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The Frobenius group is an example of a split extension. In this dissertation we study and describe the properties and structure of the group. We also describe the properties and structure of the kernel and complement, two non-trivial subgroups of every Frobenius group. Examples of Frobenius groups are included and we also describe the characters of the group. Finally we construct the Frobenius group 292 : SL(2, 5) and then compute it's Fischer matrices and character table. / Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2012.

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With the classification of finite simple groups having been completed in 1981, recent work in group theory has involved the study of the structures of simple groups. The character tables of maximal subgroups of simple groups give substantive information about these groups. Most of the maximal subgroups of simple groups are of extension type. Some of the maximal subgroups of simple groups contain constituents of the generalized symmetric groups. Here we shall be interested in discussing such groups which we may call groups associated with the generalized symmetric groups. There are several well developed methods for calculating the character tables of group extensions. However Fischer [17] has given an effective method for calculating the character tables of some group extensions including the generalized symmetric group B (m, n). Actually work on the characters of wreath products with permutation groups dates back to Specht's work [61], through the works of Osima [49] and Kerber [33]. And more recently other people have worked on characters of wreath products with symmetric groups, these amongst others include Darafshesh and Iranmanesh [14], List and Mahmoud [36], Puttaswamiah [52], Read [55, 56], Saeed-Ul-Islam [59] and Stembridge [64]. It is well known that the character table of the generalized symmetric group B(m, n), where m and n are positive integers, can be constructed in GAP [22] with B(m, n) considered as the wreath product of the cyclic group Zm of order m with the symmetric group Sn' For example Pfeiffer [50] has given programmes which compute the character tables of wreath products with symmetric groups in GAP. However it may be necessary to obtain the partial character table of a group in hand rather than its complete character table. Further due to limited workspace in GAP, the wreath product method can only be used to compute character tables of B(m, n) for small values of m and n. It is for these reasons amongst others that Fischer's method is sometimes used to construct the character tables of such groups. groups B(2, 6) and B(3, 5) of orders 46080 and 29160 is done here. We have also used Programme 5.2.4 to construct the Fischer-Clifford matrices of the groups B(2, 12) and B(4, 5) of orders 222 x 35 X 52 X 7 x 11 and 213 x 3 x 5 respectively. Due to lack of space here we have given the Fischer-Clifford matrices of B(2, 12) and B(4,5) on the compact disk submitted with this thesis. However note that these matrices are the equivalent form of the Fischer-Clifford matrices of B(2, 12) and B(4,5). In [35] R.J. List has presented a method for constructing the Fischer-Clifford matrices of group extensions of an irreducible constituent of the elementary abelian group 2n by a symmetric group. The other aim of our work is to adapt the combinatorial method in [5] to the construction of the Fischer-Clifford matrices of some group extensions associated with B(m, n), using a similar method as the one used in [35]. Examples are given on the application of this adaptation to some groups associated with the groups B(2, 6), B(3,3) and B(3, 5). In this thesis we have constructed the character tables of the groups B(2, 6) and B(3,5) and some group extensions associated with these two groups and B(3, 3). We have also constructed the character tables of the groups B(2, 12) and B(4, 5) in our work, these character tables are given on the compact disk submitted with this thesis. The correctness of all the character tables constructed in this thesis has been tested in GAP. The main working programmes (Programme 2.2.3, Programme 3.1.9, Programme 3.1.10, Programme 5.2.1, Programme 5.2.4 and Programme 5.2.2) are given on the compact disk submitted with this thesis. It is anticipated that with further improvements, a number of the programmes given here will be incorporated into GAP. Indeed with further research work the programmes given here should lead to an alternative programme for computing the character table of B(m, n). / Thesis (Ph.D.)- University of KwaZulu-Natal, Pietermaritzburg, 2005.

The purpose of this thesis is to discuss the rôle of certain types of quiver which appear in the modular representation theory of finite groups. It is our concern to study two different types of quiver. First of all we construct the ordinary quiver of certain blocks of defect 2 of the symmetric group, and then apply our results to the alternating group and to the theory of partitions. Secondly, we consider connected components of the stable Auslander-Reiten quiver of certain groups G with normal subgroup N. The main interest lies in comparing the tree class of components of N-modules, with the tree class of components of these modules induced up to G.

The thesis centres around two problems in the enumeration of p-groups. Define f_φ(p^m) to be the number of (isomorphism classes of) groups of order p^m in an isoclinism class φ. We give bounds for this function as φ is fixed and m varies and as m is fixed and φ varies. In the course of obtaining these bounds, we prove the following result. We say a group is reduced if it has no non-trivial abelian direct factors. Then the rank of the centre Z(P) and the rank of the derived factor group P|P' of a reduced p-group P are bounded in terms of the orders of P|Z(P)P' and P'∩Z(P). A long standing conjecture of Charles C. Sims states that the number of groups of order p^m is<br/> p^{²andfrasl;₂₇m³+O(m²)}. (1) We show that the number of groups of nilpotency class at most 3 and order p^m satisfies (1). We prove a similar result concerning the number of graded Lie rings of order p^m generated by their first grading.

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).

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