> > 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.