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Re: [Axiom-developer] group theory classification


From: Dylan Thurston
Subject: Re: [Axiom-developer] group theory classification
Date: Mon, 19 Jan 2004 23:21:39 -0500
User-agent: Mutt/1.5.4i

On Mon, Jan 19, 2004 at 05:32:59PM -0500, root wrote:
> Where do nilpotent groups of order 2 fit?

What is a nilpotent group of order 2?  Do you mean the length of the
lower central series is 2?  Why make a special case for order 2, and not
order N?

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

What's a free nilpotent group?  It sounds more special than a nilpotent
group, so I'm confused by your hierarchy.

This seems like a list of properties of groups, rather than
constructions of groups, so 'free' doesn't seem to quite belong.  But
maybe it does, since subgroups of free groups are automatically free.
But then why don't you have free abelian on this list?

Amalgamated and HNN groups are both best understood as special cases of
graphs of groups.  It would be nice to do the general case and unify
these two.

Aren't small cancellation groups necessarily automatic?

By the way, I object to putting finitely presented groups at the base of
the hierarchy (if I understand the diagram correctly).  There
interesting groups which are not finitely presented, which you can still
work with in practice.  Maybe it has a countable set of relations, or
maybe it's a subgroup of a finitely presented group, or maybe it's
infinitely generated.

Two infinitely generated groups that I know well is the infinite
derangement group (with only a finite number of objects displaced) and
the infinite braid group (likewise).  Both have solvable word problem
and (I think) conjugacy problem.  (The word problem, at least, is
quadratic time for the braid group.)

Several of the properties you list (like automatic) do specifically
refer to finitely presented groups, but others (like nilpotent) do not.

I think I must be misunderstanding something.  Please help clear up my
confusion!

> Among my notes I found the attached diagram:

I didn't understand the diagram.  What do the lines mean?  It doesn't
seem to be subtyping like in your earlier diagram.

> layer 1
>   FGA   finitely generated abelian 
> 
>           (+WP, +CP, +GWP, +IsoP)

What is GWP?  The others must relate to solvability of

WP = word problem
CP = conjugacy problem
IsoP = isomorphism problem

Peace,
        Dylan

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