# 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

## Contents

- 1 Statement
- 2 Related facts
- 2.1 Similar facts for irreducible representations of specific types
- 2.2 Similar facts over non-splitting fields
- 2.3 Opposite facts over non-splitting fields
- 2.4 Similar facts under action of automorphism group
- 2.5 Opposite facts under action of automorphism group
- 2.6 Similar facts in modular representation theory
- 2.7 Similar arithmetic fact

- 3 Particular cases
- 4 Facts used

## Statement

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

- The Number of conjugacy classes (?) in .
- The number of Irreducible linear representation (?)s (up to equivalence) of over .

Note that any algebraically closed field whose characteristic does not divide the order of is a splitting field, so in particular, we can always take or .

## Related facts

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

### Similar facts for irreducible representations of specific types

### 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 is a finite group and is an integer relatively prime to the order of . Suppose is a field and is a splitting field of of the form where is a primitive root of unity, with also relatively prime to (in fact, we can arrange to divide the order of because sufficiently large implies splitting). Suppose there is a Galois automorphism of that sends to . Consider the following two permutations:

- The permutation on the set of conjugacy classes of , denoted , induced by the mapping .
- The permutation on the set of irreducible representations of over , denoted , induced by the Galois automorphism of that sends to .

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 |
---|---|---|---|---|---|

-- 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 and . |

-- 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 and . |

-- 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 |

-- 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

### 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

### Similar facts in modular representation theory

- Number of irreducible Brauer characters equals number of regular conjugacy classes for any fixed prime .