Browsing by Author "Rodrigues, Bernardo Gabriel."

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Axial algebras for sporadic simple groups HS and Suz.
(2018) Shumba, Tendai M. Mudziiri.; Rodrigues, Bernardo Gabriel.; Shpectorov, Sergey.
Motivated by the construction of the Monster sporadic simple group as a group of automorphisms of an algebra and the recent development of ax- ial algebras as a generalization of Majorana representations, we construct axial algebras for the sporadic simple groups HS and Suz in different ways analogous to the Norton algebra construction. We study how these algebras decompose as direct sums of the adjoint action of an axis. Fusion rules, that is the rules with which eigenvectors from various eigenspaces of the adjoint action multiply, are found. This places these groups in a general framework of groups acting on algebras hence giving a common theme for their origin.
Codes of designs and graphs from finite simple groups.
(2002) Rodrigues, Bernardo Gabriel.; Moori, Jamshid.; Key, Jennifer Denise.
No abstract available.
Linear codes obtained from 2-modular representations of some finite simple groups.
(2012) Chikamai, Walingo Lucy.; Rodrigues, Bernardo Gabriel.
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.
On group factorizations.
(2016) Nasilele, Monde.; Rodrigues, Bernardo Gabriel.
Abstract available in PDF file.
On minimal degrees of faithful permutation representations of finite groups.
(2014) Mthethwa, Simo Sisize.; Rodrigues, Bernardo Gabriel.
Abstract available in PDF file.
On the existence of self-dual codes invariant under permutation groups.
(2014) Shumba, Tendai M. Mudziiri.; Rodrigues, Bernardo Gabriel.
Abstract available in PDF file.
On the theory and examples of group extensions.
(1999) Rodrigues, Bernardo Gabriel.; Moori, Jamshid.
The work described in this dissertation was largely motivated by the aim of producing a survey on the theory of group extensions. From the broad scope of the theory of group extensions we single out two aspects to discuss, namely the study of the split and the non-split cases and give examples of both. A great part of this dissertation is dedicated to the study of split extensions. After setting the background theory for the study of the split extensions we proceed in exploring the ramifications of this concept within the development of the group structure and consequently investigate well known products which are its derived namely the holomorph, and the wreath product. The theory of group presentations provides in principle the necessary tools that permit the description of a group by means of its generators and relators. Through this knowledge we give presentations for the groups of order pq,p2q and p3. Subsequently using a classical result of Gaschutz we investigate the split extensions of non-abelian groups in which the normal subgroup is either a non-abelian normal nilpotent group or a non-abelian normal solvable group. We also study other cases of split extensions such as the affine subgroups of the general linear and the symplectic groups. It is expected that some of the results obtained will provide a theoretical algorithm to describe these affine subgroups. A particular case of the non-split extensions is discussed as the Frattini extensions. In fact a simplest example of a Frattini extension is a non-split extension in which the kernel of an epimorphism e is an irreducible G-module.