Degree of irreducible representation divides order of group
This article gives the statement, and possibly proof, of a constraint on numerical invariants that can be associated with a finite group
This article states a result of the form that one natural number divides another. Specifically, the (degree of a linear representation) of a/an/the (irreducible linear representation) divides the (order) of a/an/the (group).
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This fact is related to: linear representation theory
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Contents
Statement
Let be a finite group and
an irreducible representation of
over an algebraically closed field of characteristic zero (or, more generally, over any splitting field of characteristic zero for
). Then, the degree of
divides the order of
.
Related facts
Other facts about degrees of irreducible representations
Further information: Degrees of irreducible representations
- Degree of irreducible representation divides index of center
- Degree of irreducible representation divides index of abelian normal subgroup
- Order of inner automorphism group bounds square of degree of irreducible representation
- Number of irreducible representations equals number of conjugacy classes
- Sum of squares of degrees of irreducible representations equals order of group
Similar fact about irreducible projective representations
See degree of irreducible projective representation divides order of group
Breakdown for a field that is not algebraically closed
Let be the cyclic group of order three and
be the field. Then, there are two irreducible representations of
over
: the trivial representation, and a two-dimensional representation given by the action by rotation by multiples of
. The two-dimensional representation has degree
, and this does not divide the order of the group, which is
.
We still have the following results:
- Degree of irreducible representation over reals divides twice the group order
- Degree of irreducible representation over any field divides product of order and Euler totient function of exponent
- Degree of irreducible representation of nontrivial finite group is strictly less than order of group
- Maximum degree of irreducible real representation is at most twice maximum degree of irreducible complex representation
Facts used
The table below lists key facts used directly and explicitly in the proof. Fact numbers as used in the table may be referenced in the proof. This table need not list facts used indirectly, i.e., facts that are used to prove these facts, and it need not list facts used implicitly through assumptions embedded in the choice of terminology and language.
Fact no. | Statement | Steps in the proof where it is used | Qualitative description of how it is used | What does it rely on? | Other applications |
---|---|---|---|---|---|
1 | Character orthogonality theorem: The part relevant for us is: for an irreducible representation over a splitting field of characteristic zero with character ![]() ![]() |
Step (1) | Equation setup that we then tinker with. | click here | |
2 | Size-degree-weighted characters are algebraic integers: This states that for an irreducible linear representation ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step (3) | Show certain parts of an expression are algebraic integers. | algebraic number theory + linear representation theory | click here |
3 | Characters are algebraic integers | Step (4) | Show certain parts of an expression are algebraic integers. | basic linear representation theory | click here |
Proof
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Given: A finite group , an irreducible linear representation
of
over a splitting field of characteristic zero for
, with character
and degree
. Note that
equals
, i.e., the value of
at the identity element of
.
To prove: divides the order of
.
Proof:
Step no. | Assertion/construction | Facts used | Given data used | Previous steps used | Explanation |
---|---|---|---|---|---|
1 | The following holds: ![]() ![]() ![]() ![]() ![]() ![]() |
Fact (1) | ![]() ![]() |
Follows from fact (1). The ![]() ![]() ![]() | |
2 | ![]() |
Step (1) | Divide both sides of step (1) by ![]() | ||
3 | Each ![]() ![]() |
Fact (2) | ![]() ![]() |
||
4 | Each ![]() ![]() |
Fact (3) | ![]() |
The complex conjugate of an algebraic integer is also an algebraic integer. | |
5 | ![]() |
Steps (3), (4) | The set of algebraic integers forms a ring, so a finite sum of products of algebraic integers is an algebraic integer. | ||
6 | ![]() |
Steps (2), (5) | By Step (5), the left side of Step (2) is an algebraic integer, hence so is the right side. | ||
7 | ![]() ![]() ![]() |
Step (6) | Both ![]() ![]() |