CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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Every irreducible representation of a finite group has degree at most $∣ G ∣$

Statement

Let $G$ be a finite group and let $V$ be an irreducible representation of $G$ over a field $k$ . Then $\deg (V) \leq ∣ G ∣ .$

Facts & Assumptions

Given: A finite group $G$ , a field $k$ , and an irreducible representation $V$ of $G$ over $k$ .

[L1]

The group algebra $k [G]$ has dimension $∣ G ∣$ over $k$ (If $G$ is finite then $\dim_{k} k [G] = ∣ G ∣$ ).

[L2]

The representation $V$ is a quotient of the regular representation (Every irreducible representation of a finite group is a quotient of the regular representation).

[L3]

In a vector space with a spanning set of size $n$ , every linearly independent set has size at most $n$ (If $V$ has a spanning set with $n$ elements, then every linearly independent subset of $V$ is finite with at most $n$ elements; in particular $V$ has no linearly independent subset equinumerous with $N$ ).

[L4]

The degree of a representation is the dimension of its underlying vector space, and that dimension is the size of any basis (A finite-dimensional representation $ρ : G \to GL (V)$ over a field, and its degree, Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis).

Proof

technique · direct

1.1L2given

By [L2], there is a surjective linear map $q : k [G] \to V$ .

2.1step 1.1L1

The images under $q$ of the $∣ G ∣$ basis vectors of $k [G]$ span $V$ , so $V$ has a spanning set with $∣ G ∣$ elements by [L1].

3.1step 2.1L3L4∎

Let $B$ be a basis of $V$ . Then $B$ is linearly independent, and step 2.1 together with [L3] shows that $∣ B ∣ \leq ∣ G ∣$ . By [L4], $∣ B ∣ = \deg (V)$ . Therefore $\deg (V) \leq ∣ G ∣$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

27 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Peter Webb, A Course in Finite Group Representation Theory, Theorem 2.1.1 and regular-module consequences (standard reference, not scraped)

Every irreducible representation of a finite group has degree at most ∣G∣∣G∣