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Group extensionsUnknown Date (has links)
"Definition 1. A group G is an extension of a group A by a group B if and only if A is a normal subgroup of G and the factor group G/A is isomorphic to B. Definition 2. Two extensions G and H of A by B are called equivalent if and only if there exists an isomorphism between G and H that on A coincides with the identity automorphism and that maps onto each other the cosets of A corresponding to one and the same element of B. Consider the following example: let G be the cyclic group of order 4, that is G = {1, a, a², a³} and let H[subscript G] = {1, a²} be a normal subgroup of G. Now let V be the Klein fourgroup, that is, V = {1, a, b, c : a²=b²=c²=1} and H[subscript V] = {1, b} a normal subgroup of V. Since H[subscript G] and H[subscript V] are cyclic groups of order 2, set H[subscript V] = H[subscript G] = H. G and V are extensions of H by itself but are not equivalent extensions since no isomorphism exists between G and V. So the question arises: what are the necessary and sufficient conditions that two extensions G and G' of a group A by a group B be equivalent?"Introduction. / Typescript. / "January 1960." / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Arts." / Advisor: Paul J. McCarthy, Professor Directing Paper. / Includes bibliographical references (leaf 37).

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On the theory of the frobenius groups.Perumal, Pragladan. January 2012 (has links)
The Frobenius group is an example of a split extension. In this dissertation we study and describe
the properties and structure of the group. We also describe the properties and structure of the
kernel and complement, two nontrivial subgroups of every Frobenius group. Examples of Frobenius
groups are included and we also describe the characters of the group. Finally we construct the
Frobenius group 292 : SL(2, 5) and then compute it's Fischer matrices and character table. / Thesis (M.Sc.)University of KwaZuluNatal, Pietermaritzburg, 2012.

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Generic Galois extensions for groups or order p³ /Blue, Meredith Patricia, January 2000 (has links)
Thesis (Ph. D.)University of Texas at Austin, 2000. / Vita. Includes bibliographical references (leaves 112116). Available also in a digital version from Dissertation Abstracts.

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On the theory and examples of group extensions.Rodrigues, Bernardo Gabriel. January 1999 (has links)
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 nonsplit 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
nonabelian groups in which the normal subgroup is either a nonabelian normal nilpotent group or a nonabelian 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 nonsplit extensions is discussed as the Frattini extensions. In fact a simplest example of a Frattini extension is a nonsplit extension in which the kernel of an epimorphism e is an irreducible Gmodule. / Thesis (M.Sc.)University of Natal, Pietermaritzburg, 1999.

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FischerClifford theory for split and nonsplit group extensions.January 2001 (has links)
The character table of a finite group provides considerable amount of information about the group, and hence is of great importance in Mathematics as well as in Physical Sciences. Most of the maximal subgroups of the finite simple groups and their automorphisms are of extensions of elementary abelian groups, so methods have been developed for calculating the character tables of extensions of elementary abelian groups. Character tables of finite groups can be constructed using various techniques. However Bernd Fischer presented a powerful and interesting technique for calculating the character tables of group extensions. This technique, which is known as the technique of the FischerClifford matrices, derives its fundamentals from the Clifford theory. If G=N.G is an appropriate extension of N by G, the method involves the construction of a nonsingular matrix for each conjugacy class of G/N~G. The character table of G can then be determined from these FischerClifford matrices and the character table of certain subgroups of G, called inertia factor groups. In this dissertation, we described the FischerClifford theory and apply it to both split and nonsplit group extensions. First we apply the technique to the split extensions 2,7:Sp6(2) and 2,8:SP6(2) which are maximal subgroups of Sp8(2) and 2,8:08+(2) respectively. This technique has also been discussed and used by many other researchers, but applied only to split extensions or to the case when every irreducible character of N can be extended to an irreducible character of its inertia group in G. However the same method can not be used to construct character tables of certain nonsplit group extensions. In particular, it can not be applied to the nonsplit extensions of the forms 3,7.07(3) and 3,7.(0,7(3):2) which are maximal subgroups of Fischer's largest sporadic simple group Fi~24 and its automorphism group Fi24 respectively. In an attempt to generalize these methods to such type of nonsplit group extensions, we need to consider the projective representations and characters. We have shown that how the technique of FischerClifford matrices can be applied to any such type of nonsplit extensions. However in order to apply this technique, the projective characters of the inertia factors must be known and these can be difficult to determine for some groups. We successfully applied the technique of FischerClifford matrices and determined the FischerClifford matrices and hence the character tables of the nonsplit extensions 3,7.0,7(3) and 3,7.(0,7(3):2). The character tables computed in this thesis have been accepted for incorporation into GAP and will be available in the latest versions. / Thesis (Ph.D)University of Natal, Pietermaritzburg, 2001.

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FischerClifford theory and character tables of group extensions.Mpono, Zwelethemba Eugene. January 1998 (has links)
The smallest Fischer sporadic simple group Fi22 is generated by a conjugacy class D of 3510 involutions called 3transpositions such that the product of any noncommuting pair is an element of order 3. In Fi22 there are exactly three conjugacy classes of involutions denoted by D, T and N and represented in the ATLAS [26] by 2A, 2B and 2C, containing 3510, 1216215 and 36486450 elements with corresponding centralizers
2·U(6,2), (2 x 2~+8:U(4,2)):2 and 25+8:(83 X 32:4) respectively. In Fi22 , we have Npi22(26) = 26:8P(6,2), where 26 is a 2Bpure group, and thus the maximal subgroup 26:8P(6, 2) of Fi22 is a 2local subgroup. The full automorphism group of Fi22 is denoted by Fi22 . In Fi22 , there are three involutory outer automorphisms of Fi22 which are denoted bye, f and 0 and
represented in the ATLAS [26] by 2D, 2F and 2E respectively. We obtain that Fi22 = Fi22 :(e) and it can be easily shown that Fi22 = Fi22 :(e) = Fi22 :(f) = Fi22 :(0). As e, f and 0 act on Fi22 , then we obtain the subgroups CPi22 (e) rv 0+(8,2):83, CPi22 (f) rv 8P(6,2) x 2 and CPi22 (()) rv 26:0(6,2) of Fi22 which are generated by CD(e), Cn(f) and CD(0) respectively.
In this thesis we are concerned with the construction of the character tables of certain groups which are associated with Fi22 and its automorphism group Fi22 . We use the technique of the FischerClifford matrices to construct the character tables of these groups, which are split extensions. These groups are 26:8P(6, 2), 26:0(6,2) and 27:8P(6, 2). The study of the group 26:8P(6, 2) is essential, as the other groups studied in this thesis are related to it. The groups 8P(6,2) and 0 (6,2) of 6 x 6
matrices over GF(2), played crucial roles in our construction of the group 8P(6, 2) as a group of 7 x 7 matrices over GF(2) which would act on 27 . Also the character table of 25:86 , the affine subgroup of 8P(6, 2) fixing a nonzero vector in 26 , is constructed by using the technique of the FischerClifford matrices. This character table is used in the construction of the character table 26:SP(6, 2). The character tables computed in this thesis have been accepted for incorporation into GAP and will be available in the latest version of GAP. / Thesis (Ph.D.)  University of Natal, Pietermaritzburg, 1998.

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Character tables of some selected groups of extension type using FischerClifford matricesMonaledi, R.L. January 2015 (has links)
>Magister Scientiae  MSc / The aim of this dissertation is to calculate character tables of group extensions. There are several well developed methods for calculating the character tables of some selected group extensions. The method we study in this dissertation, is a standard application of Clifford theory, made efficient by the use of FischerClifford matrices, as introduced by Fischer. We consider only extensions Ḡ of the normal subgroup N by the subgroup G with the property that every irreducible character of N can be extended to an irreducible character of its inertia group in Ḡ , if N is abelian. This is indeed the case if Ḡ is a split extension, by a well known theorem of Mackey. A brief outline of the classical theory of characters pertinent to this study, is followed by a discussion on the calculation of the conjugacy classes of extension groups by the method of coset analysis. The Clifford theory which provide the basis for the theory of FischerClifford matrices is discussed in detail. Some of the properties of these FischerClifford matrices which make their calculation much easier, are also given. We restrict ourselves to split extension groups Ḡ = N:G in which N is always an elementary abelian 2group. In this thesis we are concerned with the construction of the character tables (by means of the technique of FischerClifford matrices) of certain extension groups which are associated with the orthogonal group O+10(2), the automorphism groups U₆(2):2, U₆(2):3 of the unitary group U₆(2) and the smallest Fischer sporadic simple group Fi₂₂. These groups are of the type type 2⁸:(U₄(2):2), (2⁹ : L₃(4)):2, (2⁹:L₃(4)):3 and 2⁶:(2⁵:S₆).

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Fischerclifford matrices and character tables of inertia groups of maximal subgroups of finite simple groups of extension typePrins, A.L. January 2011 (has links)
Philosophiae Doctor  PhD / The aim of this dissertation is to calculate character tables of group extensions. There are several well–developed methods for calculating the character tables of group extensions. In this dissertation we study the method developed by Bernd Fischer, the so–called Fischer–Clifford matrices method, which derives its fundamentals from the Clifford theory. We consider only extensions G of the normal subgroup K by the subgroup Q with the property that every irreducible character of K can be extended to an irreducible character of its inertia group in G, if K is abelian. This is indeed the case if G is a split extension, by a wellknown theorem of Mackey. A brief outline of the classical theory of characters pertinent to this study, is followed by a discussion on the calculation of the conjugacy classes of extension groups by the method of coset analysis. The Clifford theory which provide the basis for the theory of FischerClifford matrices is discussed in detail. Some of the properties of these FischerClifford matrices which make their calculation much easier are also given. As mentioned earlier we restrict ourselves to split extension groups G in which K is always elementary abelian. In this thesis we are concerned with the construction of the character tables of certain groups which are associated with Fi₂₂ and Sp₈ (2). Both of these groups have a maximal subgroup of the form 2⁷: Sp₆ (2) but they are not isomorphic to each other. In particular we are interested in the inertia groups of these maximal subgroups, which are split extensions. We use the technique of the FischerClifford matrices to construct the character tables of these inertia groups. These inertia groups of 2⁷ : Sp₆(2), the maximal subgroup of Fi₂₂, are 2⁷ : S₈, 2⁷ : Ο⁻₆(2) and 2⁷ : (2⁵ : S₆). The inertia group of 2⁷ : Sp₆(2), the affine subgroup of Sp₈(2), is 2⁷ : (2⁵ : S₆) which is not isomorphic to the group with the same form which was mentioned earlier.

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