conjugacy classes

## Primary tabs

# conjugacy classes

Submitted by mbhatia on Fri, 09/10/2004 - 19:38

Forums:

How many groups with exactly two conjugacy classes?

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections

## Re: 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.

## Re: conjugacy classes

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.