We have conducted a systematic search for finite homomorphic images of several permutation and monomial progenitors. We have found original symmetric presentations for several important groups such as the Mathieu sporadic simple groups, Suzuki simple group, unitary group, Janko group, simplectic groups, and projective special linear groups. We have also constructed, using the technique of double coset enumeration, the following groups, L_2(11), S(4,3):2, M11, and PGL(2,11). The isomorphism class of each of the finite images is also given.

In this thesis, we investigate various problems in the representation theory of finite groups of Lie type. In Chapter 2, we hope to make sense of the last statement - we will introduce some background and notation that will be useful for the remainder of the thesis. In Chapter 3, we find bounds for the largest irreducible representation degree of a finite unitary group. In Chapter 4, we describe the block distribution and Brauer characters in cross characteristic for Sp₆(2ᵃ) in terms of the irreducible ordinary characters. This will be useful in Chapter 5 and Chapter 7, which focus primarily on the group Sp₆(2ᵃ) and contain the main results of this thesis, which we now summarize. Given a subgroup H ≤ G and a representation V for G, we obtain the restriction V|H of V to H by viewing V as an FH-module. However, even if V is an irreducible representation of G, the restriction V|H may (and usually does) fail to remain irreducible as a representation of H. In Chapter 5, we classify all pairs (V, H), where H is a proper subgroup of G = Sp₆(q) or Sp₄(q) with q even, and V is an l-modular representation of G for l ≠ 2 which is absolutely irreducible as a representation of H. This problem is motivated by the Aschbacher-Scott program on classifying maximal subgroups of finite classical groups. The local-global philosophy plays an important role in many areas of mathematics. In the representation theory of finite groups, the so-called "local-global" conjectures would relate the representation theory of G to that of certain proper subgroups, such as the normalizer of a Sylow subgroup. One might hope that these conjectures could be proven by showing that they are true for all simple groups. Though this turns out not quite to be the case, some of these conjectures have been reduced to showing that a finite set of stronger conditions hold for all finite simple groups. In Chapter 7, we show that Sp₆(q) and Sp₄(q), q even, are "good" for these reductions.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Este trabalho à baseado no artigo "Uniquely Covered Groups" de M. A. Brodie, que investiga grupos finitos que possuem uma Ãnica cobertura irredundante por subgrupos prÃprios. O resultado principal obtido por M. A. Brodie assegura que um grupo finito e nÃo nilpotente G à unicamente coberto se, e somente se, G/Z(G) à um grupo nÃo abeliano de ordem pq, onde p e q sÃo primos distantes e {x,Z(G) à cÃclico para todo x â G. Nosso propÃsito à apresentar a demonstraÃÃo e uma aplicaÃÃo deste teorema. / This work is based on the article "Uniquely Covered Groups" due to M. A. Brodie, which investigates finite groups that have a single irredundante coveraging by subgroups. The main result obtained by M. A. Brodie asserts that a non-nilpotent finite group G is uniquely covered if and only if, G/Z(G) is a non-Abelian group of order pq, where p and q are distinct primes and {x,Z(G) is cyclic for every x â G. Our purpose is to present the proof and application of this theorem.