LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-27

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P automizer condition implies fusion control

Statement

Let $G$ be a finite group, $p$ a prime and $P \in {Syl}_{p} (G)$ a Sylow $p$ -subgroup (Sylow $p$ -subgroups of a finite group). Suppose that for every subgroup $Q$ with $1 \neq Q \leq P$ the quotient

$N_{G} (Q) / C_{G} (Q)$

of the normalizer by the centralizer of $Q$ (The normalizer $N_{G} (H) = {g \in G : g H g^{- 1} = H}$ of a subgroup, The centralizer $C_{G} (H)$ of a subgroup, The quotient group $G / N$ and coset product $(g N) (h N) = g h N$ ) is a $p$ -group (A finite $p$ -group has order $p^{n}$ for a prime $p$ and some $n \in N$ ), where $C_{G} (Q) ⊴ N_{G} (Q)$ by The centralizer of a normal subgroup is normal. Then $P$ controls fusion in $P$ with respect to $G$ (Control of fusion in a sylow p subgroup): whenever $x, y \in P$ and $y = x^{g} = g x g^{- 1}$ for some $g \in G$ , there is $v \in P$ with $y = x^{v}$ .

Facts & Assumptions

Given: A finite group $G$ , a prime $p$ , a Sylow $p$ -subgroup $P \leq G$ , and the hypothesis that $N_{G} (Q) / C_{G} (Q)$ is a $p$ -group for every subgroup $Q$ with $1 \neq Q \leq P$ .

[F1]

The hypothesis is conjugation invariant: for a subgroup $R \leq G$ and $g \in G$ one has $N_{G} (R^{g}) = N_{G} (R)^{g}$ and $C_{G} (R^{g}) = C_{G} (R)^{g}$ , and conjugation by $g$ induces an isomorphism $N_{G} (R) / C_{G} (R) \to N_{G} (R^{g}) / C_{G} (R^{g})$ . Since every nontrivial $p$ -subgroup of $G$ is conjugate into $P$ (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), the hypothesis therefore holds for every nontrivial $p$ -subgroup $R \leq G$ (Conjugation $x \mapsto g x g^{- 1}$ is an automorphism, First isomorphism theorem for groups: $G / \ker f ≅ im f$ , Group isomorphisms, automorphisms and the set $Aut (G)$ , The normalizer $N_{G} (H) = {g \in G : g H g^{- 1} = H}$ of a subgroup, The centralizer $C_{G} (H)$ of a subgroup).

[F2]

If $R \leq G$ is a subgroup then $C_{G} (R) ⊴ N_{G} (R)$ , so $N_{G} (R) / C_{G} (R)$ is a quotient group of $N_{G} (R)$ (The centralizer of a normal subgroup is normal, Normal subgroup: invariance under conjugation, The quotient group $G / N$ and coset product $(g N) (h N) = g h N$ ).

[F3]

If $S$ is a nontrivial $p$ -subgroup of a finite group $H$ and $N_{H} (S) / C_{H} (S)$ is a $p$ -group, then $C_{H} (S)$ acts transitively on the Sylow $p$ -subgroups of $N_{H} (S)$ ; in particular, for each $x \in S$ , $C_{N_{H} (S)} (x)$ is transitive on those Sylow subgroups containing $x$ (The local automizer condition gives centralizer conjugacy of Sylow subgroups).

[F4]

Local Sylow conjugacy ascent: if for every nontrivial $p$ -subgroup $S \leq P$ and every $x \in S ∖ {e}$ the centralizer $C_{N_{G} (S)} (x)$ acts transitively on the Sylow $p$ -subgroups of $N_{G} (S)$ containing $x$ , then $N_{G} (P)$ controls fusion in $P$ with respect to $G$ (Local sylow conjugacy ascent for fusion, Control of fusion in a sylow p subgroup).

[F5]

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

[F6]

If $H$ is a $p$ -group then every subgroup of $H$ is a $p$ -group and $∣ H ∣$ is a power of $p$ ; the image of a $p$ -group under a homomorphism is a $p$ -group, and every subgroup of a $p$ -group is a $p$ -group (A finite $p$ -group has order $p^{n}$ for a prime $p$ and some $n \in N$ , Every subgroup of a finite $p$ -group has order a power of $p$ , The image of a group homomorphism is a subgroup and its kernel is a normal subgroup, If $[G : N]$ is finite then $∣ G / N ∣ = [G : N]$ ; for finite $G$ this equals $∣ G ∣ / ∣ N ∣$ , Lagrange's theorem: $∣ G ∣ = [G : H] ∣ H ∣$ for every subgroup $H$ of a finite group $G$ ).

[F7]

If $P \in {Syl}_{p} (G)$ then $P \leq N_{G} (P)$ , $P ⊴ N_{G} (P)$ and $P \in {Syl}_{p} (N_{G} (P))$ (The normalizer $N_{G} (H) = {g \in G : g H g^{- 1} = H}$ of a subgroup, Sylow $p$ -subgroups of a finite group, Lagrange's theorem: $∣ G ∣ = [G : H] ∣ H ∣$ for every subgroup $H$ of a finite group $G$ ).

[F8]

Conjugation laws and subgroups: $x^{g} = g x g^{- 1}$ , $(x^{a})^{b} = x^{b a}$ , equivalently $x^{a b} = (x^{b})^{a}$ ; $C_{G} (x)$ , $C_{G} (R)$ and all normalizers are subgroups, and $c \in C_{G} (R)$ centralizes every element of $R$ (Conjugation $x \mapsto g x g^{- 1}$ is an automorphism, In a group $e^{- 1} = e$ , $(g^{- 1})^{- 1} = g$ and $(g h)^{- 1} = h^{- 1} g^{- 1}$ , the order of the last product being essential, The conjugacy class ${Cl}_{G} (x)$ and centralizer $C_{G} (x)$ of an element, $C_{G} (x)$ and $N_{G} (H)$ are subgroups of $G$ , Subgroup).

Proof

technique · direct

1.1

The hypothesis holds for every nontrivial $p$ -subgroup of $G$ : if $R \leq G$ is a nontrivial $p$ -subgroup, [F1] provides $g \in G$ with $R^{g} \leq P$ , so $N_{G} (R^{g}) / C_{G} (R^{g})$ is a $p$ -group and, by [F1], $N_{G} (R) / C_{G} (R)$ is isomorphic to it, hence is a $p$ -group.

F1F6

1.2

If $P = {1}$ then the only element of $P$ is $e = e^{e}$ , so $P$ controls fusion in $P$ trivially. Assume now $P \neq {1}$ ; then the hypothesis applies to the nontrivial subgroup $Q : = P \leq P$ , so $N_{G} (P) / C_{G} (P)$ is a $p$ -group, while $C_{G} (P) ⊴ N_{G} (P)$ by [F2] and $P \in {Syl}_{p} (N_{G} (P))$ by [F7]; hence [F5] applies with $N = N_{G} (P)$ , $C = C_{G} (P)$ and the Sylow $p$ -subgroup $P$ , giving $N_{G} (P) = P C_{G} (P)$ .

F2F5F7given

1.3

Let $S \leq P$ be a nontrivial $p$ -subgroup, put $N : = N_{G} (S)$ and $C : = C_{G} (S)$ , and let $x \in S ∖ {e}$ . The hypothesis directly gives that $N / C$ is a $p$ -group, so [F3] applies to the subgroup $S \leq G$ and gives that $C_{N} (x)$ acts transitively on the Sylow $p$ -subgroups of $N$ containing $x$ .

F2F3given

2.1

Since $S \leq P$ and $x \in S ∖ {e}$ were arbitrary, step 1.3 verifies the local centralizer-transitivity hypothesis of [F4]. Thus $N_{G} (P)$ controls fusion in $P$ with respect to $G$ .

F4step 1.3

3.1

Equivalently, for every pair $a, b \in P$ with $b = a^{g}$ for some $g \in G$ , step 2.1 supplies an element $u \in N_{G} (P)$ such that $b = a^{u}$ .

F4step 2.1

4.1

Finally let $x, y \in P$ and $g \in G$ with $y = x^{g}$ . By step 3.1 there is $u \in N_{G} (P)$ with $y = x^{u}$ ; by step 1.2 write $u = v c$ with $v \in P$ and $c \in C_{G} (P)$ . Then, by the conjugation law of [F8], $y = x^{v c} = (x^{c})^{v} = x^{v}$ , because $c$ centralizes $x$ by [F8] and $v \in P$ . Hence $P$ controls fusion in $P$ with respect to $G$ . In this last step the hypothesis at $Q = P$ , not only at the smaller subgroups, is what makes the conjugation action of $N_{G} (P)$ on $P$ inner through the decomposition $N_{G} (P) = P C_{G} (P)$ of step 1.2; for $P = {1}$ the statement is vacuous (Control of fusion in a sylow p subgroup). ∎

F8step 3.1step 1.2

Depends on

Used by

Frobenius automizer criterion for p nilpotence Corollary

Dependency tree · two levels

69 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

David Craven, Finite Group Theory, Lecture 3 (standard reference, not scraped)
Hans Kurzweil and Bernd Stellmacher, The Theory of Finite Groups, §§7.1–7.2 (standard reference, not scraped)
Paul Flavell, An Introduction to Transfer and Fusion in Finite Groups, §§2–5 (standard reference, not scraped)