ExampleConstruction: Literature-sourcedVerification: Literature-sourcedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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Artin induction is tautological for a cyclic group

Example

Let $G = C_{n}$ be cyclic, and let $x \in R_{Q} (C_{n})$ . Then Artin induction can be realized with the single cyclic subgroup $C_{n}$ itself:

$x = {Ind}_{C_{n}}^{C_{n}} x .$

For the trivial character when $n = 1$ , this specializes to $1_{C_{1}} = {Ind}_{C_{1}}^{C_{1}} 1_{C_{1}}$ .

Facts & Assumptions

Given: A cyclic group $C_{n} = ⟨ g ⟩$ and an element $x \in R_{Q} (C_{n})$ .

[F1]

Every rational character is a rational linear combination of characters induced from cyclic subgroups (Artin induction for rational characters).

[F2]

The notation $⟨ g ⟩ = C_{n}$ means that $g$ generates the whole group (The subgroup $⟨ S ⟩$ generated by a subset, the cyclic subgroup $⟨ g ⟩$ , and cyclic groups).

Verification

technique · direct

1.1

By [F2], the whole group $C_{n}$ is already a cyclic subgroup of itself. Induction from a subgroup to itself is the identity construction, so ${Ind}_{C_{n}}^{C_{n}} x = x$ . Thus the Artin expression can be taken to have one summand, namely the subgroup $C_{n}$ itself.

F2givenalgebra

2.1

This realizes the conclusion of [F1] in the most degenerate possible way: no proper cyclic subgroup is needed. If $n = 1$ and $x = 1_{C_{1}}$ , then step 1.1 gives the displayed trivial-character identity.

F1step 1.1algebra∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Kay Yang, Rational Valued Characters, Theorem 11 (standard reference, not scraped)
Tammo tom Dieck, Representation Theory, Section 4.5 (standard reference, not scraped)