Two problems in the theory of finite permutation groups are considered in this thesis: A. transitive groups of degree p, where p = 4q+1 and p,q are prime, B. …
Two problems in the theory of finite permutation groups are considered in this thesis: A. transitive groups of degree p, where p = 4q+1 and p,q are prime, B. automorphism groups of 2-graphs and some related algebras. Problem A should be seen in the following context: in 1963. N.Ito began a study of insoluble, transitive groups G of degree p on a set Ω, where p = 2q+1 and p,q are prime, showing among other things, that such a group G is 3-transitive. His methods involve the modular character theory of G for both the primes p and q (developed by R.Brauer). He uses this theory to prove facts about the permutation characters of G associated with Ω (2) and Ω {2} , the sets of ordered and unordered pairs (respectively) of distinct elements of Ω. The first part of this thesis represents an attempt to extend these methods to the case p = 4q+1. The main result obtained is Theorem. Let G be an insoluble, transitive permutation group of degree p, where p = 4q+1 and p.q are prime with p>13. Then G is 3-transitive. Also some progress is made towards a proof that the groups in Problem A are 4-transitive. In the second part of this thesis (Problem B) certain algebras are defined from 2-graphs as follows: let (Ω,Δ) be a 2-graph, that is, Δ is a set of 3-subsets of a finite set Ω such that every 4-subset of Ω contains an even number of elements of Δ. Write Ω= {e 1 ....,e n }. Given any field F of characteristic 2, make FΩ into an algebra by defining [see text for continuation of abstract].
This is a transcript of a lecture course on Infinite Permutation Groups given by Peter M. Neumann (1940-2020) in Oxford during the academic year 1988-1989. The field of Infinite Permutation …
This is a transcript of a lecture course on Infinite Permutation Groups given by Peter M. Neumann (1940-2020) in Oxford during the academic year 1988-1989. The field of Infinite Permutation Groups only emerged as an independent field of study in the 1980's. Most of the results described in these notes were at the time of the lectures brand new and had either just recently appeared in print or had not appeared formally. A large part of the results described is either due to Peter himself or heavily influenced by him. These notes offer Peter's personal take on a field that he was instrumental in creating and in many cases ideas and questions that can not be found in the published literature.
‘ … it will afford me much satisfaction if, by means of this book, I shall succeed in arousing interest among English mathematicians in a branch of pure mathematics which …
‘ … it will afford me much satisfaction if, by means of this book, I shall succeed in arousing interest among English mathematicians in a branch of pure mathematics which becomes the more fascinating the more it is studied.’W. Burnside, in his preface to Theory of groups of finite order, 1897.
Permutations are fundamental to many branches of mathematics. In this chapter, we examine permutations from a group-theoretic perspective.
Permutations are fundamental to many branches of mathematics. In this chapter, we examine permutations from a group-theoretic perspective.
A finite group $G$ is called a Schur group, if any Schur ring over $G$ is the transitivity module of a point stabilizer in a subgroup of $\sym(G)$ that contains …
A finite group $G$ is called a Schur group, if any Schur ring over $G$ is the transitivity module of a point stabilizer in a subgroup of $\sym(G)$ that contains all right translations. We complete a classification of abelian $2$-groups by proving that the group $\mZ_2\times\mZ_{2^n}$ is Schur. We also prove that any non-abelian Schur $2$-group of order larger than $32$ is dihedral (the Schur $2$-groups of smaller orders are known). Finally, in the dihedral case, we study Schur rings of rank at most $5$, and show that the unique obstacle here is a hypothetical S-ring of rank $5$ associated with a divisible difference set.
In this dissertation, we consider a wide range of problems in algebraic and extremal graph theory. In extremal graph theory, we will prove that the Tree Packing Conjecture is true …
In this dissertation, we consider a wide range of problems in algebraic and extremal graph theory. In extremal graph theory, we will prove that the Tree Packing Conjecture is true for all sequences of trees that are 'almost stars'; and we prove that the Erdos-Sos conjecture is true for all graphs G with girth at least 5. We also conjecture that every graph G with minimal degree k and girth at least $2t+1$ contains every tree T of order $kt+1$ such that $\Delta(T)\leq k.$ This conjecture is trivially true for t = 1. We Prove the conjecture is true for t = 2 and that, for this value of t, the conjecture is best possible. We also provide supporting evidence for the conjecture for all other values of t. In algebraic graph theory, we are primarily concerned with isomorphism problems for vertex-transitive graphs, and with calculating automorphism groups of vertex-transitive graphs. We extend Babai's characterization of the Cayley Isomorphism property for Cayley hypergraphs to non-Cayley hypergraphs, and then use this characterization to solve the isomorphism problem for every vertex-transitive graph of order pq, where p and q distinct primes. We also determine the automorphism groups of metacirculant graphs of order pq that are not circulant, allowing us to determine the nonabelian groups of order pq that are Burnside groups. Additionally, we generalize a classical result of Burnside stating that every transitive group G of prime degree p, is doubly transitive or contains a normal Sylow p-subgroup to all $p\sp k,$ provided that the Sylow p-subgroup of G is one of a specified family. We believe that this result is the most significant contained in this dissertation. As a corollary of this result, one easily gives a new proof of Klin and Poschel's result characterizing the automorphism groups of circulant graphs of order $p\sp k,$ where p is an odd prime.
In this survey paper I will talk about the classification of the maximal subgroups of the symmetric group of degree n, which can be divided into three main categories: intransitive, …
In this survey paper I will talk about the classification of the maximal subgroups of the symmetric group of degree n, which can be divided into three main categories: intransitive, imprimitive and primitive maximal subgroups.The O'Nan-Scott theorem and its proof will be discussed.The content of this paper was inspired by two courses I gave, one was a minicourse given in December 2018 in the University of Campinas (state of S.Paulo, Brazil) during the brazilian Escola de Álgebra 2018, the other was an online PhD course given in November 2020 for the University of Padova (Italy).This paper is structured as lecture notes, in particular it contains exercises for the reader.
We classify stably/retract rational norm one tori in dimension $n-1$ for $n=2^e$ $(e\geq 1)$ as a power of $2$ and $n=12, 14, 15$. Retract non-rationality of norm one tori for …
We classify stably/retract rational norm one tori in dimension $n-1$ for $n=2^e$ $(e\geq 1)$ as a power of $2$ and $n=12, 14, 15$. Retract non-rationality of norm one tori for primitive $G\leq S_{2p}$ where $p$ is a prime number and for the five Mathieu groups $M_n\leq S_n$ $(n=11,12,22,23,24)$ is also given.
For the direct product $\cZ\times \cZ_3$ of infinite cyclic group $\cZ$ and a cyclic group $\cZ_3$ of order $3$, the schur rings over it are classified. In particular, all the …
For the direct product $\cZ\times \cZ_3$ of infinite cyclic group $\cZ$ and a cyclic group $\cZ_3$ of order $3$, the schur rings over it are classified. In particular, all the schur rings are proved to be traditional.