Guo Xiuyun. / "May 2002." / Thesis (Ph.D.)--Chinese University of Hong Kong, 2002. / Includes bibliographical references (p. 103-108) and index. / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Mode of access: World Wide Web. / Abstracts in English and Chinese.

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, August 2000. / Includes bibliographical references. Also available on the Internet.

Let <b>X</b> be a compact Hausdorff space and <b>A</b> a uniform algebra on <b>X</b>. Let if be an isometric unital representation that maps <b>A</b> into bounded linear operators on a Hilbert space. This research investigated that there is a one-to-one correspondence between the collection of maximal sets of antisymmetry for <b>A</b> and that of maximal projections of antisymmetry for π (<b>A</b>) under the extension of π if π satisfies a certain regularity property. / Ph. D.

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 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)

One might easily argue that the Classification of Finite Simple Groups is one of the most important theorems of group theory. Given that any finite group can be deconstructed into its simple composition factors, it is of great importance to have a detailed knowledge of the structure of finite simple groups. One of the classes of finite groups that appear in the classification theorem are the simple classical groups, which are matrix groups preserving some form. This thesis will shed some new light on almost simple classical groups in dimension 13, 14 and 15. In particular we will determine their maximal subgroups. We will build on the results by Bray, Holt, and Roney-Dougal who calculated the maximal subgroups of all almost simple finite classical groups in dimension less than 12. Furthermore, Aschbacher proved that the maximal subgroups of almost simple classical groups lie in nine classes. The maximal subgroups in the first eight classes, i.e. the subgroups of geometric type, were determined by Kleidman and Liebeck for dimension greater than 13. Therefore this thesis concentrates on the ninth class of Aschbacher's Theorem. This class roughly consists of subgroups which are almost simple modulo scalars and do not preserve a geometric structure. As our final result we will give tables containing all maximal subgroups of almost simple classical groups in dimension 13, 14 and 15.

The character table of any finite group provides a considerable amount of information about a group and the use of character tables is of great importance in Mathematics and Physical Sciences. Most of the maximal subgroups of finite simple groups and their automorphisms are extensions of elementary abelian groups. Various techniques have been used to compute character tables, however Bernd Fischer came up with the most powerful and informative technique of calculating character tables of group extensions. This method is known as the Fischer-Clifford Theory and uses Fischer-Clifford matrices, as one of the tools, to compute character tables. This is derived from the Clifford theory. Here G is an extension of a group N by a finite group G, that is G = N.G. We then construct a non-singular matrix for each conjugacy class of G/N =G. These matrices, together with partial character tables of certain subgroups of G, known as the inertia groups, are used to compute the full character table of G. In this dissertation, we discuss Fischer-Clifford theory and apply it to both split and non-split extensions. We first, under the guidance of Dr Mpono, studied the group 27:S8 as a maximal subgroup of 27:SP(6,2), to familiarize ourselves to Fischer-Clifford theory. We then looked at 26:A8 and 28:O+8 (2) as maximal subgroups of 28:O+8 (2) and O+10(2) respectively and these were both split extensions. Split extensions have also been discussed quite extensively, for various groups, by different researchers in the past. We then turned our attention to non-split extensions. We started with 24.S6 and 25.S6 which were maximal subgroups of HS and HS:2 respectively. Except for some negative signs in the first column of the Fischer-Clifford matrices we used the Fisher-Clifford theory as it is. The Fischer-Clifford theory, is also applied to 53.L(3, 5), which is a maximal subgroup of the Lyon's group Ly. To be able to use the Fisher-Clifford theory we had to consider projective representations and characters of inertia factor groups. This is not a simple method and quite some smart computations were needed but we were able to determine the character table of 53.L(3,5). All character tables computed in this dissertation will be sent to GAP for incorporation. / Thesis (Ph.D.)-University of KwaZulu-Natal, Pietermaritzburg, 2011.

Let Ω be the Fraïssé limit of a class of relational structures. We seek to answer the following semigroup theoretic question about Ω. What are the group H-classes, i.e. the maximal subgroups, of End(Ω)? Fraïssé limits for which we answer this question include the random graph R, the random directed graph D, the random tournament T, the random bipartite graph B, Henson's graphs G[subscript n] (for n greater or equal to 3) and the total order Q. The maximal subgroups of End(Ω) are closely connected to the automorphism groups of the relational structures induced by the images of idempotents from End(Ω). It has been shown that the relational structure induced by the image of an idempotent from End(Ω) is algebraically closed. Accordingly, we investigate which groups can be realised as the automorphism group of an algebraically closed relational structure in order to determine the maximal subgroups of End(Ω) in each case. In particular, we show that if Γ is a countable graph and Ω = R,D,B, then there exist 2[superscript aleph-naught] maximal subgroups of End(Ω) which are isomorphic to Aut(Γ). Additionally, we provide a complete description of the subsets of Q which are the image of an idempotent from End(Q). We call these subsets retracts of Q and show that if Ω is a total order and f is an embedding of Ω into Q such that im f is a retract of Q, then there exist 2[superscript aleph-naught] maximal subgroups of End(Q) isomorphic to Aut(Ω). We also show that any countable maximal subgroup of End(Q) must be isomorphic to Zⁿ for some natural number n. As a consequence of the methods developed, we are also able to show that when Ω = R,D,B,Q there exist 2[superscript aleph-naught] regular D-classes of End(Ω) and when Ω = R,D,B there exist 2[superscript aleph-naught] J-classes of End(Ω). Additionally we show that if Ω = R,D then all regular D-classes contain 2[superscript aleph-naught] group H-classes. On the other hand, we show that when Ω = B,Q there exist regular D-classes which contain countably many group H-classes.

Philosophiae Doctor - PhD (Mathematics) / The main theme of this thesis is the construction of linear codes from adjacency matrices or sub-matrices of adjacency matrices of regular graphs. We first examine the binary codes from the row span of biadjacency matrices and their transposes for some classes of bipartite graphs. In this case we consider a sub-matrix of an adjacency matrix of a graph as the generator of the code. We then shift our attention to uniform subset graphs by exploring the automorphism groups of graph covers and some classes of uniform subset graphs. In the sequel, we explore equal codes from adjacency matrices of non-isomorphic uniform subset graphs and finally consider codes generated by an adjacency matrix formed by adding adjacency matrices of two classes of uniform subset graphs.