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conjugacy classes

How many groups with exactly two conjugacy classes?


mbhatia writes:

> How many groups with exactly two conjugacy classes?

It is a result of Higman, Neumann and Neumann that every torsion-free group embeds in such a group. So the number of such groups exceeds any cardinal number.

For finite groups, there is only one: the cyclic group of order 2.

If there are only two conjugacy classes, then it is easy to see that the group must be simple, since a normal subgroup is a union of conjugacy classes. Also, every non-identity element must have the same order, and this order must be a prime (otherwise the element would have non-trivial powers of smaller order). So the group is a simple p-group, therefore cyclic of order p. But these groups are abelian, so each element forms a singleton conjugacy class. Hence p=2 is the only possibility.

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