Re: 2 questions on finitely generated solvable groups

From: A. Caranti (caranti_at_science.unitn.it)
Date: 10/21/04 

Date: 21 Oct 2004 08:15:01 -0400

It was 07:45 of Mon, 18 Oct 2004 when Yves de Cornulier wrote:

>Can a solvable, finitely generated group act 2-transitively on an
>infinite set X?

There is a construction of Philip Hall of a finitely generated soluble
group G of derived length 3 which is not residually finite. This group
has a maximal subgroup H of infinite index, so the action of G on the
cosets of H is primitive - you might try and check whether it is
actually 2-transitive.

The group was constructed in

@article {MR0110750,
    AUTHOR = {Hall, P.},
     TITLE = {On the finiteness of certain soluble groups},
   JOURNAL = {Proc. London Math. Soc. (3)},
    VOLUME = {9},
      YEAR = {1959},
     PAGES = {595--622},
   MRCLASS = {20.00},
  MRNUMBER = {MR0110750 (22 \#1618)},
MRREVIEWER = {K. Gruenberg},
}

This article can also be found in

@book {MR986732,
    AUTHOR = {Hall, Philip},
     TITLE = {The collected works of {P}hilip {H}all},
    SERIES = {Oxford Science Publications},
      NOTE = {Compiled and with a preface by K. W. Gruenberg and J. E.
              Roseblade,
              With an obituary by Roseblade},
 PUBLISHER = {The Clarendon Press Oxford University Press},
   ADDRESS = {New York},
      YEAR = {1988},
     PAGES = {xii+776},
      ISBN = {0-19-853254-7},
   MRCLASS = {01A75 (20-06)},
  MRNUMBER = {MR986732 (90b:01108)},
MRREVIEWER = {Annabelle McIver},
}

The group is also described in exercise 15.4.6 in

@book {MR648604,
    AUTHOR = {Robinson, Derek John Scott},
     TITLE = {A course in the theory of groups},
    SERIES = {Graduate Texts in Mathematics},
    VOLUME = {80},
 PUBLISHER = {Springer-Verlag},
   ADDRESS = {New York},
      YEAR = {1982},
     PAGES = {xvii+481},
      ISBN = {0-387-90600-2},
   MRCLASS = {20-01},
  MRNUMBER = {MR648604 (84k:20001)},
}

I suspected that Hall's group G could be a possible example, but it's
Francesco de Giovanni who pointed out to me that the exercise in
Robinson's book states that G has indeed a maximal subgroup H of
infinite index.

I am curious to know whether the action of G on the cosets of H is
indeed 2-transitive; I may give it a try myself, time permitting.

Andreas