# Infinite groups with only 2 conjugacy classes

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. Apr 29, 2014 at 10:05
• Thanks but I already know that construction (since it is in the same paper of Higman, Neumann, Neumann) ;) May 8, 2014 at 23:00
• Ah, sorry, I haven't looked at that paper in ages! May 9, 2014 at 8:07

## 1 Answer

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.

• It was easy! Thanks! Apr 28, 2014 at 23:01