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

Statement

In finite terms

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

1. There are only finitely many finite groups with exactly $d$ conjugacy classes of elements.
2. There are only finitely many finite groups with at most $d$ conjugacy classes of elements.
3. There exists a number $n_{d}$ dependent on $d$ such that any finite group of order more than $n_{d}$ has more than $d$ conjugacy classes.

In limit terms

Denote, for a finite group $G$, the number of conjugacy classes of $G$ by $n(G)$. Then:

$\lim _{|G|\to \infty }n(G)=\infty$

Facts used

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

$\sum _{i=1}^{d}{\frac {1}{a_{i}}}=1$

where the $a_{i}$ are (not necessarily distinct) positive integers.

Proof

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

For a finite group $G$ of order $n$ with $d$ conjugacy classes, let $a_{1}\geq a_{2}\geq \dots \geq a_{d}$ 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 $i^{th}$ conjuacy class is $|G|/a_{i}$. Since the group is a union of its conjugacy classes, we get:

${\frac {|G|}{a_{1}}}+{\frac {|G|}{a_{2}}}+\dots +{\frac {|G|}{a_{d}}}=|G|$

Canceling $|G|$ from both sides, we get:

${\frac {1}{a_{1}}}+{\frac {1}{a_{2}}}+\dots +{\frac {1}{a_{d}}}=1$

Note that $a_{1}=|G|$ so we can recover $|G|$ from the values of the $a_{i}$s. By fact (2), the number of positive integer solutions $(a_{1},\dots ,a_{d})$ to the above system is finite. Taking the maximum of the possible values of $a_{1}$ among all these gives a finite upper bound on the order of $G$, and hence a finite limit on the number of possible $G$s that have $d$ conjugacy classes.

Particular cases

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

$\sum _{i=1}^{d}{\frac {1}{a_{i}}}=1$

to arise from a group, with $a_{1}\geq a_{2}\geq \dots a_{d}$. These include:

1. All the $a_{i}$s must divide $a_{1}$, since by Lagrange's theorem, any centralizer is a subgroup and hence must divide the order of the group.
2. The number of $a_{i}$s that equal $a_{1}$ also divides $a_{1}$ (because this number is the order of the center).
3. If the number of $a_{i}$s that equal $a_{1}$ is $\alpha$, then all the $a_{i}$s are multiples of $\alpha$. For a nontrivial group, all of them are strictly bigger than $\alpha$.

Below are some examples. The list of Egyptian fraction representations is exhaustive for $d=1,2,3$, but there are many other cases for $d=4$ that we have not listed here.

Number of conjugacy classes $d$	Egyptian fraction representation of unity	Groups where this representation is realized
1	1	trivial group
2	$1/2+1/2$	cyclic group:Z2
3	$1/3+1/3+1/3$	cyclic group:Z3
3	$1/4+1/4+1/2$	not realized for any group -- condition (3) violated.
3	$1/6+1/3+1/2$	symmetric group:S3
4	$1/4+1/4+1/4+1/4$	cyclic group:Z4, Klein four-group
4	$1/6+1/3+1/4+1/4$	not realized for any group -- condition (1) violated
4	$1/6+1/6+1/3+1/3$	not realized for any group -- condition (3) violated
4	$1/6+1/6+1/6+1/2$	not realized for any group -- condition (3) violated
4	$1/8+1/8+1/4+1/2$	not realized for any group -- condition (3) violated
4	$1/10+1/5+1/5+1/2$	dihedral group:D10
4	$1/12+1/4+1/3+1/3$	alternating group:A4
4	$1/12+1/12+1/3+1/2$	not realized for any group -- condition (3) violated