TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-sol)audited 2026-09-27

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Frobenius kernel theorem

Statement

Let $G$ be a finite Frobenius group with complement $H$ and let $N = (G ∖ ⋃_{x \in G} x H x^{- 1}) \cup {1}$ be the associated kernel set. Then $N$ is a normal subgroup of $G$ .

Facts & Assumptions

Given: A finite group $G$ with Frobenius complement ${1} < H < G$ and the kernel set $N$ with its associated family $I$ of nontrivial irreducible characters of $H$ .

[F1]

$N = ⋂_{φ \in I} \ker \tilde{φ}$ for a nonempty family ${\tilde{φ} : φ \in I}$ of irreducible complex characters of $G$ (Frobenius kernel is an intersection of character kernels).

[F2]

If $ρ$ is a finite-dimensional complex representation of $G$ with character $χ$ , then $\ker χ = \ker ρ ⊴ G$ (The kernel of a complex character agrees with the kernel of any representation affording it).

[F3]

The intersection of a nonempty family of normal subgroups of $G$ is again a normal subgroup of $G$ (The intersection of a nonempty family of normal subgroups is normal).

Proof

technique · direct

1.1

By [F1], $N$ is the intersection over the nonempty family of kernels $\ker \tilde{φ}$ of irreducible characters $\tilde{φ}$ of $G$ .

F1given

2.1

Each $\ker \tilde{φ}$ occurring in [F1] is a normal subgroup of $G$ : it is the kernel of the representation $ρ$ affording the character $\tilde{φ}$ , hence equals $\ker ρ$ , which is normal by [F2].

F2step 1.1

3.1

Therefore $N$ is an intersection of a nonempty family of normal subgroups of $G$ , so by [F3] it is a subgroup of $G$ and is normal in $G$ . In particular the kernel set is closed under products and inverses, a fact that the counting argument of Frobenius kernel cardinality could not supply. ∎

F1F3step 1.1step 2.1

Depends on

Used by

Frobenius semidirect product decomposition Corollary
Frobenius groups and fixed point free actions Proposition
Frobenius kernel closure is the content of the theorem Remark

Dependency tree · two levels

14 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

Alex Bartel, Introduction to Representation Theory of Finite Groups, §6.1 (standard reference, not scraped)