# Number of conjugacy classes in a subgroup may be more than in the whole group

## Contents

## Statement

It is possible to have a finite group and a subgroup of such that the Number of conjugacy classes (?) in is more than in .

## Related facts

### Similar facts

- Commuting fraction in subgroup is at least as much as in whole group: This says that the number of conjugacy classes in the subgroup is
*at least*as much as the number of conjugacy classes in the whole group divided by the index of the subgroup.

### Opposite facts

## Proof

### Example of the dihedral group of degree five

`Further information: dihedral group:D10, cyclic group:Z5`

The smallest pair of examples is where the group is , the dihedral group of degree five and order ten, and is the subgroup is cyclic group:Z5. The group has four conjugacy classes: one of involutions, one identity element, and two conjugacy classes in the cyclic subgroup of order five. On the other hand, is an abelian group of order five hence has five conjugacy classes.

### Other dihedral examples

`Further information: element structure of dihedral groups`

More generally, for odd , the dihedral group of order and degree has conjugacy classes, and the cyclic subgroup of order has conjugacy classes. For , the cyclic subgroup has more conjugacy classes than the whole group.

For even , the dihedral group of order has conjugacy classes and the cyclic subgroup of order has conjugacy classes. For , the cyclic subgroup has more conjugacy classes than the whole group. The first example of this is cyclic group:Z8 in dihedral group:D16.