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Local normal p complements force control of fusion

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 every nontrivial $p$ -local normalizer $N_{G} (Q)$ , $1 \neq Q \leq P$ , has a normal $p$ -complement (P local normalizer for normal complement theory, Normal p complement and p nilpotent group). 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 $N_{G} (Q)$ has a normal $p$ -complement for every subgroup $Q$ with $1 \neq Q \leq P$ .

[F1]

Normal $p$ -complement structure: if a finite group $H$ has a normal $p$ -complement $K$ , then $K ⊴ H$ , $p ∤ ∣ K ∣$ , $[H : K]$ is a power of $p$ , and for every Sylow $p$ -subgroup $S$ of $H$ one has $[H : K] = ∣ S ∣$ , $S \cap K = {1}$ and $H = K S = S K$ ; thus every $h \in H$ can be written $h = s k$ with $s \in S$ , $k \in K$ (Normal p complement and p nilpotent group, Sylow $p$ -subgroups of a finite group, Normal subgroup: invariance under conjugation).

[F2]

Conjugation and commutators: $x^{g} = g x g^{- 1}$ , $(x^{a})^{b} = x^{b a}$ , equivalently $x^{a b} = (x^{b})^{a}$ ; $[a, b] = a b a^{- 1} b^{- 1}$ ; if $N ⊴ H$ , $a \in N$ and $h \in H$ , then $h a h^{- 1} \in N$ , so $[h, a] = h a h^{- 1} a^{- 1} \in N$ as well; and $[a, b]$ , as a product of $a$ and $b$ , lies in every subgroup containing both $a$ and $b$ (The conjugacy class ${Cl}_{G} (x)$ and centralizer $C_{G} (x)$ of an element, 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, Commutators $[g, h] = g h g^{- 1} h^{- 1}$ and the commutator subgroup $[G, G]$ , Normal subgroup: invariance under conjugation, Conjugation $x \mapsto g x g^{- 1}$ is an automorphism, Subgroup).

[F3]

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 $G$ ).

[F4]

$T \leq N_{H} (T)$ for every subgroup $T \leq H$ : every element of $T$ normalizes $T$ (The normalizer $N_{G} (H) = {g \in G : g H g^{- 1} = H}$ of a subgroup, $C_{G} (x)$ and $N_{G} (H)$ are subgroups of $G$ ).

[F5]

If a Sylow $p$ -subgroup $T$ of a finite group $H$ controls fusion in $T$ , then its normalizer also controls fusion there, and Fusion control and centralizer transitivity are equivalent says this is equivalent to $C_{H} (x)$ acting transitively on Sylow $p$ -subgroups of $H$ containing every $x \in T ∖ {e}$ . Local Sylow conjugacy ascent (Local sylow conjugacy ascent for fusion) needs this centralizer transitivity only for $x \in S ∖ {e}$ , for each nontrivial $S \leq P$ and $H = N_{G} (S)$ .

[F6]

Subgroup and order facts: a subgroup of a finite $p$ -group is a finite $p$ -group; subgroups of finite groups have order dividing the group order (Every subgroup of a finite $p$ -group has order a power of $p$ , A finite $p$ -group has order $p^{n}$ for a prime $p$ and some $n \in N$ , Lagrange's theorem: $∣ G ∣ = [G : H] ∣ H ∣$ for every subgroup $H$ of a finite group $G$ ).

Proof

technique · direct

1.1

(A $p$ -nilpotent group is controlled by its Sylow subgroups.) Let $H$ be a finite group with a normal $p$ -complement $K$ , and let $S \in {Syl}_{p} (H)$ . Let $x, y \in S$ and $h \in H$ with $y = x^{h}$ . By [F1], $H = K S$ , so we may write $h = k s$ with $k \in K$ , $s \in S$ , and by [F2] $y = x^{k s} = (x^{s})^{k}$ . Put $a : = x^{s} \in S$ , so that $y = a^{k} = k a k^{- 1}$ and hence $[k, a] = k a k^{- 1} a^{- 1} = y a^{- 1}$ : this element lies in $S$ , because $y, a \in S$ , and it also equals $k (a k^{- 1} a^{- 1})$ , which lies in $K$ because $a k^{- 1} a^{- 1} \in K$ by normality of $K$ and $k \in K$ . Therefore $[k, a] \in S \cap K = {1}$ by [F1], so $y a^{- 1} = 1$ and $y = a = x^{s}$ with $s \in S$ . Thus every $H$ -conjugacy between elements of $S$ is realized inside $S$ : $S$ controls fusion in $S$ with respect to $H$ .

F1F2given

2.1

Let $S \leq P$ be a nontrivial subgroup; since $P$ is a finite $p$ -group, the subgroup $S$ is a finite $p$ -group by [F6]. Let $T \in {Syl}_{p} (N_{G} (S))$ . The hypothesis gives that $N : = N_{G} (S)$ has a normal $p$ -complement, so by step 1.1 applied to $H : = N$ and the Sylow $T$ of $N$ , the group $T$ controls fusion in $T$ with respect to $N$ ; by [F4] we have $T \leq N_{N} (T)$ , so the normalizer $N_{N_{G} (S)} (T)$ controls fusion in $T$ with respect to $N_{G} (S)$ as well.

F4F6givenstep 1.1

2.2

If $P \neq {1}$ , then $P$ is a nontrivial subgroup of $P$ , so the hypothesis gives that $N_{G} (P)$ has a normal $p$ -complement; by [F3] the subgroup $P$ is a Sylow $p$ -subgroup of $N_{G} (P)$ , and by step 1.1 applied to $H : = N_{G} (P)$ with the Sylow $P$ , the group $P$ controls fusion in $P$ with respect to $N_{G} (P)$ .

F3givenstep 1.1

3.1

Let $S \leq P$ be nontrivial and $x \in S ∖ {e}$ . Since $S \leq N_{G} (S)$ , some Sylow $T$ of $N_{G} (S)$ contains $S$ and hence $x$ . By step 2.1, $T$ controls fusion in $T$ with respect to $N_{G} (S)$ , so [F5] gives centralizer transitivity for $x$ . As $S$ and $x$ were arbitrary, the hypothesis of the local Sylow conjugacy ascent in [F5] is satisfied; therefore $N_{G} (P)$ controls fusion in $P$ with respect to $G$ .

F5step 2.1

4.1

Let $x, y \in P$ and $g \in G$ with $y = x^{g}$ . If $P \neq {1}$ , step 3.1 provides $u \in N_{G} (P)$ with $y = x^{u}$ , and then step 2.2 provides $v \in P$ with $y = x^{v}$ . If $P = {1}$ then $x = y = e$ and $y = x^{e}$ with $e \in P$ , so $P$ controls fusion in $P$ in this case too. ∎

step 2.2step 3.1

Depends on

Used by

Frobenius normal p complement theorem Theorem

Dependency tree · two levels

47 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)