Finitely presented infinite simple groups

<p>This thesis is mainly concerned with the development of a procedure to construct finitely presented infinite simple groups. </p> <p>The finitely presented groups, G<sub>n,i</sub>, constructed by G.Higman [5) are extended by certain subgroups, H, of the inverse lim...

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Detalles Bibliográficos
Autor Principal: Scott, EA
Outros autores: Higman, G
Formato: Thesis
Idioma:English
Publicado: 1981
Descripción
Summary:<p>This thesis is mainly concerned with the development of a procedure to construct finitely presented infinite simple groups. </p> <p>The finitely presented groups, G<sub>n,i</sub>, constructed by G.Higman [5) are extended by certain subgroups, H, of the inverse limit of wreath products, S<sub>n-1</sub>(wr s<sub>n-1</sub>)<sup>w</sup>. It is shown that if H is defined by a finite set of expansion rules (see Chapter 2) then the group <G<sub>n</sub>,1,H> is finitely presented if H is finitely presented. The commutator subgroup, <G<sub>n</sub>,1,H>, is shown to be a simple group and it is proved that an integer m and a set of expansion rules defining H can be chosen such that <Gm,1,H> is a finitely presented infinite simple group.</p> <p>Let GL(n,𝕫) be the general linear group of nxn dimensional matrices over the integers. The finitely presented simple group, <G<sub>m,1</sub>,𝕫<sup>n</sup>.GL(n,𝕫)<sup>1</sup>, is constructed, where 𝕫<sup>n</sup> is the free Abelian group of finite rank, n. So GL(n,𝕫) can be embedded in a finitely presented simple group, for all n. </p> <p>Let X<sub>p</sub> be the ring of rationals whose denominators are divisible only by primes belonging to the finite set P. If A is a countable Abelian group whose torsion factor group is a subgroup of a finitely generated torsion free X<sub>p</sub>-module, then it is shown that A can be embedded in a finitely presented simple group.</p> <p>It is also noted, at the end of the discussion of linear groups, that if there exists a finitely presented linear group, over the integers, with unsolvable conjugacy or orbit problem, then a finitely presented infinite simple group with unsolvable conjugacy problem can be constructed.</p>