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

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A subgroup containing the normalizer of a Sylow subgroup is self-normalizing

Statement

Let $P$ be a Sylow $p$ -subgroup of a finite group $G$ . If $N_{G} (P) \leq H \leq G$ , then $N_{G} (H) = H$ . In particular, $N_{G} (N_{G} (P)) = N_{G} (P)$ . 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).

Proof

technique · direct

1.1

For $N_{G} (P) \leq H \leq G$ and $x \in N_{G} (H)$ , the groups $P$ and $x P x^{- 1}$ are Sylow in $H$ .

L1L2givenalgebra

2.1

Conjugate them inside $H$ ; the resulting element puts $x$ in $H$ .

step 1.1givenalgebra

3.1

Specialize to $H = N_{G} (P)$ to obtain $N_{G} (N_{G} (P)) = N_{G} (P)$ . This proves the stated claim.

step 2.1givenalgebra∎

Depends on

Used by

A finite group is nilpotent if and only if all Sylow subgroups are normal, if and only if it is their internal direct product Lemma

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 34 results over 12 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

Keith Conrad, Consequences of the Sylow Theorems, Sections 1-5 (standard reference, not scraped)