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Topic: combinatorial group theroy
Replies: 5   Last Post: Feb 27, 2004 10:30 AM

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Posts: 401
Registered: 12/3/04
Re: combinatorial group theroy
Posted: Feb 26, 2004 2:00 PM
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> Two independent questions.
> 1) Does there exist an infinite, finitely presented, simple group G,
> and a two-transitive action of G?
> 2) Does there exist a residually finite, finitely generated group,
> satisfying a nontrivial relation, but not virtually solvable?
> Thanks in advance,
> --
> Yves de Cornulier

2). Yes. By the Neumann-Neumann proof of the Higman-Neumann-Neumann
embedding theorem, a finite p-group G can be embedded in a finite
2-generator group H, such that H is a subgroup of a repeated wreath
product GwrCwrD, with C and D cyclic. Then the 2nd derived subgroup H''
of H has exponent p. If p \ge 5, we may choose G to have arbitrarily
large derived length. Let F be a free group of rank 2, and let N be the
intersection of all normal subgroups K of F such that F/K is finite and
has 2nd derived subgroup of exponent p. By the above, F/N is infinite,
2-generated, and insoluble, but each finite factor group of F/N is
soluble, therefore F/N is not virtually soluble.

Avinoam Mann

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