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

Field	Corresponding notion to irreducible representation	Corresponding notion to conjugacy class	Statement
$\mathbb{Q}$ -- field of rational numbers	irreducible representation over rational numbers (need not be absolutely irreducible)	rational conjugacy class, or equivalently, equivalence class under the equivalence relation of generating conjugate cyclic subgroups	Number of irreducible representations over rationals equals number of rational conjugacy classes which in turn equals the number of conjugacy classes of cyclic subgroups.
$\mathbb{R}$ -- field of real numbers	irreducible representation over real numbers (need not be absolutely irreducible)	real conjugacy class, i.e., the union of a conjugacy class and the conjugacy class of its inverse elements	Number of irreducible representations over reals equals number of real conjugacy classes

Similar facts under action of automorphism group

• 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 number of equivalence classes of irreducible representations up to automorphisms of the group equals the number of equivalence classes of elements under group automorphisms.

Opposite facts under action of automorphism group

• Number of orbits of irreducible representations need not equal number of orbits of conjugacy classes under automorphism group

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	element structure of general linear group of degree two
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	element structure of special linear group of degree two
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	element structure of special linear group of degree two
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	element structure of projective general linear group of degree two

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Facts used

1. Splitting implies characters span class functions