From: rusin@washington.math.niu.edu (Dave Rusin)
Newsgroups: sci.math.research
Subject: Infinite group with finite number of conjugacy classes?
Date: 12 Sep 1995 17:29:50 GMT
Remember this question?
In post <42n3fh$ij0@nef.ens.fr>, Philippe Crocy wrote:
>Is it possible to give an explicit upper bound to the order of a finite
>group, depending only on the number of conjugacy classes of this group?
>(that is, to give a function f such that the order of any finite group
>containing x conjugacy classes should be at most f(x) ?)
I posted an answer which used the assumption that the group is finite.
Our Fearless Moderator enquired if the condition that the group be
finite is in fact necessary. I thought about and got nowhere.
Is there an infinite group with only finitely many conjugacy classes?
There would be quite a few restrictions on the group, and in fact there
would be a simple infinite group with the same property, having
only bounded torsion, no subgroups of finite index, etc.
Let me be even bolder: can an infinite group have just 2 classes? I tried
to construct an example of a sequence Z < G1 < G2 < ... in which
all non-identity elements of G_i are conjugate in G_(i+1) (hoping
to be able to take a union) but this got kind of messy :-).
There are groups called FC-groups but they're the other extreme: each
conjugacy class is finite.
dave
==============================================================================
From: Albert Goodman
Date: Tue, 12 Sep 1995 20:38:36 -0500
To: rusin@washington.math.niu.edu (Dave Rusin)
Subject: Re: number of conjugacy classes in finite groups
Newsgroups: sci.math.research
In article <42qfmf$16f@watson.math.niu.edu> you wrote:
: In article <42n3fh$ij0@nef.ens.fr>,
: Philippe Crocy wrote:
: >Is it possible to give an explicit upper bound to the order of a finite
: >group, depending only on the number of conjugacy classes of this group?
: [...]
: I guess if Erdos and Turan are willing to give up when they
: get this far, then I should too. However, there are papers by Bertam (1974)
: and Sherman (1978) which get better mileage by discussing "almost all"
: groups, and nilpotent groups, respectively.
There's a fairly recent (I think 1992) paper by Pyber which I think gives
an even better answer (I forget the details right now but I could easily
look it up if you want).
-- Albert Goodman
==============================================================================
From: william@pdh.com (Andy Grosso)
Date: Wed, 13 Sep 95 08:35:36 -0700
To: rusin@math.niu.edu (Dave Rusin)
Subject: Re: What was the answer ?
Thanks for your reply.
About infinite groups: You can construct infinite groups
that have only finitely many conjugacy classes. However,
all of the constructions I come up with (as well as one
Stallings sent me by e-mail) are infinite presentations.
Which is sort of interesting. The extra bonus challenge
question is whether it can be done with a finite presentation....
Cheers,
Andy
==============================================================================
From: william@pdh.com (Andy Grosso)
Date: Wed, 13 Sep 95 10:46:31 -0700
To: rusin@math.niu.edu (Dave Rusin)
Subject: Re: What was the answer ?
Well, I'd send you my groups but they're messy
(I viewed the problem as an interesting combinatorial
question). Here's what Stallings sent me by e-mail:
>Some antique Higman Neumann Neumann type of stuff,
>I'd say. Just take the infinite cyclic group. HNN two
>elements together, and in the resulting group those two
>elements are conjugate, and all the elts have infinite order.
>HNN two more together, etc. Transfinitely. You can
>stop at the first uncountable ordinal, and all the elements
>but the identity will be conjugate. maybe you can stop
>earlier if you really try hard.
As I said, it's not a finite presentation :=)
My "presentations" also involved taking limits.
Cheers,
Andy
Begin forwarded message:
Date: Wed, 13 Sep 95 12:22:45 CDT
From: rusin@math.niu.edu (Dave Rusin)
To: william@pdh.com
Subject: Re: What was the answer ?
Well, I'd like to see the construction -- I have another respondent who
says no such group can exist.
Can you even have an infinite group with just two classes?
dave
==============================================================================
From: [Permission pending]
Date: Thu, 14 Sep 1995 14:14:06 -0600
To: rusin@washington.math.niu.edu
Subject: reply sci.math.research
>Is there an infinite group with only finitely many conjugacy classes?
>Let me be even bolder: can an infinite group have just 2 classes?
Start with any torsion-free nontrivial group G_0 (for
example, one could take G_0 to be an infinite cyclic group). Then construct
the tower of groups inductively G_iI tried
>to construct an example of a sequence Z < G1 < G2 < ... in which
>all non-identity elements of G_i are conjugate in G_(i+1) (hoping
>to be able to take a union) but this got kind of messy :-).
The construction I gave above is canonical. Thus a presentation for
G_{i+1} includes one for G_i, new generators t_{a,b} for all pairs
a,b \ne 1 in G_i, and new relations
t_{a,b}a=bt_{a,b}
for each such pair.
[sig deleted -- djr]
==============================================================================
From: mareg@csv.warwick.ac.uk (Dr D F Holt)
Newsgroups: sci.math.research
Subject: Re: Infinite group with finite number of conjugacy classes?
Date: 14 Sep 1995 10:56:45 +0100
In article <436clrINNm3v@PASCAL.MATH.YALE.EDU>,
kuperberg-greg@MATH.YALE.EDU (Greg Kuperberg) writes:
>In article <434g2e$k3r@watson.math.niu.edu> rusin@washington.math.niu.edu (Dave Rusin) writes:
>>I posted an answer which used the assumption that the group is finite.
>>Our Fearless Moderator enquired if the condition that the group be
>>finite is in fact necessary. I thought about and got nowhere.
>
>Actually, that was me, and I'm not the moderator on duty at the
>moment. You should check the "Approved" or "Originator" lines on
>recent postings to find out who is running the show.
>
(Sorry, I missed the original post, hence the double quote.)
Every torsion free group can be embedded in a group with only two
conjugacy classes (see, for example, Exercise 12.63, p. 338 of Rotman, "An
Introduction to the Theory of Groups", 3rd Edition).
The trick is to use the Higman-Neumann-Neumann (HNN) extension construction
repeatedly.
Let the given group be G_1 = {1,g_1,g_2, ..., g_n, ... }
By assumption, the g_i all have infinite order, so the are all isomorphic
and infinite cyclic. Thus we can form the (repeated) HNN extension
G_2 = < G_1, h_2, h_3, .... | h_i^-1 g_1 h_i = g_i (2 <= i) >.
Then G_1 is a subgroup of G_2, and all non-identity elements of G_1 are
conjugate in G_2. Furthermore, G_2 is torsion free (from the general theory of
HNN extensions).
So we can repeat the construction on G_2 and form the chain
G_1 < G_2 < G_3 < ...
in which all non-identity elements of G_i are conjugate in G_{i+1}.
So the union G of the G_i has the required property of having only
2 conjugacy classes (all non-identity elements are conjuagte).
Question: Is there a finitely generated infinite group having finitely
many conjugacy classes?
Derek Holt.
==============================================================================
From: mvs@unlinfo.unl.edu (Mark V. Sapir)
Newsgroups: sci.math.research
Subject: Re: Infinite group with finite number of conjugacy classes?
Date: 14 Sep 1995 21:36:10 GMT
mareg@csv.warwick.ac.uk (Dr D F Holt) writes:
>Question: Is there a finitely generated infinite group having finitely
>many conjugacy classes?
>Derek Holt.
Yes, see Theorem 41.2 in Olshansky, "Geometry of defining relations in groups":
for every sufficiently large prime p there exists a 2-generated infinite
group of exponent p which has exactly p conjugacy classes. This result was
proved by S. Ivanov.
Mark Sapir
==============================================================================
From: jmccarro@barrow.uwaterloo.ca (James McCarron)
Newsgroups: sci.math.research
Subject: Re: Infinite group with finite number of conjugacy classes?
Date: Fri, 15 Sep 1995 03:54:32 -0400
>
>Let me be even bolder: can an infinite group have just 2 classes? I tried
>to construct an example of a sequence Z < G1 < G2 < ... in which
>all non-identity elements of G_i are conjugate in G_(i+1) (hoping
>to be able to take a union) but this got kind of messy :-).
Well, it's not *too* messy if you use the construction of Higman, Neumann
and Neumann ("HNN extension"). In fact, (and I think the result is in the 1949
paper of HN&N), one can embed any countable group in a group with the property
that any two elements of equal order are conjugate. So, using that result,
you need only begin with your favourite (nontrivial) torsion free group,
and you get the example you asked for. The proof of the HN&N theorem
(at least, the proof I know) uses your idea of taking an ascending union.
At each step, you can use the HNN extension construction to embed G_k in
a group G_{k+1}, in which any two elements of G_k are conjugate provided only
that their orders are equal (it may be, however, that two elements of G_{k+1}
with infinite orders are not conjugate in G_{k+1}). After forming the union
of the chain {G_k} of groups, any two members with the same order belong
to some G_n, and they are then conjugate in G_{n+1}.
James
__
James McCarron
Department of Pure Mathematics // University of Waterloo
Waterloo, Ontario CANADA N2L 3G1
==============================================================================
From: Ron Ji
Newsgroups: sci.math.research
Subject: Re: Infinite group with finite number of conjugacy classes?
Date: 14 Sep 1995 21:22:48 GMT
It is well-known that any countable torsion free discrete group can be embedded
into a group of 2 generators which has only 2 conjugacy classes. This can be
found in textbooks containing HNN extensions in the content. Using HNN
extension
it is not hard to construct a infinte group of 2 conjugacy classes.
==============================================================================
From: mvs@unlinfo.unl.edu (Mark V. Sapir)
Newsgroups: sci.math.research
Subject: Re: Infinite group with finite number of conjugacy classes?
Date: 17 Sep 1995 12:54:20 GMT
Ron Ji writes:
>It is well-known that any countable torsion free discrete group can be embedded
>into a group of 2 generators which has only 2 conjugacy classes. This can be
>found in textbooks containing HNN extensions in the content. Using HNN
>extension
>it is not hard to construct a infinte group of 2 conjugacy classes.
You probably mixed two results here: every group is embeddable into
a 2-generated group and every torsion free group is embeddable into
a group with 2 conjugacy classes. No infinite groups with two generators and
two conjugacy classes are known. There were some announcements but there
are no proofs. Notice that a group with two conjugacy classes must be full
(every element has roots of arbitrary degrees). The first finitely
generated full group was constructed by Victor Guba in 1986. Before that it was
a well known open problem.
Mark Sapir
==============================================================================
From: mann@kineret.huji.ac.il (Avinoam Mann)
Newsgroups: sci.math.research
Subject: Re: Infinite group with finite number of conjugacy classes?
Date: 18 Sep 1995 01:04 IST
In article <438u8t$gkl@crocus.csv.warwick.ac.uk>, mareg@csv.warwick.ac.uk
(Dr D F Holt) writes...
>>In article <434g2e$k3r@watson.math.niu.edu> rusin@washington.math.niu.edu
(Dave Rusin) writes:
>>>I posted an answer which used the assumption that the group is finite.
>>>Our Fearless Moderator enquired if the condition that the group be
>>>finite is in fact necessary. I thought about and got nowhere.
>>>
(The original question was to give a bound for the order of a finite group
with a given number of conjugacy classes.)
>Every torsion free group can be embedded in a group with only two
>conjugacy classes (see, for example, Exercise 12.63, p. 338 of Rotman, "An
>Introduction to the Theory of Groups", 3rd Edition).
>
>The trick is to use the Higman-Neumann-Neumann (HNN) extension construction
>repeatedly.
>
>Let the given group be G_1 = {1,g_1,g_2, ..., g_n, ... }
>By assumption, the g_i all have infinite order, so the are all isomorphic
>and infinite cyclic. Thus we can form the (repeated) HNN extension
>
>G_2 = < G_1, h_2, h_3, .... | h_i^-1 g_1 h_i = g_i (2 <= i) >.
>
>Then G_1 is a subgroup of G_2, and all non-identity elements of G_1 are
>conjugate in G_2. Furthermore, G_2 is torsion free (from the general theory of
>HNN extensions).
>
>So we can repeat the construction on G_2 and form the chain
>G_1 < G_2 < G_3 < ...
>in which all non-identity elements of G_i are conjugate in G_{i+1}.
>
>So the union G of the G_i has the required property of having only
>2 conjugacy classes (all non-identity elements are conjuagte).
>
>
>Question: Is there a finitely generated infinite group having finitely
>many conjugacy classes?
>
>
>Derek Holt.
>
A Tarski monster is an infinite group all of whose proper subgroups have
prime order. Such monsters exist (see Olshanski's book), and among them some
have all proper subgroups of the same order, say p. Obviously a TM is generated
by two elements. I was told by E.Rips that they can be constructed to have all
proper subgroups conjugate. I'm not sure if we can have all non-identity
elements conjugate, but in any case a TM with all proper subgroups conjugate
has at most p conjugacy classes.
Avinoam Mann
==============================================================================
From: ab431@cleveland.Freenet.Edu (Bruce Ikenaga)
Newsgroups: sci.math.research
Subject: Re: Infinite group with finite number of conjugacy classes?
Date: 18 Sep 1995 15:44:00 GMT
In a previous article, rusin@washington.math.niu.edu (Dave Rusin) says:
>Remember this question?
>
>In post <42n3fh$ij0@nef.ens.fr>, Philippe Crocy wrote:
>>Is it possible to give an explicit upper bound to the order of a finite
>>group, depending only on the number of conjugacy classes of this group?
>>(that is, to give a function f such that the order of any finite group
>>containing x conjugacy classes should be at most f(x) ?)
>
>I posted an answer which used the assumption that the group is finite.
>Our Fearless Moderator enquired if the condition that the group be
>finite is in fact necessary. I thought about and got nowhere.
>
>Is there an infinite group with only finitely many conjugacy classes?
>There would be quite a few restrictions on the group, and in fact there
>would be a simple infinite group with the same property, having
>only bounded torsion, no subgroups of finite index, etc.
G. Higman, B. H. Neumann, and H. Neumann, Embedding theorems
for groups, J. London Math. Soc. 24(1949), 247--254.
This is the paper in which HNN extensions were first defined.
They used iterated HNN extensions to show that every torsion
free group embeds in a group in which all elements other than
the identity are conjugate. They also note that if G has torsion
and exactly two conjugacy classes, then G is Z_2.
V. S. Guba (A finitely generated complete group, Math. USSR
Izvestiya, 29(1987), 233--277) constructed a noncyclic two generator
group in which every element is conjugate to a power of the
first generator --- but I don't know whether this is a torsion
group (so that there would be only finitely many conjugacy
classes).
==============================================================================