Double coset enumeration

AbstractCoset enumeration is the principal method for solving the word problem in finitely presented groups. The technique has a long history and was one of the first applications of electronic computers to pure mathematics. Present applications are limited by the space available to store the coset table in computer memory. Enumerating double cosets offers a substantial saving of space in suitable cases. An algorithm is described with some notes on implementation

Symmetric Presentations and Double Coset Enumeration

In this project, we demonstrate our discovery of original symmetric presentations and constructions of important groups, including nonabelian simple groups, and groups that have these as factor groups. The target nonabelian simple groups include alternating, linear, and sporadic groups. We give isomorphism types for each finite homomorphic image that has been found. We present original symmetric presentations of $M_{12}$, $M_{21}:(2 \times 2)$, $L_{3}(4):2^2$, $2:^{\cdot}L_{3}(4):2$, $S(4,3)$, and $S_{7}$ as homomorphism images of the progenitors $2^{*20}$ $:$ $A_{5}$, $2^{*10}$ $:$ $PGL(2,9)$, $2^{*10}$ $:$ $Aut(A_{6})$, $2^{*10}$ $:$ $A_{6}$, $2^{*10}$ $:$ $A_{5}$, and $2^{*24}$ $:$ $S_{5}$, respectively. We also construct $M_{12}$, $M_{21}:(2 \times 2)$, $L_{3}(4):2^2$, $L_{3}(4):2^2$, $2:^{\cdot}L_{3}(4):2$, $S(4,3)$, and $S_{7}$ over $A_{5}$, $PGL(2,9)$, $Aut(A_{6})$, $A_{6}$, $A_{5}$, and $S_{5}$, respectively, using our technique of double coset enumeration. All of the symmetric presentations given are original to the best of our knowledge

Homomorphic images of semi-direct products

The main purpose of this thesis is to describe methods of constructing computer-free proofs of existence of finite groups and give useful techniques to perform double coset enumeration of groups with symmetric presentations over their control groups

Symmetric generation of finite homomorphic images?

The purpose of this thesis was to present the technique of double coset enumeration and apply it to construct finite homomorphic images of infinite semidirect products. Several important homomorphic images include the classical groups, the Projective Special Linear group and the Derived Chevalley group were constructed

Progenitors Involving Simple Groups

I will be going over writing representations of both permutation and monomial progenitors, which include 2^{*4} : D_4, 2^(*7) :L_2 (7) as permutation progenitors, and monomial progenitors 7^(*2) :_m S_3 \times 2, 11^{*2} :_m (5:2)^{*}5, 11^{*3} :_m (25:3), 11^{*4} :_m (4 : 5)^{*}5. Also, the images of these different progenitors at both lower and higher fields and orders. \\ We will also do the double coset enumeration of S5 over D6, S6 over 5 : 4, A_5 x A_5 over (5:2)^{*}5, and go on to also do the double coset enumeration over maximal subgroups for larger constructions. We will also do the construction of sporadic group M22 over maximal subgroup A7, and also J1 with the monomial representation 7^(*2) :_m S_3 \times 2 over maximal subgroup PSL(2,11). We will also look at different extension problems of composition factors of different groups, and determine the isomorphism types of each extension

A study of finite symmetrical groups

This study investigated finite homomorphic images of several progenitors, including 2*⁵ : S₅, 2*⁶ : A₆, and 3*⁵ : C₅ The technique of manual of double coset enumeration is used to construct several groups by hand and computer-based proofs are given for the isomorphism types of the groups that are not constructed

Symmetrically generated groups

This thesis constructs several groups entirely by hand via their symmetric presentations. In particular, the technique of double coset enumeration is used to manually construct J₃ : 2, the automorphism group of the Janko group J₃, and represent every element of the group as a permutation of PSL₂ (16) : 4, on 120 letters, followed by a word of length at most 3

Symmetric representation of elements of finite groups

The purpose of the thesis is to give an alternative and more efficient method for working with finite groups by constructing finite groups as homomorphic images of progenitors. The method introduced can be applied to all finite groups that possess symmetric generating sets of involutions. Such groups include all finite non-abelian simple groups, which can then be constructed by the technique of manual double coset enumeration

Symmetric generation of finite groups

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