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

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The local automizer condition gives centralizer conjugacy of Sylow subgroups

Statement

Let $H$ be a finite group, $p$ a prime, and $S \leq H$ a nontrivial $p$ -subgroup. Put $N = N_{H} (S)$ and $C = C_{H} (S)$ (The normalizer $N_{G} (H) = {g \in G : g H g^{- 1} = H}$ of a subgroup, The centralizer $C_{G} (H)$ of a subgroup). If $N / C$ is a $p$ -group, then any two Sylow $p$ -subgroups of $N$ are conjugate by an element of $C$ . In particular, for every $x \in S$ the centralizer $C_{N} (x)$ acts transitively on the Sylow $p$ -subgroups of $N$ containing $x$ .

Facts & Assumptions

Given: A finite group $H$ , a prime $p$ , a nontrivial $p$ -subgroup $S \leq H$ , and the hypothesis that $N_{H} (S) / C_{H} (S)$ is a $p$ -group.

[F1]

$C = C_{H} (S)$ is normal in $N = N_{H} (S)$ , so $N / C$ is a quotient group (The centralizer of a normal subgroup is normal, The centralizer $C_{G} (H)$ of a subgroup, The normalizer $N_{G} (H) = {g \in G : g H g^{- 1} = H}$ of a subgroup, Normal subgroup: invariance under conjugation).

[F2]

If $C ⊴ N$ and $N / C$ is a $p$ -group, then $N = T C = C T$ for every $T \in {Syl}_{p} (N)$ (Sylow times normal subgroup covers when the index is a p-power).

[F3]

Any two Sylow $p$ -subgroups of $N$ are conjugate in $N$ , and conjugation by a member of a subgroup fixes that subgroup (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, Sylow $p$ -subgroups of a finite group, Conjugation $x \mapsto g x g^{- 1}$ is an automorphism).

Proof

technique · direct

1.1

Let $T_{1}, T_{2} \in {Syl}_{p} (N)$ . By [F3] there is $n \in N$ with $T_{2} = n T_{1} n^{- 1}$ . Since $C ⊴ N$ by [F1] and $N / C$ is a $p$ -group by hypothesis, [F2] gives $N = C T_{1}$ ; write $n = c t$ with $c \in C$ and $t \in T_{1}$ .

F1F2F3given

2.1

Then $T_{2} = (c t) T_{1} (c t)^{- 1} = c (t T_{1} t^{- 1}) c^{- 1} = c T_{1} c^{- 1}$ , since $t \in T_{1}$ . Thus $C$ acts transitively on ${Syl}_{p} (N)$ .

F3step 1.1

3.1

Every element of $C = C_{H} (S)$ centralizes every $x \in S$ , and $C \leq N$ ; hence $C \leq C_{N} (x)$ for each $x \in S$ . The transitivity in step 2.1 therefore implies the claimed transitivity by $C_{N} (x)$ on the subcollection of Sylow $p$ -subgroups containing $x$ . ∎

F1step 2.1

Depends on

Used by

P automizer condition implies fusion control Lemma

Dependency tree · two levels

29 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

Paul Flavell, An Introduction to Transfer and Fusion in Finite Groups, §§2–5 (standard reference, not scraped)
Hans Kurzweil and Bernd Stellmacher, The Theory of Finite Groups, §§7.1–7.2 (standard reference, not scraped)