by Lam Chak Ming. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2001. / Includes bibliographical references (leaves 76-79). / Abstracts in English and Chinese. / Chapter 0 --- Introduction --- p.5 / Chapter 1 --- Background Knowledge of Group Theory --- p.7 / Chapter 1.1 --- Some Basic Results --- p.7 / Chapter 1.2 --- Solvable Groups --- p.13 / Chapter 1.3 --- Nilpotent groups and some useful results --- p.15 / Chapter 2 --- Theory of Formations --- p.21 / Chapter 2.1 --- Some Basic Results of Formations --- p.21 / Chapter 2.2 --- X-covering subgroups and X-projectors --- p.23 / Chapter 2.3 --- The Conjugacy of F-Covering Subgroups --- p.32 / Chapter 2.4 --- F-Normalizers --- p.36 / Chapter 3 --- Local Formations --- p.47 / Chapter 3.1 --- The Construction of Local Formations --- p.47 / Chapter 3.2 --- The Stability of Formations --- p.51 / Chapter 3.3 --- The Complements of F-coradicals --- p.57 / Chapter 3.4 --- Minimal non-F-groups --- p.59 / Chapter 4 --- Finite Groups With Given Minimal Subgroups --- p.66 / Chapter 4.1 --- C-normality of Groups --- p.66 / Chapter 4.2 --- A Generalized Version of Ito's Theorem --- p.70 / Bibliography --- p.76

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Thesis--Wisconsin. / Vita. Includes bibliographical references (leaf 100).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1969. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliography.

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A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. December 2014. / Let G be a finite group and let r ∈ N. A coloring of G is any mapping : G −→ {1, 2, 3, ..., r}. Colorings of G, and are equivalent if there exists an element g in G such that (xg−1) = (x) for all x in G. A coloring of a finite group G is called symmetric with respect to an element g in G if (gx−1g) = (x) for all x ∈ G. We derive formulae for computing the number of symmetric colorings and the number of equivalence classes of symmetric colorings for some classes of finite groups

In Chapter 2 we develop the concept of total orthogonality. A number of necessary conditions are derived. Necessary and sufficient conditions for total orthogonality are obtained for 2-groups and for split extensions of elementary abelian 2-groups. A complete description is given for totally orthogonal groups whose character degrees are bounded by 2. Brauer's problem is reduced for Frobenius groups to the corresponding problems for Frobenius kernels and complements. In Chapter 3 classes of examples are presented illustrating the concepts and results of Chapter 2. It is shown, in particular, that 2-Sylow subgroups of finite reflection groups, and of alternating groups, are totally orthogonal.

Let G be a connected reductive algebraic group defined over an algebraic closure of the finite field of prime order p > 0, which we assume to be good for G. We denote by F : G → G a Frobenius endomorphism of G and by G the corresponding Fq-rational structure. If Irr(G) denotes the set of ordinary irreducible characters of G then by work of Lusztig and Geck we have a well defined map ΦG : Irr(G) → {F-stable unipotent conjugacy classes of G} where ΦG(χ) is the unipotent support of χ. Lusztig has given a classification of the irreducible characters of G and obtained their degrees. In particular he has shown that for each χ ∈ Irr(G) there exists an integer nχ such that nχ · χ(1) is a monic polynomial in q. Given a unipotent class O of G with representative u ∈ G we may define AG(u) to be the finite quotient group CG(u)/CG(u)◦. If the centre Z(G) is connected and G/Z(G) is simple then Lusztig and H´ezard have independently shown that for each F-stable unipotent class O of G there exists χ ∈ Irr(G) such that ΦG(χ) = O and nχ = |AG(u)|, (in particular the map ΦG is surjective). The main result of this thesis extends this result to the case where G is any simple algebraic group, (hence removing the assumption that Z(G) is connected). In particular if G is simple we show that for each F-stable unipotent class O of G there exists χ ∈ Irr(G) such that ΦG(χ) = O and nχ = |AG(u)F| where u ∈ OF is a well-chosen representative. We then apply this result to prove, (for most simple groups), a conjecture of Kawanaka’s on generalised Gelfand–Graev representations (GGGRs). Namely that the GGGRs of G form a Z-basis for the Z-module of all unipotently supported class functions of G. Finally we obtain an expression for a certain fourth root of unity associated to GGGRs in the case where G is a symplectic or special orthogonal group.

The main purpose of this thesis is to obtain surprising identities by counting the representations of quivers over finite fields. A classical result states that the dimension vectors of the absolutely indecomposable representations of a quiver ?? are in one-to-one correspondence with the positive roots of a root system ??, which is infinite in general. For a given dimension vector ?? ??? ??+, the number A??(??, q), which counts the isomorphism classes of the absolutely indecomposable representations of ?? of dimension ?? over the finite field Fq, turns out to be a polynomial in q with integer coefficients, which have been conjectured to be nonnegative by Kac. The main result of this thesis is a multi-variable formal identity which expresses an infinite series as a formal product indexed by ??+ which has the coefficients of various polynomials A??(??, q) as exponents. This identity turns out to be a qanalogue of the remarkable Weyl-Macdonald-Kac denominator identity modulus a conjecture of Kac, which asserts that the multiplicity of ?? is equal to the constant term of A??(??, q). An equivalent form of this conjecture is established and a partial solution is obtained. A new proof of the integrality of A??(??, q) is given. Three Maple programs have been included which enable one to calculate the polynomials A??(??, q) for quivers with at most three nodes. All sample out-prints are consistence with Kac???s conjectures. Another result of this thesis is as follows. Let A be a finite dimensional algebra over a perfect field K, M be a finitely generated indecomposable module over A ???K ??K. Then there exists a unique indecomposable module M??? over A such that M is a direct summand of M??? ???K ??K, and there exists a positive integer s such that Ms = M ??? ?? ?? ?? ??? M (s copies) has a unique minimal field of definition which is isomorphic to the centre of End ??(M???) rad (End ??(M???)). If K is a finite field, then s can be taken to be 1.

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