Infinite groups with only 2 conjugacy classes

Question

In "Embedding Theorems for Groups", Higman, Neumann, Neumann state that "the only group containing elements of finite order, with only two classes of conjugacte elements is the cyclic groups of order two".

In the finite case this is easy. In the infinite case I was just able to prove that such a group must be a $2$-group. Of course it is also simple. What I miss?

You might be interested in this old question. It constructs an infinite group with precisely two conjugacy classes. — user1729, Apr 29 '14 at 10:05
Thanks but I already know that construction (since it is in the same paper of Higman, Neumann, Neumann) ;) — W4cc0, May 08 '14 at 23:00

score 3 · Accepted Answer · answered Apr 28 '14 at 22:50

3

As you noted, any such group is a 2-group. Since all non-identity elements are conjugate, they must all have order 2, and so the group is abelian.

answered Apr 28 '14 at 22:50

Cardboard Box

4,997

It was easy! Thanks! – W4cc0 Apr 28 '14 at 23:01

Infinite groups with only 2 conjugacy classes

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