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[Axiom-developer] group theory classification
From: |
root |
Subject: |
[Axiom-developer] group theory classification |
Date: |
Mon, 19 Jan 2004 17:32:59 -0500 |
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
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3 NIL HYPER
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2 FN HNN OR AUTO AMAL SC F
| | | | | | |
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--------------------------------------------------
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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
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5 FGABP | FGS FPRF
| | | |
| | | |
4 FGABN | SA FPS |
| | | | | |
| | | | | |
----------- ------- | |
| | | |
| | | |
3 FPDSL3 FGM A P FPRN
| | | | | |
| | | | | |
---------- -------------- |
| | |
| | |
2 FPM FGN |
| | |
------------------ |
| |
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1 FGA FPRF
Tim
address@hidden
address@hidden
- [Axiom-developer] group theory classification,
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