In this dissertation, we study four methods for constructing codes and designs from finite groups. The first method was developed by Carmichael and Ernst in the nineteen thirty's. The second method is a generalization of the first one by D.R. Hughes in the nineteen sixty's. These first two methods use t-transitive groups to construct t-designs. The last methods arc two recent techniques developed by J .D. Key and J. Moori (2002). they use primitive finite groups to build l-designs. We will apply these methods to simple groups, and use the incidence matrix of the constructed designs to generate codes. / Thesis (Msc. in Mathematics) North-West University, Mafikeng Campus, 2013

This thesis contains methods of finding new presentations of finite groups, particularly nonabelian simple groups. We have presented several progenitors such as 2^{*8}:Z_4 wr Z_2, 3^{*3}:_m L(2,7), 2^{*4}:[2:2^2], 2^{*11}:D_{11} and many more on which we've found the mathieu group M12 and 2*[M21:2^2] among their homomorphic images. We give the full monomial automorphism groups of Aut(3^{*2}), Aut(3^{*3}), and Aut(5^{*2}). Included is a proof showing that the full monomial automorphism group of Aut(m^{*n}) is isomorphic to U(m) wr S_n. In addition we have constructed the Cayley Diagrams of PGL(2,7), [3 x A_5]:2, 3:[A_6:2], and 2 x [(3 x L(2,11)):2] using the process of double coset enumeration.

No description available.

No description available.

A finite cover C of a group G is a finite collection of proper subgroups of G such that G is equal to the union of all of the members of C. Such a cover is called minimal if it has the smallest cardinality among all finite covers of G. The covering number of G, denoted by σ(G), is the number of subgroups in a minimal cover of G. Here we determine the covering numbers of the projective special unitary groups U3(q) for q ≤ 5, and give upper and lower bounds for the covering number of U3(q) when q > 5. We also determine the covering number of the McLaughlin sporadic simple group, and verify previously known results on the covering numbers of the Higman-Sims and Held groups. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2019. / FAU Electronic Theses and Dissertations Collection

Thesis (M. Sc.)--University of Adelaide, Dept. of Pure Mathematics, 1984. / Includes bibliographical references (leaves 87-88).

A group G is said to be 2-generated if G = (x, y), for some non-trivial elements x, y E G. In this thesis we investigate three special types of 2-generations of the sporadic simple groups. A group G is a (l, rn, n )-generated group if G is a quotient group of the triangle group T(l, rn, n) = (x, y, zlx1 = ym = zn = xyz = la). Given divisors l, rn, n of the order of a sporadic simple group G, we ask the question: Is G a (l, rn, n)-generated group? Since we are dealing with simple groups, we may assume that III +l/rn + l/n < 1. Until recently interest in this type of generation had been limited to the role it played in genus actions of finite groups. The problem of determining the genus of a finite simple group is tantamount to maximizing the expression III +l/rn +Iln for which the group is (l,rn,n)-generated. Secondly, we investigate the nX-complementary generations of the finite simple groups. A finite group G is said to be nX-complementary generated if, given an arbitrary non-trivial element x E G, there exists an element y E nX such that G = (x, y). Our interest in this type of generation is motivated by a conjecture (Brenner-Guralnick-Wiegold [18]) that every finite simple group can be generated by an arbitrary non-trivial element together with another suitable element. It was recently proved by Woldar [181] that every sporadic simple group G is pAcomplementary generated, where p is the largest prime divisor of IGI. In an attempt to further the theory of X-complementary generations of the finite simple groups, we pose the following problem. Which conjugacy classes nX of the sporadic simple groups are nX-complementary generated conjugacy classes. In this thesis we provide a complete solution to this problem for the sporadic simple groups HS, McL, C03, Co2 , Jt , J2 , J3 , J4 and Fi 22 · We partially answer the question on (l, rn, n)-generation for the said sporadic groups. A finite non-abelian group G is said to have spread r iffor every set {Xl, X2, ' , "xr } of r non-trivial distinct elements, thpre is an element y E G such that G = (Xi, y), for all i. Our interest in this type of 2-generation comes from a problem by BrennerWiegold [19] to find all finite non-abelian groups with spread 1, but not spread 2. Every sporadic simple group has spread 1 (Woldar [181]) and we show that every sporadic simple group has spread 2. / Thesis (Ph.D.)-University of Natal, Pietermaritzburg, 1997.

No description available.

The purpose of this thesis is to develop original symmetric presentations of finite non-abelian simple groups, particularly the sporadic simple groups. We have found original symmetric presentations for the Janko group J1, the Mathieu group M12, the Symplectic groups S(3,4) and S(4,5), a Lie type group Suz(8), and the automorphism group of the Unitary group U(3,5) as homomorphic images of the progenitors 2*60 : (2 x A5), 2*60 : A5, 2*56 : (23 : 7), and 2*28 : (PGL(2,7):2), respectively. We have also discovered the groups 24 : A5, 34 : S5, PSL(2,31), PSL(2,11), PSL(2,19), PSL(2,41), A8, 34 : S5, A52, 2• A52, 2 : A62, PSL(2,49), 28 : A5, PGL(2,19), PSL(2,71), 24 : A5, 24 : A6, PSL(2,7), 3 x PSL(3,4), 2• PSL(3,4), PSL(3,4), 2• (M12 : 2), 37:S7, 35 : S5, S6, 25 : S6, 35 : S6, 25 : S5, 24 : S6, and M12 as homomorphic images of the permutation progenitors 2*60 : (2 x A5), 2*60 : A5, 2*21 : (7: 3), 2*60 : (2 x A5), 2*120 : S5, and 2*144 : (32 : 24). We have given original proof of the 2*n Symmetric Presentation Theorem. In addition, we have also provided original proof for the Extension of the Factoring Lemma (involutory and non-involutory progenitors). We have constructed S5, PSL(2,7), and U(3,5):2 using the technique of double coset enumeration and by way of linear fractional mappings. Furthermore, we have given proofs of isomorphism types for 7 x 22, U(3,5):2, 2•(M12 : 2), and (4 x 2) :• 22.

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.