The Lyons group denoted by Ly is a Sporadic Simple Group of order 51765179004000000 = 28 37 56 7 11 31 37 67. It(Ly) has a trivial Schur Multiplier and a trivial Outer Automorphism Group. Its maximal subgroups are G2(5) of order 5859000000 and index 8835156, 3 McL:2 of order 5388768000 and index 9606125, 53 L3(5) of order 46500000 and index 1113229656, 2 A11 of order 29916800 and index 1296826875, 51+4 + :4S6 of order 9000000 and index 5751686556, 35:(2 M11) of order 3849120 and index 13448575000, 32+4:2 A5 D8 of order 699840 and index 73967162500, 67:22 of order 1474 and index 35118846000000 and 37:18 of order 666 and index 77725494000000. Its existence was suggested by Richard Lyons. Lyons characterized its order as the unique possible order of any nite simple group where the centralizer of some involution is isomorphic to the nontrivial central extension of the alternating group of degree 11 by the cyclic group of order 2. Sims proved the existence of this group and its uniqueness using permutations and machine calculations. In this dissertation, we compute the (p; q; t)-generations of the Lyons group for dis- tinct primes p, q and t which divide the order of Ly such that p < q < t. For computations, we made use of the Computer Algebra System GAP / Mathematical Sciences / M.Sc. (Mathematics)

Part I of this thesis studies P[subscript(G)](d), the probability of generating a nonabelian simple group G with d randomly chosen elements, and extends this idea to consider the conditional probability P[subscript(G,Soc(G))](d), the probability of generating an almost simple group G by d randomly chosen elements, given that they project onto a generating set of G/Soc(G). In particular we show that for a 2-generated almost simple group, P[subscript(G,Soc(G))](2) 53≥90, with equality if and only if G = A₆ or S₆. Furthermore P[subscript(G,Soc(G))](2) 9≥10 except for 30 almost simple groups G, and we specify this list and provide exact values for P[subscript(G,Soc(G))](2) in these cases. We conclude Part I by showing that for all almost simple groups P[subscript(G,Soc(G))](3)≥139/150. In Part II we consider a related notion. Given a probability ε, we wish to determine d[superscript(ε)] (G), the number of random elements needed to generate a finite group G with failure probabilty at most ε. A generalisation of a result of Lubotzky bounds d[superscript(ε)](G) in terms of l(G), the chief length of G, and d(G), the minimal number of generators needed to generate G. We obtain bounds on the chief length of permutation groups in terms of the degree n, and bounds on the chief length of completely reducible matrix groups in terms of the dimension and field size. Combining these with existing bounds on d(G), we obtain bounds on d[superscript(ε)] (G) for permutation groups and completely reducible matrix groups.

We derive basic properties of groups which can be approximated with matrices. These include closure of classes of such groups under group theoretic constructions including direct and inverse limits and free products. We show that metric ultraproducts of projective linear groups over fields of different characteristics are not isomorphic. We further prove that the lattice of normal subgroups in ultraproducts of compact simple groups is distributive. It is linearly ordered in the case of finite simple groups or Lie groups of bounded rank.

The Lyons group denoted by Ly is a Sporadic Simple Group of order 51765179004000000 = 28 37 56 7 11 31 37 67. It(Ly) has a trivial Schur Multiplier and a trivial Outer Automorphism Group. Its maximal subgroups are G2(5) of order 5859000000 and index 8835156, 3 McL:2 of order 5388768000 and index 9606125, 53 L3(5) of order 46500000 and index 1113229656, 2 A11 of order 29916800 and index 1296826875, 51+4 + :4S6 of order 9000000 and index 5751686556, 35:(2 M11) of order 3849120 and index 13448575000, 32+4:2 A5 D8 of order 699840 and index 73967162500, 67:22 of order 1474 and index 35118846000000 and 37:18 of order 666 and index 77725494000000. Its existence was suggested by Richard Lyons. Lyons characterized its order as the unique possible order of any nite simple group where the centralizer of some involution is isomorphic to the nontrivial central extension of the alternating group of degree 11 by the cyclic group of order 2. Sims proved the existence of this group and its uniqueness using permutations and machine calculations. In this dissertation, we compute the (p; q; t)-generations of the Lyons group for dis- tinct primes p, q and t which divide the order of Ly such that p < q < t. For computations, we made use of the Computer Algebra System GAP / Mathematical Sciences / M.Sc. (Mathematics)

The main goal of this thesis is a contribution to the classification of all incidence geometries<p>of rank two on which some group PSL(2,q), q a prime power, acts flag-transitively.<p>Actually we require that the action be RWPRI (residually weakly primitive) and (2T)1<p>(doubly transitive on every residue of rank one). In fact our definition of RWPRI requires<p>the geometry to be firm (each residue of rank one has at least two elements) and RC<p>(residually connected).<p><p>The main goal is achieved in this thesis.<p>It is stated in our "Main Theorem". The proof of this theorem requires more than 60pages.<p><p>Quite surprisingly, our proof in the direction of the main goal uses essentially the classification<p>of all subgroups of PSL(2,q), a famous result provided in Dickson’s book "Linear groups: With an exposition of the Galois field theory", section 260, in which the group is called Linear Fractional Group LF(n, pn).<p><p>Our proof requires to work with all ordered pairs of subgroups up to conjugacy.<p><p>The restrictions such as RWPRI and (2T)1 allow for a complete analysis.<p><p>The geometries obtained in our "Main Theorem" are bipartite graphs; and also locally 2-arc-transitive<p>graphs in the sense of Giudici, Li and Cheryl Praeger. These graphs are interesting in their own right because of<p>the numerous connections they have with other fields of mathematics. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

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We derive basic properties of groups which can be approximated with matrices. These include closure of classes of such groups under group theoretic constructions including direct and inverse limits and free products. We show that metric ultraproducts of projective linear groups over fields of different characteristics are not isomorphic. We further prove that the lattice of normal subgroups in ultraproducts of compact simple groups is distributive. It is linearly ordered in the case of finite simple groups or Lie groups of bounded rank.

Let q be a nontrivial odd prime power, and let 𝑛 ≥ 2 be a natural number with (𝑛, 𝑞) ≠ (2, 3). We characterize the groups 𝑃𝑆𝐿𝑛(𝑞) and 𝑃𝑆𝑈𝑛(𝑞) by their 2-fusion systems. This contributes to a programme of Aschbacher aiming at a simplified proof of the classification of finite simple groups.

Dans le Chapitre 1 nous étudions les groupes localement compacts lacunaires hyperboliques. Nous caractérisons les groupes ayant un cône asymptotique qui est un arbre réel et dont l'action naturelle est focale. Nous étudions également la structure des groupes lacunaires hyperboliques, et montrons que dans le cas unimodulaire les sous-groupes ne satisfont pas de loi. Nous appliquons au Chapitre 2 les résultats précédents pour résoudre le problème de l'existence de points de coupure dans un cône asymptotique dans le cas des groupes de Lie connexes. Dans le Chapitre 3 nous montrons que le groupe de Neretin est compactement présenté et donnons une borne supérieure sur sa fonction de Dehn. Nous étudions également les propriétés métriques du groupe de Neretin, et prouvons que certains sous-groupes remarquables sont quasi-isométriquement plongés. Nous étudions dans le Chapitre 4 une famille de groupes agissant sur un arbre, et dont l'action locale est prescrite par un groupe de permutations. Nous montrons entre autres que ces groupes ont la propriété (PW), et exhibons des groupes simples au sein de cette famille. Dans le Chapitre 5 nous introduisons l'éventail des relations d'un groupe de type fini, qui est l'ensemble des longueurs des relations non engendrées par des relations plus courtes. Nous établissons un lien entre la simple connexité d'un cône asymptotique et l'éventail des relations du groupe, et donnons une grande classe de groupes dont l'éventail des relations est aussi grand que possible. / In Chapter 1 we investigate the class of locally compact lacunary hyperbolic groups. We characterize locally compact groups having one asymptotic cone that is a real tree and whose natural isometric action is focal. We also study the structure of lacunary hyperbolic groups, and prove that in the unimodular case subgroups cannot satisfy a law. We apply the previous results in Chapter 2 to solve the problem of the existence of cut-points in asymptotic cones for connected Lie groups. In Chapter 3 we prove that Neretin's group is compactly presented and give an upper bound on its Dehn function. We also study metric properties of Neretin's group, and prove that some remarkable subgroups are quasi-isometrically embedded. In Chapter 4 we study a family of groups acting on a tree, and whose local action is prescribed by some permutation group. We prove among other things that these groups have property (PW), and exhibit some simple groups in this family. In Chapter 5 we introduce the relation range of a finitely generated group, which is the set of lengths of relations that are not generated by relations of smaller length. We establish a link between simple connectedness of asymptotic cones and the relation range of the group, and give a large class of groups having a relation range as large as possible.