# There are finitely many finite groups with bounded number of conjugacy classes

## Statement

### In finite terms

Let be any positive integer. Then, the following hold:

- There are only finitely many finite groups with exactly conjugacy classes of elements.
- There are only finitely many finite groups with
*at most*conjugacy classes of elements. - There exists a number dependent on such that any finite group of order more than has more than conjugacy classes.

### In limit terms

Denote, for a finite group , the number of conjugacy classes of by . Then:

## Facts used

- Size of conjugacy class equals index of centralizer (we use it to derive a primitive version of the class equation of a group)
- Number of Egyptian fraction representations of unity with bounded number of fractions is finite: For fixed , there are only finitely many solutions to:

where the are (not necessarily distinct) positive integers.

## Proof

We prove that there is a bound on the order of any finite group with exactly conjugacy classes.

For a finite group of order with conjugacy classes, let be the sizes of the centralizers of representatives of each of the conjugacy classes, arranged in descending order. By fact (1), the size of the conjuacy class is . Since the group is a union of its conjugacy classes, we get:

Canceling from both sides, we get:

Note that so we can recover from the values of the s. By fact (2), the number of positive integer solutions to the above system is finite. Taking the maximum of the possible values of among all these gives a finite upper bound on the order of , and hence a finite limit on the number of possible s that have conjugacy classes.

## Particular cases

Note that there are a number of additional necessary (but not sufficient) conditions for a decomposition:

to arise from a group, with . These include:

- All the s must divide , since by Lagrange's theorem, any centralizer is a subgroup and hence must divide the order of the group.
- The number of s that equal also divides (because this number is the order of the center).
- If the number of s that equal is , then all the s are multiples of . For a nontrivial group, all of them are
*strictly*bigger than .

Below are some examples. The list of Egyptian fraction representations is exhaustive for , but there are many other cases for that we have not listed here.

Number of conjugacy classes | Egyptian fraction representation of unity | Groups where this representation is realized |
---|---|---|

1 | 1 | trivial group |

2 | cyclic group:Z2 | |

3 | cyclic group:Z3 | |

3 | not realized for any group -- condition (3) violated. | |

3 | symmetric group:S3 | |

4 | cyclic group:Z4, Klein four-group | |

4 | not realized for any group -- condition (1) violated | |

4 | not realized for any group -- condition (3) violated | |

4 | not realized for any group -- condition (3) violated | |

4 | not realized for any group -- condition (3) violated | |

4 | dihedral group:D10 | |

4 | alternating group:A4 | |

4 | not realized for any group -- condition (3) violated |