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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| Formaat: | Thesis |
| Taal: | English |
| Gepubliceerd in: |
1981
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| _version_ | 1826311577752567808 |
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| author | Scott, EA |
| author2 | Higman, G |
| author_facet | Higman, G Scott, EA |
| author_sort | Scott, EA |
| collection | OXFORD |
| description | <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> |
| first_indexed | 2024-03-07T08:11:49Z |
| format | Thesis |
| id | oxford-uuid:e88535db-9d0c-4f67-8e89-da163c88747b |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T08:11:49Z |
| publishDate | 1981 |
| record_format | dspace |
| spelling | oxford-uuid:e88535db-9d0c-4f67-8e89-da163c88747b2023-11-28T10:06:21ZFinitely presented infinite simple groupsThesishttp://purl.org/coar/resource_type/c_db06uuid:e88535db-9d0c-4f67-8e89-da163c88747bEnglishHyrax Deposit1981Scott, EAHigman, G<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> |
| spellingShingle | Scott, EA Finitely presented infinite simple groups |
| title | Finitely presented infinite simple groups |
| title_full | Finitely presented infinite simple groups |
| title_fullStr | Finitely presented infinite simple groups |
| title_full_unstemmed | Finitely presented infinite simple groups |
| title_short | Finitely presented infinite simple groups |
| title_sort | finitely presented infinite simple groups |
| work_keys_str_mv | AT scottea finitelypresentedinfinitesimplegroups |