| 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>
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