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Symmetric Presentations, Representations, and Related TopicsManriquez, Adam 01 June 2018 (has links)
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.
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Sobre a influência dos centralizadores dos automorfismos de ordem dois em grupos de ordem ímpar / Centralizers of involutory automorphisms of groups of odd orderRojas, Yerko Contreras 05 July 2013 (has links)
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Previous issue date: 2013-07-05 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This document presents an approach and development of some of the results of
Shumyatsky in [14, 15, 16, 17, 18], where he worked with automorphisms of order two
in finite groups of odd order, mainly showing the influence that the structure of the
centralizer has on that of Group. Let G be a group with odd order, and ϕ an automorphism
on G, of order two, where G = [G,ϕ], and given a limitation in the order of the centralizer
of ϕ regard to G, CG(ϕ), which induces a limitation in the order of derived group G′ of
group G, and we also verified that G has a normal subgroup H that is ϕ-invariant, such
that H′ ≤ Gϕ and its index [G : H] is bounded with the initial limitation. With the same
hypothesis of the group G and with the same limitation of the order of the centralizer of
the automorphism, let V a abelian p-group such that G⟨ϕ⟩ act faithful and irreductible
on V, then there is a bounded constant k, limitated by a function depending only on the
parameter m, where m is tha limitation in the order of CG(ϕ), and elements x1, ...xk ∈ G−ϕ
such that V = ρϕx
1,...,xk(V−ϕ). / O trabalho baseia-se na apresentação e desenvolvimento de alguns resultados expostos
por Shumyatsky em [14, 15, 16, 17, 18], onde trabalha com automorfismos de ordem
dois em grupos de ordem ímpar, mostrando fundamentalmente a influência da estrutura
do centralizador do automorfismo na estrutura do grupo. Seja G um grupo de ordem
ímpar e ϕ um automorfismo de G, de ordem dois, tal que G = [G,ϕ], dada uma limitação
na ordem do centralizador de ϕ em G, CG(ϕ), a mesma induz uma limitação na ordem do
grupo derivado G′ do grupo G, além disso verificamos que G tem um subgrupo H normal
ϕ-invariante, tal que H′ ≤ Gϕ e o índice [G : H] é limitado dependendo da limitação
inicial de CG(ϕ). Nas mesmas hipóteses do grupo G e com a mesma limitação da ordem
do centralizador do automorfismo, seja V um p-grupo abeliano, tal que G⟨ϕ⟩ age fiel e
irredutivelmente sobre V, então existe uma constante k, limitada por uma função que
depende só da limitação de CG(ϕ), e elementos x1, ...xk ∈ G−ϕ, tal que V = ρϕx
1,...,xk(V−ϕ).
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