Number of irreducible representations equals number of conjugacy classes

This article gives a proof/explanation of the equivalence of multiple definitions for the term number of conjugacy classes
View a complete list of pages giving proofs of equivalence of definitions

Statement

Consider a finite group $G$ and a splitting field $K$ for $G$. Then, the following two numbers are equal:

1. The Number of conjugacy classes (?) in $G$.
2. The number of Irreducible linear representation (?)s (up to equivalence) of $G$ over $K$.

Note that any algebraically closed field whose characteristic does not divide the order of $G$ is a splitting field, so in particular, we can always take $K = \mathbb{C}$ or $K = \overline{\mathbb{Q}}$.

Related facts

For more facts about the degrees of irreducible representations, see degrees of irreducible representations.

Similar facts over non-splitting fields

The key starting fact is this:

Application of Brauer's permutation lemma to Galois automorphism on conjugacy classes and irreducible representations (follows in turn from Brauer's permutation lemma): Suppose $G$ is a finite group and $r$ is an integer relatively prime to the order of $G$. Suppose $K$ is a field and $L$ is a splitting field of $G$ of the form $K(\zeta)$ where $\zeta$ is a primitive $d^{th}$ root of unity, with $d$ also relatively prime to $r$ (in fact, we can arrange $d$ to divide the order of $G$ because sufficiently large implies splitting). Suppose there is a Galois automorphism of $L/K$ that sends $\zeta$ to $\zeta^r$. Consider the following two permutations:

• The permutation on the set of conjugacy classes of $G$, denoted $C(G)$, induced by the mapping $g \mapsto g^r$.
• The permutation on the set of irreducible representations of $G$ over $L$, denoted $I(G)$, induced by the Galois automorphism of $L$ that sends $\zeta$ to $\zeta^r$.

Then, these two permutations have the same cycle type. In particular, they have the same number of cycles, and the same number of fixed points, as each other.

Using this fact, we can deduce various corollaries:

Field Applicable groups Corresponding notion to irreducible representation Corresponding notion to conjugacy class Statement Roughly how it is proved
$\mathbb{R}$ -- field of real numbers any finite group irreducible representation over real numbers (need not be absolutely irreducible) equivalence class under real conjugacy, i.e., union of a conjugacy class and the conjugacy class of its inverse elements Number of irreducible representations over reals equals number of equivalence classes under real conjugacy use above application of Brauer's permutation lemma and look at the number of cycles for the permutations on $C(G)$ and $I(G)$.
$\mathbb{R}$ -- field of real numbers any finite group irreducible representation over complex numbers taking real character values conjugacy class of real elements, i.e., a conjugacy class of elements that are conjugate to their inverses Number of irreducible representations over complex numbers with real character values equals number of conjugacy classes of real elements use above application of Brauer's permutation lemma and look at the number of fixed points for the permutations of $C(G)$ and $I(G)$.
$\mathbb{Q}$ -- field of rational numbers any finite group irreducible representation over rational numbers equivalence class under rational conjugacy, i.e., relation where two elements that generate conjugate cyclic subgroups are considered equivalent Number of irreducible representations over rationals equals number of equivalence classes under rational conjugacy Combine Brauer's permutation lemma with the orbit-counting theorem (Burnside's lemma), in the following form: the character of permutation representation determines number of orbits
$\mathbb{Q}$ -- field of rational numbers any finite group whose splitting field is a cyclic extension of the rationals. This includes any odd-order p-group irreducible representation over complex numbers with real-valued characters conjugacy class of rational elements number of irreducible representations over complex numbers with rational character values equals number of conjugacy classes of rational elements for any finite group whose cyclotomic splitting field is a cyclic extension of the rationals application of Brauer's permutation lemma

Similar facts under action of automorphism group

The key facts are:

Particular cases

Families

Family Order of group Degrees of irreducible representations, indexing set for them Conjugacy class sizes, indexing set for them Number of conjugacy classes = number of irreducible representations More information on linear representations More information on conjugacy classes
finite abelian group of order $n$ $n$ 1 ($n$ times) 1 ($n$ times) $n$
dihedral group of even degree $n$ $2n$ 1 (4 times), 2 ($(n - 2)/2$ times) 1 (2 times), 2 ($(n - 2)/2$ times), $n/2$ (2 times) $(n + 6)/2$ linear representation theory of dihedral groups element structure of dihedral groups
dihedral group of odd degree $n$ $2n$ 1 (2 times), 2 ($(n - 1)/2$ times) 1 (1 time), 2 ($(n - 1)/2$ times), $n$ (1 time) $(n + 3)/2$ linear representation theory of dihedral groups element structure of dihedral groups
symmetric group of degree $n$ $n!$ indexed by partitions (see linear representation theory of symmetric groups), described in terms of Young diagram for a partition indexed by partitions, via cycle type (see cycle type determines conjugacy class) $p(n)$ the number of unordered integer partitions of $n$ linear representation theory of symmetric groups element structure of symmetric groups
general linear group of degree two over field of size $q$ $q(q+1)(q-1)^2$ 1 ($q - 1$ times), $q$ ($q - 1$ times), $q + 1$ ($(q-1)(q-2)/2$ times), $q - 1$ ($q(q-1)/2$ times) 1 ($q - 1$ times), $q(q - 1)$ ($q(q-1)/2$ times), $q(q+1)$ ($(q - 1)(q - 2)/2$ times), $q^2 - 1$ ($q - 1$ times) $q^2 - 1$ linear representation theory of general linear group of degree two over a finite field element structure of general linear group of degree two over a finite field
special linear group of degree two over field of size $q$, $q$ odd $q(q+1)(q-1)$  ? 1 (2 times), $(q^2 - 1)/2$ (4 times), $q(q-1)$ ($(q - 1)/2$ times), $q(q + 1)$ ($(q - 3)/2$ times) $q + 4$ linear representation theory of special linear group of degree two over a finite field element structure of special linear group of degree two over a finite field
special linear group of degree two over field of size $q$, $q$ a power of 2 $q(q+1)(q-1)$  ? PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] $q + 1$ linear representation theory of special linear group of degree two over a finite field element structure of special linear group of degree two over a finite field
projective general linear group of degree two over field of size $q$, $q$ odd $q(q+1)(q-1)$  ? PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] $q + 2$ linear representation theory of projective general linear group of degree two over a finite field element structure of projective general linear group of degree two over a finite field
projective special linear group of degree two over field of size $q$, $q$ odd $q(q+1)(q-1)/2$  ? PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] $(q + 5)/2$ linear representation theory of projective special linear group of degree two over a finite field element structure of projective special linear group of degree two over a finite field

Facts used

1. Splitting implies characters span class functions