The work described in this dissertation was largely motivated by the aim of producing a survey on the theory of group extensions. From the broad scope of the theory of group extensions we single out two aspects to discuss, namely the study of the split and the non-split cases and give examples of both. A great part of this dissertation is dedicated to the study of split extensions. After setting the background theory for the study of the split extensions we proceed in exploring the ramifications of this concept within the development of the group structure and consequently investigate well known products which are its derived namely the holomorph, and the wreath product. The theory of group presentations provides in principle the necessary tools that permit the description of a group by means of its generators and relators. Through this knowledge we give presentations for the groups of order pq,p2q and p3. Subsequently using a classical result of Gaschutz we investigate the split extensions of non-abelian groups in which the normal subgroup is either a non-abelian normal nilpotent group or a non-abelian normal solvable group. We also study other cases of split extensions such as the affine subgroups of the general linear and the symplectic groups. It is expected that some of the results obtained will provide a theoretical algorithm to describe these affine subgroups. A particular case of the non-split extensions is discussed as the Frattini extensions. In fact a simplest example of a Frattini extension is a non-split extension in which the kernel of an epimorphism e is an irreducible G-module. / Thesis (M.Sc.)-University of Natal, Pietermaritzburg, 1999.

This thesis primarily discusses the results of two papers, [Hu] and [HaHu]. The first is an overview of algebraic-geometric techniques for integrable systems in which the AKS theorem is proven. Under certain conditions, this theorem asserts the commutatvity and (potential) non-triviality of the Hamiltonian flow of Ad*-invariant functions once they're restricted to subalgebras. This theorem is applied to the case of coadjoint orbits on loop algebras, identifying the flow with a spectral curve and a line bundle via the Lax equation. These results play an important role in the discussion of [HaHu], wherein we consider three levels of spaces, each possessing a linear family of Poisson spaces. It is shown that there exist Poisson mappings between these levels. We consider the two cases where the underlying Riemann surface is an elliptic curve, as well as its degeneration to a Riemann sphere with two points identified (the trigonometric case). Background in necessary areas is provided.

The character table of a finite group provides considerable amount of information about the group, and hence is of great importance in Mathematics as well as in Physical Sciences. Most of the maximal subgroups of the finite simple groups and their automorphisms are of extensions of elementary abelian groups, so methods have been developed for calculating the character tables of extensions of elementary abelian groups. Character tables of finite groups can be constructed using various techniques. However Bernd Fischer presented a powerful and interesting technique for calculating the character tables of group extensions. This technique, which is known as the technique of the Fischer-Clifford matrices, derives its fundamentals from the Clifford theory. If G=N.G is an appropriate extension of N by G, the method involves the construction of a nonsingular matrix for each conjugacy class of G/N~G. The character table of G can then be determined from these Fischer-Clifford matrices and the character table of certain subgroups of G, called inertia factor groups. In this dissertation, we described the Fischer-Clifford theory and apply it to both split and non-split group extensions. First we apply the technique to the split extensions 2,7:Sp6(2) and 2,8:SP6(2) which are maximal subgroups of Sp8(2) and 2,8:08+(2) respectively. This technique has also been discussed and used by many other researchers, but applied only to split extensions or to the case when every irreducible character of N can be extended to an irreducible character of its inertia group in G. However the same method can not be used to construct character tables of certain non-split group extensions. In particular, it can not be applied to the non-split extensions of the forms 3,7.07(3) and 3,7.(0,7(3):2) which are maximal subgroups of Fischer's largest sporadic simple group Fi~24 and its automorphism group Fi24 respectively. In an attempt to generalize these methods to such type of non-split group extensions, we need to consider the projective representations and characters. We have shown that how the technique of Fischer-Clifford matrices can be applied to any such type of non-split extensions. However in order to apply this technique, the projective characters of the inertia factors must be known and these can be difficult to determine for some groups. We successfully applied the technique of Fischer-Clifford matrices and determined the Fischer-Clifford matrices and hence the character tables of the non-split extensions 3,7.0,7(3) and 3,7.(0,7(3):2). The character tables computed in this thesis have been accepted for incorporation into GAP and will be available in the latest versions. / Thesis (Ph.D)-University of Natal, Pietermaritzburg, 2001.

The smallest Fischer sporadic simple group Fi22 is generated by a conjugacy class D of 3510 involutions called 3-transpositions such that the product of any noncommuting pair is an element of order 3. In Fi22 there are exactly three conjugacy classes of involutions denoted by D, T and N and represented in the ATLAS [26] by 2A, 2B and 2C, containing 3510, 1216215 and 36486450 elements with corresponding centralizers 2·U(6,2), (2 x 2~+8:U(4,2)):2 and 25+8:(83 X 32:4) respectively. In Fi22 , we have Npi22(26) = 26:8P(6,2), where 26 is a 2B-pure group, and thus the maximal subgroup 26:8P(6, 2) of Fi22 is a 2-local subgroup. The full automorphism group of Fi22 is denoted by Fi22 . In Fi22 , there are three involutory outer automorphisms of Fi22 which are denoted bye, f and 0 and represented in the ATLAS [26] by 2D, 2F and 2E respectively. We obtain that Fi22 = Fi22 :(e) and it can be easily shown that Fi22 = Fi22 :(e) = Fi22 :(f) = Fi22 :(0). As e, f and 0 act on Fi22 , then we obtain the subgroups CPi22 (e) rv 0+(8,2):83, CPi22 (f) rv 8P(6,2) x 2 and CPi22 (()) rv 26:0-(6,2) of Fi22 which are generated by CD(e), Cn(f) and CD(0) respectively. In this thesis we are concerned with the construction of the character tables of certain groups which are associated with Fi22 and its automorphism group Fi22 . We use the technique of the Fischer-Clifford matrices to construct the character tables of these groups, which are split extensions. These groups are 26:8P(6, 2), 26:0-(6,2) and 27:8P(6, 2). The study of the group 26:8P(6, 2) is essential, as the other groups studied in this thesis are related to it. The groups 8P(6,2) and 0- (6,2) of 6 x 6 matrices over GF(2), played crucial roles in our construction of the group 8P(6, 2) as a group of 7 x 7 matrices over GF(2) which would act on 27 . Also the character table of 25:86 , the affine subgroup of 8P(6, 2) fixing a nonzero vector in 26 , is constructed by using the technique of the Fischer-Clifford matrices. This character table is used in the construction of the character table 26:SP(6, 2). The character tables computed in this thesis have been accepted for incorporation into GAP and will be available in the latest version of GAP. / Thesis (Ph.D.) - University of Natal, Pietermaritzburg, 1998.

Integrity bases for tensors of type (GAMMA)(,r) whose components are polynomials in the components of tensors of type (GAMMA)(,5) ((GAMMA)(,6) for ('(d))O) are given explicitely for the double tetrahedral and octahedral point groups (('(d))T and ('(d))O), where the main axis of symmetry is trigonal. We formulate analytic basis states for the decomposition of SU(2) through the chain ('(d))T (R-HOOK) ('(d))C(,3) (R-HOOK) ('(d))C(,1) and use them to construct the Racah algebra. / A method is given for deriving branching rules, in the form of generating functions, for the decomposition of representations of SU(3) into representations of its finite subgroups. Interpreted in terms of an integrity basis, the generating functions define analytic polynomial basis states for SU(3) which respect the finite subgroup.

Thesis (M.S.)--State University of New York at Binghamton, Computer Science Department, 2006. / Includes bibliographical references.

Thesis (M.S.)--University of Akron, Dept. of Mathematics, 2008. / "May, 2008." Title from electronic thesis title page (viewed 07/12/2008) Advisor, Jeffrey M. Riedl; Faculty Readers, Ethel Wheland, Stuart Clay; Department Chair, Joseph Wilder; Dean of the College, Ronald F. Levant; Dean of the Graduate School, George R. Newkome. Includes bibliographical references.

Thesis (M.S.)--University of Akron, Dept. of Mathematics, 2008. / "May, 2008." Title from electronic thesis title page (viewed 07/12/2008) Advisor, Jeffrey M. Riedl; Faculty Readers, William S. Clary, Ethel R. Wheland; Department Chair, Joseph W. Wilder; Dean of the College, Ronald F. Levant; Dean of the Graduate School, George R. Newkome. Includes bibliographical references.

Thesis (Ph. D.)--Michigan State University. Dept. of Mathematics, 2008. / Title from PDF t.p. (viewed on July 24, 2009) Includes bibliographical references (p. 83-84). Also issued in print.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1967. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliography.