Thesis (Ph. D.)--University of Wisconsin--Madison, 1937. / Typescript. Includes abstract and vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaf [50]).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1982. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 52-53).

Rational homotopy theory is the study of uniquely divisible homotopy invariants. For each nilpotent space X the association X ——» minimal algebra for X is a complete determination of these invariants. If X is a space and Mx its minimal algebra, the algebraic group Aut Mx and the representation Aut Mx ——» Gl(Mx) have considerable influence on the structure of Mx . This thesis contains a systematic study of this interaction. Chapter I contains preliminary results from algebraic group theory and general topology. In Chapter II I define and study inverse limits of algebraic groups. I prove that many of the known structural properties of algebraic groups remain valid in this more general setting. Emphasis is placed on the conjugacy theorems that are particularly useful for studying minimal algebras. Chapter III is the main part of the thesis where I develop a structure theory for minimal algebras which relates toroidal symmetry to retracts. Precisely, if M is a minimal algebra, then there exists a 1-parameter subgroup λ : Q* ——> Aut Mx such that λ extends to λ : Q——» End Mx λ: (0) = e = e²: Mx——» Mx Further if e so chosen is minimal then it is uniquely determined up to conjugation by Aut Mx . In the interesting case where e = 0m I give a pro-algebraic group theoretic proof of uniqueness of coproduct and product decompositions in the appropriate homotopy category. / Science, Faculty of / Mathematics, Department of / Graduate

Thesis (M. Phil.)--University of Hong Kong, 2001. / Includes bibliographical references (leaves 17-18).

published_or_final_version / Mathematics / Master / Master of Philosophy

No description available.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1980. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND SCIENCE. / Bibliography: leaves 112-113. / by Barbara R. Peskin. / Ph.D.

In this thesis we explore the problem of computing automorphisms of finite groups, eventually focusing on some group product constructions. Roughly speaking, the automorphism group of a group gives the nature of its internal symmetry. In general, determination of the automorphism group requires significant computational effort and it is advantageous to find situations in which this may be reduced. The two main results give descriptions of the automorphism groups of finite direct products and split metacyclic p-groups. Given a direct product G = H x K where H and K have no common direct factor, we give the order and structure of Aut G in terms of Aut H, Aut K and the central homomorphism groups Hom (H, Z(K)) and Hom (K, Z(H)). A similar result is given for the the split metacyclic p-group, in the case where p is odd. Implementations of both of these results are given as functions for the computational algebra system GAP, which we use extensively throughout. An account of the literature and relevant standard results on automorphisms is given. In particular we mention one of the more esoteric constructions, the automorphism tower. This is defined as the series obtained by repeatedly taking the automorphism group of some starting group G₀. There is interest as to whether or not this series terminates, in the sense that some group is reached that is isomorphic to its group of automorphisms. Besides a famous result of Wielandt in 1939, there has not been much further insight gained here. We make use of the technology to construct several examples, demonstrating their complex and varied behaviour. For the main results we introduce a 2 x 2 matrix description for the relevant automorphism groups, where the entries come from the homorphism groups mentioned previously. In the case of the direct product, this is later generalised to an n x n matrix (when we consider groups with any number of direct factors) and the common direct factor restriction is relaxed to the component groups not having a common abelian direct factor. In the case of the split metacyclic p-group, our matrices have entries that are not all homomorphisms, but are similar. We include the code for our GAP impementation of these results, which we show significantly expedites computation of the automorphism groups. We show that this matrix language can be used to describe automorphisms of any semidirect product and certain central products too, although these general cases are much more complicated. Specifically, multiplication is no longer defined in such a natural way as is seen in the previous cases and the matrix entries are mappings much less well-behaved than homomorphisms. We conclude with some suggestion of types of semidirect products for which our approach may yield a convenient description of the automorphisms.

abstract / Mathematics / Doctoral / Doctor of Philosophy

published_or_final_version / abstract / toc / Mathematics / Doctoral / Doctor of Philosophy