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 for irreducible representations of specific types

• Number of one-dimensional representations equals order of abelianization

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

Opposite facts over non-splitting fields

• Number of irreducible representations over complex numbers with rational character values need not equal number of conjugacy classes of rational elements

Similar facts under action of automorphism group

The key facts are:

• Application of Brauer's permutation lemma to group automorphism on conjugacy classes and irreducible representations
• Cyclic quotient of automorphism group by class-preserving automorphism group implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group: In particular, if the quotient of the automorphism group by the group of class-preserving automorphisms is cyclic, then the orbit sizes both for the set of conjugacy classes and for the set of irreducible representations are equal.
• Number of orbits of irreducible representations equals number of orbits of conjugacy classes under any subgroup of automorphism group
• Number of orbits of irreducible representations equals number of orbits under automorphism group

Opposite facts under action of automorphism group

• Orbit sizes for irreducible representations may differ from orbit sizes for conjugacy classes under action of automorphism group

Similar facts in modular representation theory

• Number of irreducible Brauer characters equals number of regular conjugacy classes for any fixed prime $p$.

Similar arithmetic fact

• Sum of squares of degrees of irreducible representations equals order of group

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] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	$q+1$	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] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	$q+2$	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] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	$(q+5)/2$	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