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Local sylow conjugacy ascent for fusion

Statement

Let $G$ be a finite group, $p$ a prime and $P \in {Syl}_{p} (G)$ . Suppose that for every nontrivial $p$ -subgroup $S \leq P$ and every $x \in S$ with $x \neq e$ , the centralizer $C_{N_{G} (S)} (x)$ acts transitively on the Sylow $p$ -subgroups of $N_{G} (S)$ containing $x$ (P local normalizer for normal complement theory). Then $N_{G} (P)$ controls fusion in $P$ with respect to $G$ : any two $G$ -conjugate elements of $P$ are conjugate by an element of $N_{G} (P)$ (Control of fusion in a sylow p subgroup).

Facts & Assumptions

Given: A finite group $G$ , a prime $p$ , a Sylow $p$ -subgroup $P \leq G$ , and the hypothesis that 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 $x$ .

[F1]

Sylow facts in a finite group $X$ : Sylow $p$ -subgroups exist, every $p$ -subgroup lies in one, and any two are conjugate (Sylow I: every finite group has a Sylow $p$ -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).

[F2]

The local hypothesis says that for every nontrivial $p$ -subgroup $S \leq P$ , every $x \in S ∖ {e}$ and all Sylow $p$ -subgroups $T_{1}, T_{2}$ of $N_{G} (S)$ containing $x$ , there is $c \in C_{N_{G} (S)} (x)$ with $T_{2} = T_{1}^{c}$ (P local normalizer for normal complement theory).

[F3]

If $S$ is a proper subgroup of a finite $p$ -group $Q$ , then $S < N_{Q} (S) \leq Q$ (Proper subgroup of a finite p group is properly normalized local, The normalizer $N_{G} (H) = {g \in G : g H g^{- 1} = H}$ of a subgroup).

[F4]

Conjugation $z \mapsto g z g^{- 1}$ is an automorphism, $(z^{a})^{b} = z^{b a}$ , equivalently $z^{a b} = (z^{b})^{a}$ ; $C_{G} (x)$ and all normalizers are subgroups; $v \in N_{G} (T)$ gives $T^{v} = T$ , and $v \in C_{G} (x)$ gives $x^{v} = x$ ; also $S \leq N_{G} (S)$ (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, $C_{G} (x)$ and $N_{G} (H)$ are subgroups of $G$ , The centralizer $C_{G} (H)$ of a subgroup, The conjugacy class ${Cl}_{G} (x)$ and centralizer $C_{G} (x)$ of an element, Subgroup).

[F5]

Orders: all Sylow $p$ -subgroups of a finite group $X$ have the same order, equal to the exact power of $p$ dividing $∣ X ∣$ ; a subgroup's order divides the group's order (Sylow $p$ -subgroups of a finite group, Lagrange's theorem: $∣ G ∣ = [G : H] ∣ H ∣$ for every subgroup $H$ of a finite group $G$ , Every subgroup of a finite $p$ -group has order a power of $p$ ).

[F6]

Strong induction on the positive integer $m (Q, R) : = ∣ P ∣ / ∣ Q \cap R ∣$ , for Sylow $p$ -subgroups $Q, R$ of $G$ : $∣ Q \cap R ∣$ divides $∣ Q ∣ = ∣ P ∣$ by [F5], so $m (Q, R)$ is a positive integer, and $m (Q, R) = 1$ exactly when $Q = R$ (Strong (complete) induction, Lagrange's theorem: $∣ G ∣ = [G : H] ∣ H ∣$ for every subgroup $H$ of a finite group $G$ ).

Proof

technique · direct

1.1

Let $x \in P$ , $x \neq e$ , and let $Ω$ be the set of Sylow $p$ -subgroups of $G$ containing $x$ . We prove by strong induction on $m (Q, R)$ that any two members $Q, R \in Ω$ are conjugate by an element of $C_{G} (x)$ ; the case $m (Q, R) = 1$ , that is $Q = R$ , is trivial by [F6].

F6given

1.2

The hypothesis extends to every nontrivial $p$ -subgroup of $G$ , not only to those inside $P$ : let $S \leq G$ be a nontrivial $p$ -subgroup and choose $d \in G$ with $S^{d} \leq P$ by [F1], so that $N_{G} (S^{d}) = N_{G} (S)^{d}$ and conjugation by $d$ carries Sylow $p$ -subgroups of $N_{G} (S)$ to Sylow $p$ -subgroups of $N_{G} (S^{d})$ . If $x \in S ∖ {e}$ and $T_{1}, T_{2}$ are Sylow $p$ -subgroups of $N_{G} (S)$ containing $x$ , then $x^{d} \in S^{d} ∖ {e}$ and $T_{1}^{d}, T_{2}^{d}$ are Sylow $p$ -subgroups of $N_{G} (S^{d})$ containing $x^{d}$ , so [F2] applied to the nontrivial $p$ -subgroup $S^{d} \leq P$ provides $c^{'} \in C_{N_{G} (S^{d})} (x^{d})$ with $(T_{1}^{d})^{c^{'}} = T_{2}^{d}$ . Then $c : = c^{' d^{- 1}}$ normalizes $S$ and centralizes $x$ , so $c \in C_{N_{G} (S)} (x)$ , and conjugating the displayed equality by $d^{- 1}$ gives $T_{1}^{c} = T_{2}$ ; hence the form [F2] of the hypothesis holds for $S$ .

F1F2F4algebra

2.1

For the induction step let $Q, R \in Ω$ with $S : = Q \cap R$ and $m (Q, R) > 1$ , so that $S < Q$ and $S < R$ by [F6]; note $x \in S$ , so $S$ is a nontrivial $p$ -subgroup, being a subgroup of the $p$ -group $Q$ by [F5]. By [F3] there are $N_{Q} (S)$ and $N_{R} (S)$ with $S < N_{Q} (S) \leq Q$ and $S < N_{R} (S) \leq R$ .

F3F5F6step 1.1

3.1

Put $N : = N_{G} (S)$ , which is a $p$ -local normalizer; by [F1] there is a Sylow $p$ -subgroup $T_{Q}$ of $N$ with $N_{Q} (S) \leq T_{Q}$ and a Sylow $p$ -subgroup $T_{R}$ of $N$ with $N_{R} (S) \leq T_{R}$ ; then $x \in S < N_{Q} (S) \leq T_{Q}$ and $x \in S < N_{R} (S) \leq T_{R}$ by [F4]. By [F2] in the form of step 1.2, applied to the nontrivial $p$ -subgroup $S$ of step 2.1, the element $x \in S$ and the Sylows $T_{Q}, T_{R}$ of $N_{G} (S)$ containing $x$ , there is $c \in C_{N_{G} (S)} (x)$ with $T_{Q}^{c} = T_{R}$ .

F1F2F4step 2.1step 1.2

4.1

By [F1] choose Sylow $p$ -subgroups $Q^{*}, R^{*}$ of $G$ with $T_{Q} \leq Q^{*}$ and $T_{R} \leq R^{*}$ . Then $x \in T_{Q} \leq Q^{*}$ and $x \in T_{R} \leq R^{*}$ , so $Q^{*}, R^{*} \in Ω$ . Moreover $S < N_{Q} (S) \leq Q \cap Q^{*}$ and $S < N_{R} (S) \leq R \cap R^{*}$ , and since $T_{R} = T_{Q}^{c} \leq (Q^{*})^{c}$ also $T_{R} \leq (Q^{*})^{c} \cap R^{*}$ with $∣ T_{R} ∣ \geq ∣ N_{R} (S) ∣ > ∣ S ∣$ . Hence $m (Q, Q^{*}) < m (Q, R)$ , $m (R, R^{*}) < m (Q, R)$ and $m ((Q^{*})^{c}, R^{*}) < m (Q, R)$ by [F5] and [F6].

F1F5F6step 2.1step 3.1

5.1

Also $(Q^{*})^{c} \in Ω$ because $x^{c} = x$ and $x \in Q^{*}$ , and $c \in C_{G} (x)$ , so $(Q^{*})^{c}$ is $C_{G} (x)$ -conjugate to $Q^{*}$ .

F4step 3.1step 4.1

6.1

The induction hypothesis of step 1.1 applies to the pairs $(Q, Q^{*})$ , $((Q^{*})^{c}, R^{*})$ and $(R^{*}, R)$ , all of whose measures are smaller than $m (Q, R)$ : $Q$ is $C_{G} (x)$ -conjugate to $Q^{*}$ , $(Q^{*})^{c}$ to $R^{*}$ , and $R^{*}$ to $R$ . Since $C_{G} (x)$ -conjugacy is an equivalence relation, and since $(Q^{*})^{c}$ is $C_{G} (x)$ -conjugate to $Q^{*}$ by step 5.1, the element $Q$ is $C_{G} (x)$ -conjugate to $R$ , completing the induction.

step 1.1step 4.1step 5.1algebra

7.1

We have shown that for every $x \in P$ with $x \neq e$ , $C_{G} (x)$ acts transitively on the Sylow $p$ -subgroups of $G$ containing $x$ ; by Fusion control and centralizer transitivity are equivalent applied with $H : = G$ , the normalizer $N_{G} (P)$ controls fusion in $P$ with respect to $G$ . ∎

step 1.1step 6.1

Depends on

Used by

Dependency tree · two levels

59 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)