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Designs and codes from certain finite simple groups / George Ferdinand Randriafanomezantsoa-RodeheryRandriafanomezantsoa-Rodehery, George Ferdinand January 2013 (has links)
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
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Some finite simple groupsFletcher, L. R. January 1971 (has links)
No description available.
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The Covering Numbers of Some Finite Simple GroupsUnknown Date (has links)
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
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An elementary characterization of the simple groups PSL (3, 3) and M 11 in terms of the centralizer of an involution /Doyle, John. January 1984 (has links) (PDF)
Thesis (M. Sc.)--University of Adelaide, Dept. of Pure Mathematics, 1984. / Includes bibliographical references (leaves 87-88).
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2-generations pf the sporadic simple groups.Ganief, Moegamad Shahiem. January 1997 (has links)
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.
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Linear codes obtained from 2-modular representations of some finite simple groups.Chikamai, Walingo Lucy. January 2012 (has links)
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.
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Subgroups of the symmetric group of degree n containing an n-cycle /Charlebois, Joanne January 1900 (has links)
Thesis (M.Sc.) - Carleton University, 2002. / Includes bibliographical references (p. 40-43). Also available in electronic format on the Internet.
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Verifying Huppert's Conjecture for the Simple Groups of Lie Type of Rank TwoWakefield, Thomas Philip 30 May 2008 (has links)
No description available.
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Symmetrically generated groupsNguyen, Benny 01 January 2005 (has links)
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.
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The maximal subgroups of the classical groups in dimension 13, 14 and 15Schröder, Anna Katharina January 2015 (has links)
One might easily argue that the Classification of Finite Simple Groups is one of the most important theorems of group theory. Given that any finite group can be deconstructed into its simple composition factors, it is of great importance to have a detailed knowledge of the structure of finite simple groups. One of the classes of finite groups that appear in the classification theorem are the simple classical groups, which are matrix groups preserving some form. This thesis will shed some new light on almost simple classical groups in dimension 13, 14 and 15. In particular we will determine their maximal subgroups. We will build on the results by Bray, Holt, and Roney-Dougal who calculated the maximal subgroups of all almost simple finite classical groups in dimension less than 12. Furthermore, Aschbacher proved that the maximal subgroups of almost simple classical groups lie in nine classes. The maximal subgroups in the first eight classes, i.e. the subgroups of geometric type, were determined by Kleidman and Liebeck for dimension greater than 13. Therefore this thesis concentrates on the ninth class of Aschbacher's Theorem. This class roughly consists of subgroups which are almost simple modulo scalars and do not preserve a geometric structure. As our final result we will give tables containing all maximal subgroups of almost simple classical groups in dimension 13, 14 and 15.
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