A classification of all undirected trees with automorphism group isomorphic to $(Z_2)^l$ is given in terms of a vertex partition called a refined star partition. Recently the notion of a quantum automorphism group has been defined by T. Banica and J. Bichon. The quantum automorphism group is similar to the classical automorphism group, but has relaxed commutivity. The classification of all undirected trees with automorphism group isomorphic to $(Z_2)^l$ along with a similar classification of all undirected asymmetric trees is used to give some insight into the structure of the quantum automorphism group for such graphs. / Ph. D.

This paper will be concerned mainly with automorphisms of groups. The concept of a group endomorphism will be used at various points in this paper.

This thesis examines the connections between the motivations of ante rem structuralism and the problem of automorphism. Ante rem structuralists are led to the problem of automorphism because of their commitment to the thesis of structure-relative identity. Ladyman's and Button's solutions to the problem are both unsatisfactory. The problem can be solved only if ante rem structuralists drop the thesis of structure-relative identity. Besides blocking the problem of automorphism, there are further reasons why the thesis has to be dropped. (i) The purported metaphysical and epistemic purchase of adopting the thesis can be put into doubt. (ii) Primitive identity within a mathematical structure is more in line with ante rem structuralist's commitment to the faithfulness constraint and to the ontological priority of structure over positions. However, the cost of dropping the thesis is that ante rem structuralists cannot provide a satisfactory solution to Benacerraf's problem of multiple reductions of arithmetic.

A not necessarily continuous, linear or multiplicative function £c from an algebra A into itself is called a local automorphism if £c agrees with an automorphism of A at each point in $A$. In this paper, we study the question when a local automorphism of a semisimple Banach algebra, is a Jordan isomorphism. Also a algebra is not necessary unital, but be implicitly assumed to be associative.

In this thesis we study the chamber graphs of the geometries ΓpA2nΓ1q, Γp3A7q, ΓpL2p11qq and ΓpL2p25qq which are related to the Petersen graph [4, 13]. In Chapter 2 we look at the chamber graph of ΓpA2nΓ1q and see what minimal paths between chambers look like. Chapter 3 finds and proves the diameter of these chamber graphs and we see what two chambers might look like if they are as far apart as possible. We discover the full automorphism group of the chamber graph. Chapters 4, 5 and 6 focus on the chamber graphs of ΓpL2p11qq,ΓpL2p25qq and Γp3A7q respectively. We ask questions such as what two chambers look like if they are as far apart as possible, and we find the automorphism groups of the chamber graphs.

In 1993, Muzychuk showed that the rational S-rings over a cyclic group Z_n are in one-to-one correspondence with sublattices of the divisor lattice of n, or equivalently, with sublattices of the lattice of subgroups of Z_n. This idea is easily extended to show that for any finite group G, sublattices of the lattice of characteristic subgroups of G give rise to rational S-rings over G in a natural way. Our main result is that any finite group may be represented as the automorphism group of such a rational S-ring over an abelian p-group. In order to show this, we first give a complete description of the automorphism classes and characteristic subgroups of finite abelian groups. We show that for a large class of abelian groups, including all those of odd order, the lattice of characteristic subgroups is distributive. We also prove a converse to the well-known result of Muzychuk that two S-rings over a cyclic group are isomorphic if and only if they coincide; namely, we show that over a group which is not cyclic, there always exist distinct isomorphic S-rings. Finally, we show that the automorphism group of any S-ring over a cyclic group is abelian.

It may be said of certain pairs of elements of a set that one element precedes the other. If the collection of all such pairs of elements in a given set exhibits certain properties, the set and the collection of pairs is said to constitute a partially ordered set. The purpose of this paper is to explore some of the properties of partially ordered sets.

The symmetric automorphism group of a free product is a group rich in algebraic structure and with strong links to geometric configuration spaces. In this thesis I describe in detail and for the first time the (co)homology of the symmetric automorphism groups. To this end I construct a classifying space for the Fouxe-Rabinovitch automorphism group, a large normal subgroup of the symmetric automorphism group. This classifying space is a moduli space of 'cactus products', each of which has the homotopy type of a wedge product of spaces. To study this space we build a combinatorial theory centred around 'diagonal complexes' which may be of independent interest. The diagonal complex associated to the cactus products consists of the set of forest posets, which in turn characterise the homology of the moduli spaces of cactus products. The machinery of diagonal complexes is then turned towards the symmetric automorphism groups of a graph product of groups. I also show that symmetric automorphisms may be determined by their categorical properties and that they are in particular characteristic of the free product functor. This goes some way to explain their occurence in a range of situations. The final chapter is devoted to a class of configuration spaces of Euclidean n-spheres embedded disjointly in (n+2)-space. When n = 1 this is the configuration space of unknotted, unlinked loops in 3-space, which has been well studied. We continue this work for higher n and find that the fundamental groups remain unchanged. We then consider the homology and the higher homotopy groups of the configuration spaces. Our last contribution is an epilogue which discusses the place of these groups in the wider field of mathematics. It is the functoriality which is important here and using this new-found emphasis we argue that there should exist a generalised version of the material from the final chapter which would apply to a far wider range of configuration spaces.

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.

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