[Axiom-developer] group theory classification

[root]

Gilbert, Chuck,

I've been looking at the classification scheme of finitely presented
and finitely generated groups so I can implement the proper category
hierarchy in Axiom. I'm now looking for references that will give me
the axioms which define each group. Any help would be appreciated.
I'd like to state the axioms that are added at each point in the
lattice.

The "finitely presented simple group (+WP)" is hanging out unclassified.

Where do nilpotent groups of order 2 fit?

There are a list of groups that need classification. From a discussion
with Gilbert I find a very bushy tree of the form:

layer 1
  FPG   finitely presented group

layer 2
  FN    free nilpotent
  HNN   HNN group
  OR    one relator 
  AUTO  automatic
  AMAL  amalgamated
  SC    small cancellation
  F     free

layer 3
  HYPER hyperbolic
  NIL   nilpotent

layer 4
  ABEL  abelian

4 ABEL
   |
   |
3 NIL                    HYPER
   |                       |
   |                       |
2 FN     HNN     OR      AUTO     AMAL     SC       F
   |      |       |        |        |       |       |
   |      |       |        |        |       |       |
   --------------------------------------------------
                              |
                              |
1                            FPG



Among my notes I found the attached diagram:

layer 1
  FGA   finitely generated abelian 
          (+WP, +CP, +GWP, +IsoP)
  FPRF  finitely presented residually free
          (+WP, ?CP, -GWP, ?IsoP)

layer 2
  FPM   finitely presented metabelian 
          (+WP, +CP, +GWP, ?IsoP)
  FGN   finitely generated nilpotent
          (+WP, +CP, +GWP, +IsoP)

layer 3
  FPSDL3 finitely presented solvable derived length 3
          (-WP, -CP, -GWP, -IsoP)
  FGM    finitely generated metabelian
          (+WP, +CP, +GWP, ?IsoP)
  FPRN   finitely presented residually nilpotent
          (+WP, -CP, -GWP, -IsoP)
  P      polycyclic
          (+WP, +CP, +GWP, +IsoP)
  A      arithmetic
          (+WP, +CP, -GWP, ?IsoP)

layer 4
  FGABN finitely generated abelian-by-nilpotent
          (+WP, ?CP, +GWP, ?IsoP)
  SA    S-arithmetic
          (+WP, +CP, -GWP, ?IsoP)
  FPS   finitely presented subgroups
          (+WP, ?CP, -GWP, ?IsoP)

layer 5
  FGABP finitely generated abelian-by-polycyclic
          (+WP, ?CP, ?GWP, ?IsoP)
  FGSGL finitely generated subgroups of GL(n,Z)
          (+WP, -CP, -GWP, -IsoP)
  FPRF  finitely presented residually finite
          (+WP, -CP, -GWP, -IsoP)

layer 6
  FGL   finitely generated linear
          (+WP, -CP, -GWP, -IsoP)
  FPH   finitely generated hopfian
          (-WP, -CP, -GWP, -IsoP)




6                           FGL                FPH
                             |                  |
                             |                  |
5                FGABP       |          FGS    FPRF
                   |         |           |       |
                   |         |           |       |
4                FGABN       |   SA     FPS      |
                   |         |    |     | |      |
                   |         |    |     | |      |
                   -----------    ------- |      |
                            |       |     |      |
                            |       |     |      |
3               FPDSL3     FGM      A     P     FPRN
                   |        |       |     |      | |
                   |        |       |     |      | |
                   ----------       -------------- |  
                        |                |         |
                        |                |         |
2                      FPM              FGN        |
                        |                |         |
                        ------------------         |
                            |                      |
                            |                      |
1                          FGA                    FPRF


Tim
address@hidden
address@hidden