[Formulae and special characters in this field can only be approximated. See PDF version for accurate reproduction] In this thesis we investigate the properties of k-closures of certain finite permutation groups. Given a permutation group G on a finite set Ω, for k ≥ 1, the k-closure G(k) of G is the largest subgroup of Sym(Ω) with the same orbits as G on the set Ωk of k-tuples from Ω. The first problem in this thesis is to study the 3-closures of affine permutation groups. In 1992, Praeger and Saxl showed if G is a finite primitive group and k ≥ 2 then either G(k) and G have the same socle or (G(k),G) is known. In the case where the socle of G is an elementary abelian group, so that G is a primitive group of affine transformations of a finite vector space, the fact that G(k) has the same socle as G gives little information about the relative sizes of the two groups G and G(k). In this thesis we use Aschbacher’s Theorem for subgroups of finite general linear groups to show that, if G ≤ AGL(d, p) is an affine permutation group which is not 3-transitive, then for any point α ∈ Ω, Gα and (G(3) ∩ AGL(d, p))α lie in the same Aschbacher class. Our results rely on a detailed analysis of the 2-closures of subgroups of general linear groups acting on non-zero vectors and are independent of the finite simple group classification. In addition, modifying the work of Praeger and Saxl in [47], we are able to give an explicit list of affine primitive permutation groups G for which G(3) is not affine. The second research problem is to give a partial positive answer to the so-called Polycirculant Conjecture, which states that every transitive 2-closed permutation group contains a semiregular element, that is, a permutation whose cycles all have the same length. This would imply that every vertex-transitive graph has a semiregular automorphism. In this thesis we make substantial progress on the Polycirculant Conjecture by proving that every vertex-transitive, locally-quasiprimitive graph has a semiregular automorphism. The main ingredient of the proof is the determination of all biquasiprimitive permutation groups with no semiregular elements. Publications arising from this thesis are [17, 54].

Part I of this thesis presents methods for finding the primitive permutation groups of degree d, where 2500 ≤ d < 4096, using the O'Nan-Scott Theorem and Aschbacher's theorem. Tables of the groups G are given for each O'Nan-Scott class. For the non-affine groups, additional information is given: the degree d of G, the shape of a stabiliser in G of the primitive action, the shape of the normaliser N in S[subscript(d)] of G and the rank of N. Part II presents a new algorithm NormaliserGL for computing the normaliser in GL[subscript(n)](q) of a group G ≤ GL[subscript(n)](q). The algorithm is implemented in the computational algebra system MAGMA and employs Aschbacher's theorem to break the problem into several cases. The attached CD contains the code for the algorithm as well as several test cases which demonstrate the improvement over MAGMA's existing algorithm.

A transitive decomposition of a graph is a partition of the arc set such that there exists a group of automorphisms of the graph which preserves and acts transitively on the partition. This turns out to be a very broad idea, with several striking connections with other areas of mathematics. In this thesis we first develop some general theory of transitive decompositions, and in particular we illustrate some of the more interesting connections with certain combinatorial and geometric structures. We then give complete, or nearly complete, structural characterisations of certain classes of transitive decompositions preserved by a group with a rank 3 action on vertices (such a group has exactly two orbits on ordered pairs of distinct vertices). The main classes of rank 3 groups we study (namely those which are imprimitive, or primitive of grid type) are derived in some way from 2-transitive groups (that is, groups which are transitive on ordered pairs of distinct vertices), and the results we achieve make use of the classification by Sibley in 2004 of transitive decompositions preserved by a 2-transitive group.

No abstract available. / Thesis (Ph.D.)-University of Natal, Pietermaritzburg, 2002.

Discrete mathematics has had many applications in recent years and this is only one reason for its increasing dynamism. The study of finite structures is a broad area which has a unity not merely of description but also in practice, since many of the structures studied give results which can be applied to other, apparently dissimilar structures. Apart from the applications, which themselves generate problems, internally there are still many difficult and interesting problems in finite geometry and combinatorics. There are still many puzzling features about sub-structures of finite projective spaces, the minimum weight of the dual codes of polynomial codes, as well as about finite projective planes. Finite groups are an ever strong theme for several reasons. There is still much work to be done to give a clear geometric identification of the finite simple groups. There are also many problems in characterizing structures which either have a particular group acting on them or which have some degree of symmetry from a group action. Codes obtained from permutation representations of finite groups have been given particular attention in recent years. Given a representation of group elements of a group G by permutations we can work modulo 2 and obtain a representation of G on a vector space V over lF2 . The invariant subspaces (the subspaces of V taken into themselves by every group element) are then all the binary codes C for which G is a subgroup of Aut(C). Similar methods produce codes over arbitrary fields. Through a module-theoretic approach, and based on a study of monomial actions and projective representations, codes with given transitive permutation group were determined by various authors. Starting with well known simple groups and defining designs and codes through the primitive actions of the groups will give structures that have this group in their automorphism groups. For each of the primitive representations, we construct the permutation group and form the orbits of the stabilizer of a point. Taking these ideas further we have investigated the codes from the primitive permutation representations of the simple alternating and symplectic groups of odd characteristic in their natural rank-3 primitive actions. We have also investigated alternative ways of constructing these codes, and these have come about by noticing that the codes constructed from the primitive permutations of the groups could also be obtained from graphs. We achieved this by constructing codes from the span of adjacency matrices of graphs. In particular we have constructed codes from the triangular graphs and from the graphs on triples. The simple symplectic group PSp2m(q), where m is at least 2 and q is any prime power, acts as a primitive rank-3 group of degree q2m-1/q-1 on the points of the projective (2m-1)-space PG2m-1(IFq ). The codes obtained from the primitive rank-3 action of the simple projective symplectic groups PSp2m(Q), where Q= 2t with t an integer such that t ≥ 1, are the well known binary subcodes of the projective generalized Reed-Muller codes. However, by looking at the simple symplectic groups PSp2m(q), where q is a power of an odd prime and m ≥ 2, we observe that in their rank-3 action as primitive groups of degree q2m-1/q-1 these groups have 2-modular representations that give rise to self-orthogonal binary codes whose properties can be linked to those of the underlying geometry. We establish some properties of these codes, including bounds for the minimum weight and the nature of some classes of codewords. The knowledge of the structures of the automorphism groups has played a key role in the determination of explicit permutation decoding sets (PD-sets) for the binary codes obtained from the adjacency matrix of the triangular graph T(n) for n ≥ 5 and similarly from the adjacency matrices of the graphs on triples. The successful decoding came about by ordering the points in such a way that the nature of the information symbols was known and the action of the automorphism group apparent. Although the binary codes of the triangular graph T(n) were known, we have examined the codes and their duals further by looking at the question of minimum weight generators for the codes and for their duals. In this way we find bases of minimum weight codewords for such codes. We have also obtained explicit permutation-decoding sets for these codes. For a set Ω of size n and Ω{3} the set of subsets of Ω of size 3, we investigate the binary codes obtained from the adjacency matrix of each of the three graphs with vertex set Ω{3}1 with adjacency defined by two vertices as 3-sets being adjacent if they have zero, one or two elements in common, respectively. We show that permutation decoding can be used, by finding PD-sets, for some of the binary codes obtained from the adjacency matrix of the graphs on (n3) vertices, for n ≥ 7. / Thesis (Ph.D.)-University of Natal, Pietermaritzburg, 2002.

Let F be a finite field of q elements and G be a primitive group on a finite set . Then there is a G-action on , namely a map G ! , (g; !) 7! !g = g!; satisfying !gg0 = (gg0)! = g(g0!) for all g; g0 2 G and all ! 2 , and that !1 = 1! = ! for all ! 2 : Let F = ff j f : ! Fg, be the vector space over F with basis . Extending the G-action on linearly, F becomes an FG-module called an FG- permutation module. We are interested in finding all G-invariant FG-submodules, i.e., codes in F . The elements f 2 F are written in the form f = P !2 a! ! where ! is a characteristic function. The natural action of an element g 2 G is given by g P !2 a! ! = P !2 a! g(!): This action of G preserves the natural bilinear form defined by * X a! !; X b! ! + = X a!b!: In this thesis a program is proposed on how to determine codes with given primitive permutation group. The approach is modular representation theoretic and based on a study of maximal submodules of permutation modules F defined by the action of a finite group G on G-sets = G=Gx. This approach provides the advantage of an explicit basis for the code. There appear slightly different concepts of (linear) codes in the literature. Following Knapp and Schmid [83] a code over some finite field F will be a triple (V; ; F), where V = F is a free FG-module of finite rank with basis and a submodule C. By convention we call C a code having ambient space V and ambient basis . F is the alphabet of the code C, the degree n of V its length, and C is an [n; k]-code if C is a free module of dimension k. In this thesis we have surveyed some known methods of constructing codes from primitive permutation representations of finite groups. Generally, our program is more inclusive than these methods as the codes obtained using our approach include the codes obtained using these other methods. The designs obtained by other authors (see for example [40]) are found using our method, and these are in general defined by the support of the codewords of given weight in the codes. Moreover, this method allows for a geometric interpretation of many classes of codewords, and helps establish links with other combinatorial structures, such as designs and graphs. To illustrate the program we determine all 2-modular codes that admit the two known non-isomorphic simple linear groups of order 20160, namely L3(4) and L4(2) = A8. In the process we enumerate and classify all codes preserved by such groups, and provide the lattice of submodules for the corresponding permutation modules. It turns out that there are no self-orthogonal or self-dual codes invariant under these groups, and also that the automorphism groups of their respective codes are in most cases not the prescribed groups. We make use of the Assmus Matson Theorem and the Mac Williams identities in the study of the dual codes. We observe that in all cases the sets of several classes of non-trivial codewords are stabilized by maximal subgroups of the automorphism groups of the codes. The study of the codes invariant under the simple linear group L4(2) leads as a by-product to a unique flag-transitive, point primitive symmetric 2-(64; 28; 12) design preserved by the affi ne group of type 26:S6(2). This has consequently prompted the study of binary codes from the row span of the adjacency matrices of a class of 46 non-isomorphic symmetric 2-(64; 28; 12) designs invariant under the Frobenius group of order 21. Codes obtained from the orbit matrices of these designs have also been studied. The thesis concludes with a discussion of codes that are left invariant by the simple symplectic group S6(2) in all its 2-modular primitive permutation representations. / Thesis (Ph.D.)-University of KwaZulu-Natal, Westville, 2012.

The current thesis is focused on synchronizing permutation groups and on graph endo- morphisms. Applying the implicit classification of rank 3 groups, we provide a bound on synchronizing ranks of rank 3 groups, at first. Then, we determine the singular graph endomorphisms of the Hamming graph and related graphs, count Latin hypercuboids of class r, establish their relation to mixed MDS codes, investigate G-decompositions of (non)-synchronizing semigroups, and analyse the kernel graph construction used in the theorem of Cameron and Kazanidis which identifies non-synchronizing transformations with graph endomorphisms [20]. The contribution lies in the following points: 1. A bound on synchronizing ranks of groups of permutation rank 3 is given, and a complete list of small non-synchronizing groups of permutation rank 3 is provided (see Chapter 3). 2. The singular endomorphisms of the Hamming graph and some related graphs are characterised (see Chapter 5). 3. A theorem on the extension of partial Latin hypercuboids is given, Latin hyper- cuboids for small values are counted, and their correspondence to mixed MDS codes is unveiled (see Chapter 6). 4. The research on normalizing groups from [3] is extended to semigroups of the form < G, T >, and decomposition properties of non-synchronizing semigroups are described which are then applied to semigroups induced by combinatorial tiling problems (see Chapter 7). 5. At last, it is shown that all rank 3 graphs admitting singular endomorphisms are hulls and it is conjectured that a hull on n vertices has minimal generating set of at most n generators (see Chapter 8).

The theory of infinite multiplication has been studied in the case of the Hawaiian earring group, and has been seen to simplify the description of that group. In this paper we try to extend the theory of infinite multiplication to other groups and give a few examples of how this can be done. In particular, we discuss the theory as applied to symmetric groups and braid groups. We also give an equivalent definition to K. Eda's infinitary product as the fundamental group of a modified wedge product.

Advantages of the double coset enumeration technique include its use to represent group elements in a convenient shorter form than their usual permutation representations and to find nice permutation representations for groups. In this thesis we construct, by hand, several groups, including U₃(3) : 2, L₂(13), PGL₂(11), and PGL₂(7), represent their elements in the short form (symmetric representation) and produce their permutation representations.

Every transitive permutation representation of a finite group is the representation of the group in its action on the cosets of a particular subgroup of the group. The group has a certain rank for each of these representations. We first find almost all rank-3 and rank-4 transitive representations of the projective special linear group P SL(2, q) where q = pm and p is an odd prime. We also determine the rank of P SL (2, p) in terms of p on the cosets of particular given subgroups. We then investigate the construction of rank-3 transitive and primitive extensions of a simple group, such that the extension group formed is also simple. In the latter context we present a new, group theoretic construction of the famous Hoffman-Singleton graph as a rank-3 graph. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2015 / FAU Electronic Theses and Dissertations Collection