs of the following groups, 25 : S5 over 2∗5 : S5, PSL(2, 8) over 2∗7 : D14, M12 over a maximal subgroup, 2XS5. We have developed a lemma using relations to factor permutation progenitors of the form m∗n : N to give an isomorphism of mn : N . Motivated by Robert T. Curtis’ research, we will present a program using MAGMA that, when given a target finite non-abelian simple group, the program will generate possible control groups to write progenitors that will give the given finite non-abelian simple group. Iwasawa’s lemma is also discussed and used to prove PSL(2, 8) and M12 to be simple groups.

A group G is called an FC-group if the conjugacy class of every element is finite. G is called a minimal non-FC-group if G is not an FC-group, but every proper subgroup of G is an FC-group. The first part of this thesis is on minimal non-FC-groups and their finitary permutational representations. Belyaev proved in 1998 that, every perfect locally finite minimal non-FC-group has non-trivial finitary permutational representation. In Chapter 3, we write the proof of Belyaev in detail. Recall that a group G is called quasi-simple if G is perfect and G/Z(G) is simple. The second part of this thesis is on finite quasi-simple groups and their coprime automorphisms. In Chapter 4, the result of Parker and Quick is written in detail: Namely / if Q is a quasi-simple group and A is a non-trivial group of coprime automorphisms of Q satisfying |Q: C_{Q}(A)| &lt / n then |Q| &lt / n3, that is |Q| is bounded by a function of n.

Cette thèse porte sur les plongements d'espaces homogènes sphériques sur un corps quelconque. Dans une première partie, on aborde la classification de ces plongements, dans la lignée des travaux de Demazure et bien d'autres sur les variétés toriques, et de Luna, Vust et Knop sur les variétés sphériques. Dans une seconde partie, on généralise en caractéristique positive certains résultats obtenus par Bien et Brion portant sur les plongements complets et lisses qui sont log homogènes, c'est-à-dire dont le bord est un diviseur à croisements normaux et le fibré tangent logarithmique associé est engendré par ses sections globales. Dans une dernière partie, on construit par éclatements successifs une compactification lisse et log homogène explicite du groupe linéaire (différente de celle obtenue par Kausz). En prenant dans cette compactification les points fixes de certains automorphismes, on en déduit alors la construction de compactifications lisses et log homogènes de certains groupes semi-simples classiques. / This thesis is devoted to the study of embeddings of spherical homogeneous spaces over an arbitrary field. In the first part, we address the classification of such embeddings, in the spirit of Demazure and many others in the setting of toric varieties and of Luna, Vust and Knop in the setting of spherical varieties. In the second part, we generalize in positive characteristics some results obtained by Bien and Brion on those complete smooth embeddings that are log homogeneous, i.e., whose boundary is a normal crossing divisor and the associated logarithmic tangent bundle is generated by its global sections. In the last part, we construct an explicit smooth log homogeneous compactification of the general linear group by successive blow-ups (different from the one obtained by Kausz). By taking fixed points of certain automorphisms on this compactification, one gets smooth log homogeneous compactifications of some classical semi-simple groups.

Let C be the class of finite nilpotent, solvable, symmetric, simple or semi-simple groups and n be a positive integer. We discuss the following question on group laws: What is the length of the shortest non-trivial law holding for all finite groups from the class C of order less than or equal to n?:Introduction 0 Essentials from group theory 1 The two main tools 1.1 The commutator lemma 1.2 The extension lemma 2 Nilpotent and solvable groups 2.1 Definitions and basic properties 2.2 Short non-trivial words in the derived series of F_2 2.3 Short non-trivial words in the lower central series of F_2 2.4 Laws for finite nilpotent groups 2.5 Laws for finite solvable groups 3 Semi-simple groups 3.1 Definitions and basic facts 3.2 Laws for the symmetric group S_n 3.3 Laws for simple groups 3.4 Laws for finite linear groups 3.5 Returning to semi-simple groups 4 The final conclusion Index Bibliography / Sei C die Klasse der endlichen nilpotenten, auflösbaren, symmetrischen oder halbeinfachen Gruppen und n eine positive ganze Zahl. We diskutieren die folgende Frage über Gruppengesetze: Was ist die Länge des kürzesten nicht-trivialen Gesetzes, das für alle endlichen Gruppen der Klasse C gilt, welche die Ordnung höchstens n haben?:Introduction 0 Essentials from group theory 1 The two main tools 1.1 The commutator lemma 1.2 The extension lemma 2 Nilpotent and solvable groups 2.1 Definitions and basic properties 2.2 Short non-trivial words in the derived series of F_2 2.3 Short non-trivial words in the lower central series of F_2 2.4 Laws for finite nilpotent groups 2.5 Laws for finite solvable groups 3 Semi-simple groups 3.1 Definitions and basic facts 3.2 Laws for the symmetric group S_n 3.3 Laws for simple groups 3.4 Laws for finite linear groups 3.5 Returning to semi-simple groups 4 The final conclusion Index Bibliography