The purpose of this paper was to find a general formula to count the number of automorphisms of any finite abelian group. These groups were separated into five different types. For each of the first three types, theorems were proven, and formulas were derived based on the theorems. A formula for the last two types of groups was derived from a theorem based on a conjecture which was proven in only one direction. Then it was shown that a count found from any of the first three formulas could also be found using the last formula. The result of these comparisons gave credence to the conjecture. Thus we found that the last formula is a general formula to count the number of automorphisms of finite abelian groups. / Department of Mathematical Sciences

Dated 1983 / Bibliography: leaves 143-145 / 145 leaves ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, 1984

Thesis (M.Phil.)--Hong Kong University of Science and Technology, 2009. / Includes bibliographical references (p. 47-48).

Thesis--Bonn. Extra t. p. with thesis statement inserted. / Bibliography: p. 45.

No description available.

In this dissertation, we describe the automorphism groups for the strong Bruhat orders A_n-1, B_n, and D_n. In particular, the automorphism group of A_n-1 for n ≥ 3 is isomorphic to the dihedral group of order eight, D_4; the automorphism group of B_n for n ≥ 3 is isomorphic to C_2 x C_2 where C_2 is the cyclic group of order two; the automorphism group of D_n for n > 5 and n even is isomorphic to C_2 x C_2 x C_2; and the automorphism group of D_n for n ≥ 5 and n odd is isomorphic to the dihedral group D_4.

Free group outer automorphisms were shown by Bestvina and Randell to have fixed subgroups whose rank is bounded in terms of the rank of the underlying group. We consider the case where this upper bound is achieved and obtain combinatorial results about such outer automorphisms thus extending the work of Collins and Turner. We go on to show that such automorphisms can be represented by certain graph of group isomorphisms called Dehn Twists and also solve the conjuagacy problem in a restricted case, thus reproducing the work of Cohen and Lustig, but with different methods. We rely heavily on the relative train tracks of Bestvina and Randell and in fact go on to use an analogue of these for free product automorphisms developed by Collins and Turner. We prove an index theorem for such automorphisms which counts not only the group elements which are fixed but also the points which are fixed at infinity - the infinite reduced words. Two applications of this theorem are considered, first to irreducible free group automorphisms and then to the action of an automorphism on the boundary of a hyperbolic group. We reduce the problem of counting the number of points fixed on the. boundary to the case where the hyperbolic group is indecomposable and provide an easy application to virtually free groups.

In this thesis, we consider the properties of diffeomorphisms of R3 called trace maps. We begin by introducing the definition of the trace map. The group B3 acts by trace maps on R3. The first two chapters deal with the action of a specific element of B3,called αn. In particular, we study the fixed points of αn lying on a topological subspace contained in R3, called T . We investigate the duality of the fixed points of the action ofαn, which will be defined later in the thesis.Chapter 3 involves the study of the fixed points of an element called γnm, and it generalizes the results of chapter 2. Chapter 4 involves a study of the period two points of γnm. Chapters 5-8 deal with surfaces and curves induced by trace maps, in a manner described in chapter 5. Trace maps define surfaces, and we study the intersection of those surfaces. In particular, we classify each such possible intersection.

No description available.

In the first part of this thesis we investigate the automorphism groups of regular trees. In the second part we look at the action of the automorphism group of a locally finite graph on the ends of the graph. The two part are not directly related but trees play a fundamental role in both parts. Let T_n be the regular tree of valency n. Put G := Aut(T_n) and let G₀ be the subgroup of G that is generated by the stabilisers of points. The main results of the first part are : Theorem 4.1 Suppose 3 ≤ n < N₀ and α ϵ T_n. Then G_α (the stabiliser of α in G) contains 2^2N0 subgroups of index less than 2^2N0. Theorem 4.2 Suppose 3 ≤ n < N₀ and H ≤ G with G : H |< 2^N0. Then H = G or H = G₀ or H fixes a point or H stabilises an edge. Theorem 4.3 Let n = N₀ and H ≤ G with | G : H |< 2^N0. Then H = G or H = G₀ or there is a finite subtree ϕ of T_n such that G(_ϕ) ≤ H ≤ G{_ϕ}. These are proved by finding a concrete description of the stabilisers of points in G, using wreath products, and also by making use of methods and results of Dixon, Neumann and Thomas [Bull. Lond. Math. Soc. 18, 580-586]. It is also shown how one is able to get short proofs of three earlier results about the automorphism groups of regular trees by using the methods used to prove these theorems. In their book Groups acting on graphs, Warren Dicks and M. J. Dunwoody [Cambridge University Press, 1989] developed a powerful technique to construct trees from graphs. An end of a graph is an equivalence class of half-lines in the graph, with two half-lines, L₁ and L₂, being equivalent if and only if we can find the third half-line that contains infinitely many vertices of both L₁ and L₂. In the second part we point out how one can, by using this technique, reduce questions about ends of graphs to questions about trees. This allows us both to prove several new results and also to give simple proofs of some known results concerning fixed points of group actions on the ends of a locally finite graph (see Chapter 10). An example of a new result is the classification of locally finite graphs with infinitely many ends, whose automorphism group acts transitively on the set of ends (Theorem 11.1).