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Inducing the trivial representation gives the permutation representation on $G / H$

Statement

Let $G$ be a finite group and let $H \leq G$ . Inducing the trivial complex representation of $H$ to $G$ gives the permutation representation of $G$ on the left coset set $G / H$ .

Facts & Assumptions

Given: A finite group $G$ , a subgroup $H \leq G$ , and the trivial complex representation $1_{H}$ of $H$ .

[F1]

The induced module consists of the functions $f : G \to C$ satisfying $f (g h) = h^{- 1} \cdot f (g)$ , with $G$ acting by $(x \cdot f) (g) = f (x^{- 1} g)$ (The induced $R$ -linear $G$ -module ${Ind}_{H}^{G} W$ as $H$ -covariant functions on $G$ ).

[F2]

The left cosets of $H$ are the subsets $g H$ , and the permutation representation on a finite $G$ -set has basis vectors indexed by that set (Left and right cosets $g H$ and $H g$ of a subgroup, The trivial representation, the regular representation, and permutation representations from finite $G$ -sets).

Proof

technique · direct

1.1

In the trivial representation of $H$ , every $h \in H$ acts as the identity on $C$ . So the covariance condition of [F1] becomes $f (g h) = f (g)$ for all $g \in G$ and $h \in H$ . Therefore $f$ is constant on each left coset $g H$ .

F1given

2.1

Define $Φ : {Ind}_{H}^{G} 1_{H} ⟶ C^{(G / H)}$ by $Φ (f) (g H) : = f (g)$ . Step 1.1 makes this well defined, and every function on $G / H$ pulls back uniquely to an $H$ -covariant function on $G$ , so $Φ$ is a linear bijection.

F2step 1.1construct

3.1

For $x \in G$ , one has $Φ (x \cdot f) (g H) = (x \cdot f) (g) = f (x^{- 1} g) = Φ (f) (x^{- 1} g H)$ , which is exactly the left permutation action of $G$ on the coset set $G / H$ from [F2]. Hence $Φ$ is $G$ -equivariant.

F1F2step 2.1algebra

4.1

The bijection of step 2.1 and the equivariance of step 3.1 identify ${Ind}_{H}^{G} 1_{H}$ with the permutation representation of $G$ on $G / H$ .

step 2.1step 3.1∎

Depends on

Used by

Inducing the trivial character of a subgroup of order two in S₃ gives 1 plus an irreducible degree-two character Example

Dependency tree · two levels

12 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, Example 4.3.4 (standard reference, not scraped)
Pavel Etingof et al., Introduction to Representation Theory, Section 4.8 (standard reference, not scraped)

Inducing the trivial representation gives the permutation representation on G/HG/H