Re: The number of conjugacy classes of infinite group?


Gvnaena Pura wrote:
Thanks!

While going through old qualify exams, I saw a problem that ask the
students to classify all the groups that has 3 conjugacy classes. I
was able to classify all of them in finite case, but for infinite
groups, I don't even know where to start. Any hints will be
appreciated.

Well I would bet that whoever set that exam question was thinking of
finite groups, but forgot to say so!

I do not believe that it is possible to classify infinite groups with
3 conjugacy classes, because it is possible to construct all sorts of
examples.

The basic idea of the construction is the HNN extension: given any
group H with isomorphic subgroups H1 and H2, it is possible to
construct a larger group, containing G as a subgroup, in which H1 and
H2 are conjugate.
So, for example, by repeating this construction on all pairs of
elements of the same order, you can embed a group G1 in a group G2 in
which all elements of the same order in G1 are conjugate in G2.
Then by repeating this on G2, you can get an ascending chain of groups

G1 < G2 < G3 ...

and, letting G be the union of the groups in the chain, all elements
of the same order in G are conjugate.

So, for example, if you started with G1 a direct product of an
infinite cyclic group and a group of order 2, then G would have
exactly three conjugacy classes, containing the element sof orders 1,2
and infinity.

Derek Holt.
On Aug 16, 2:42 pm, Derek Holt <ma...@xxxxxxxxxxxxx> wrote:
I think that to get any useful answers you will need to ask more
specific questions.

Of course, most infinite groups that you are likely to encounter have
infinitely many conjugacy classes, but it is possible to construct
infinite groups with finitely many. In fact examples in which all non-
identity elements are conjugate can be constructed.

Derek Holt.

.

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