TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-17

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Frattini argument: if $N ⊴ G$ and $P$ is Sylow in $N$ , then $G = N N_{G} (P)$

Statement

If $N ⊴ G$ is finite and $P$ is a Sylow $p$ -subgroup of $N$ , then $G = N N_{G} (P) .$ See Sylow II: in a finite group every $p$ -subgroup lies in a conjugate of any Sylow $p$ -subgroup, and the Sylow $p$ -subgroups form a single conjugacy class.

Facts & Assumptions

Given: The hypotheses and objects in the Statement.

[L1]

Let $G$ be finite, let $P$ be a Sylow $p$ -subgroup, and let $H \leq G$ be a $p$ -subgroup. There is $g \in G$ with $H \leq g P g^{- 1}$ . In particular, for every Sylow $p$ -subgroup $Q$ there is $g \in G$ with $Q = g P g^{- 1}$ , so the Sylow $p$ -subgroups form one conjugacy class. (Sylow II: in a finite group every $p$ -subgroup lies in a conjugate of any Sylow $p$ -subgroup, and the Sylow $p$ -subgroups form a single conjugacy class).

[L2]

Let $H \leq G$ be a subgroup (def-subgroup). The normalizer of $H$ in $G$ is $N_{G} (H) : = {g \in G : g H g^{- 1} = H} .$ Thus $g \in N_{G} (H)$ exactly when the conjugation automorphism $c_{g}$ preserves $H$ setwise (thm-conjugation-is-an-automorphism). The subgroup property is proved in lem-centralizers-and-normalizers-are-subgroups. (The normalizer $N_{G} (H) = {g \in G : g H g^{- 1} = H}$ of a subgroup).

[L3]

If $H \leq G$ and $N ⊴ G$ , then $H N$ is a subgroup and $H \cap N ⊴ H$ . Here $H N : = {h n : h \in H, n \in N}$ . (If $H \leq G$ and $N ⊴ G$ , then $H N$ is a subgroup and $H \cap N ⊴ H$ ).

Proof

technique · direct

1.1

For each $g \in G$ , normality makes $g P g^{- 1}$ Sylow in $N$ .

L1L2L3givenalgebra

2.1

Sylow II inside $N$ supplies $n \in N$ conjugating it back to $P$ , so $n g \in N_{G} (P)$ and $g \in N N_{G} (P)$ .

step 1.1givenalgebra

3.1

The reverse containment is immediate. This proves the stated claim.

step 2.1givenalgebra∎

Depends on

Used by

Nilpotence lifts over the Frattini subgroup of a finite group Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 41 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

David A. Craven, Finite Group Theory, Sections 1.4 and 2.3 (standard reference, not scraped)

Frattini argument: if N⊴GN⊴G and PP is Sylow in NN, then G=NNG(P)G=NNG​(P)