Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 35 (1994), No. 1, 95-99.

On Alperin’s Fusion Theorem

Bernd Stellmacher

Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel Ludewig-Meyn-Str. 4, D - 24118 Kiel

The purpose of this note is to give a graph theoretic interpretation of Alperin’s Fusion Theorem and its variations by Alperin, Gorenstein and Goldschmidt (see [1], section 2 of [2] and [3]) and then to use this interpretation for a short proof of a result (Theorem 1) which includes all of these theorems.

Let C be a non–empty set and R a symmetric relation on C. Then Γ = (C, R) is called a graph with vertex set C; and two elements a, b ∈ C are said to be adjacent if (a, b) ∈ R. An (n + 1)–tuple (c_{0},…, c_{n}) of vertices is a path of Γ, if c_{i} is adjacent to c_{i+1} for i = 0,…, n − 1.

The group G operates (transitively) on Γ if G operates (transitively) on C and R is invariant under the induced operation of G on C × C. For c ∈ C the stabilizer of c in G is denoted by G_{c}.

Lemma 1. Let Γ = (C, R) be a graph, c ∈ C and G be a group which operates transitively on Γ. Suppose that there exist h ∈ G and a non–empty subset A of G such that A, A^{h} ⊆ G_{c}. Then the following two properties are equivalent:

(a) There exists n ≥ 1 and a path (c_{0},…, c_{n}) such that c_{0} = c, c_{n} = ch^{−1} and A ⊆ G_{c_{k}} for k = 1,…, n.

(b) There exists n ≥ 1 and elements y_{1},…, y_{n} ∈ G such that (b_{1}) h = y_{1} ⋯ y_{n}s for a suitable s ∈ G_{c}, (b_{2}) c is adjacent to cy_{k} for k = 1,…, n, (b_{3}) A^{y_{1}···y_{k}} ⊆ G_{c} for k = 1,…, n.

Proof. (a) ⟹ (b): Since G is transitive on C there exist x_{1},…, x_{n} ∈ G such that c_{k} = cx_{1} ⋯ x_{k} for k = 1,…, n. We define x_{0} = 1 and

y_{k}^{−1} = x_{k}^{(x_{0}···x_{k−1})^{−1}} for k = 1,…, n.

It follows that x_{1} ⋯ x_{k} = y_{k}^{−1} ⋯ y_{1}^{−1} for k = 1,…, n. In particular, cy_{n}^{−1} ⋯ y_{1}^{−1} = c_{n} = ch^{−1} and thus h = y_{1} ⋯ y_{n}s for some s ∈ G_{c}.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1994 Heldermann Verlag, Berlin

Since A ⊆ G_{c_{k}} = G_{c}^{x_{1}···x_{k}} = G_{c}^{y_{k}^{−1}···y_{1}^{−1}} we also get A^{y_{1}···y_{k}} ⊆ G_{c}. In addition, (c_{k−1}, c_{k}) ∈ R implies (c, c(x_{0} ⋯ x_{k−1})x_{k}(x_{0} ⋯ x_{k−1})^{−1}) = (c, cy_{k}^{−1}) ∈ R. Hence, (b_{1}) – (b_{3}) are satisfied.

(b) ⟹ (a): Define c_{0} = c and c_{k} = cy_{k}^{−1} ⋯ y_{1}^{−1} for k = 1,…, n. Then

c_{n} = cy_{n}^{−1} ⋯ y_{1}^{−1} = csy_{n}^{−1} ⋯ y_{1}^{−1} = ch^{−1}.

From (b_{2}) we get that (cy_{k}^{−1}, c) ∈ R. Hence

(cy_{k}^{−1} ⋯ y_{1}^{−1}, cy_{k−1}^{−1} ⋯ y_{1}^{−1}) = (c_{k}, c_{k−1}) ∈ R,

and (c_{0},…, c_{n}) is a path of Γ.

Since by (b_{3}) A ⊆ G_{c}^{(y_{1}···y_{k})^{−1}} and c_{k} = c(y_{1} ⋯ y_{k})^{−1} we also get that A ⊆ G_{c_{k}}.

Lemma 2. Suppose that Γ and G satisfy (b) of Lemma 1 for some c ∈ C, h ∈ G and A ⊆ G_{c}. Then for every choice of b_{1},…, b_{n} ∈ G_{c} the elements g_{i} = (b_{i}y_{i})^{b_{i}···b_{1}}, i = 1,…, n, satisfy (b_{1}) – (b_{3}) in place of y_{1},…, y_{n}.

Proof. Define z_{i} = b_{i}y_{i}, b_{i} ∈ G_{c}. Then y_{i} = b_{i}^{−1}z_{i} and y_{1} ⋯ y_{k} = b_{1}^{−1}z_{1} ⋯ b_{k}^{−1}z_{k} = g_{1} ⋯ g_{k}b_{1}^{−1} ⋯ b_{k}^{−1}. Now (b_{1}) – (b_{3}) are easy to check.

Remark. Property (b) in Lemma 1 (or its seemingly stronger variation in Lemma 2) can be viewed as a graph theoretic version of Alperin’s Fusion Theorem. The crucial observation is that this version is equivalent to a very natural graph theoretic condition:

(∗) Any set of elements (automorphisms) fixing two vertices also fixes a path joining these two vertices.

In the following let G be a finite group and p a prime which divides |G|. A map W from the set of p–subgroups of G into itself is said to be a conjugacy functor, if for every p–subgroup P of G:

(i) W (P) ≤ P, and W (P) ≠ 1 if P ≠ 1.

(ii) W (P)^{g} = W (P^{g}) for every g ∈ G.

For every pair of p–subgroups (P_{0}, P) with P_{0} ≤ P we define W_{0}(P_{0}, P) = P_{0} = H_{0}(P_{0}, P) and

H_{i}(P_{0}, P) = N_{P} (W_{i−1}(P_{0}, P)) and W_{i}(P_{0}, P) = W (H_{i}(P_{0}, P)).

Because of (ii) there exists an integer n such that H_{n}(P_{0}, P) = P. The next Lemma and its Corollary are trivial.

Lemma 3. Let (P_{0}, P) be a pair of p–subgroups with P_{0} ≤ P, set N_{0} = P_{0} and N_{i} = N_{G}(W_{i−1}(P_{0}, P)) for i ≥ 1 and let k ≥ 0 be an integer such that H_{i}(P_{0}, P) ∈ Syl_{p}(N_{i}) for i = 0,…, k. Then there exists a p–subgroup T containing H_{k+1}(P_{0}, P) such that H_{k+1}(P_{0}, T) ∈ Syl_{p}(N_{k+1}).

Corollary 1. Let P_{0} be a p–subgroup. Then there exists P ∈ Syl_{p}(G) such that P_{0} ≤ P and

(∗)  H_{i}(P_{0}, P) ∈ Syl_{p}(N_{G}(W_{i−1}(P_{0}, P))) for every i ≥ 1.

Definition. Let P_{0} ≤ P ∈ Syl_{p}(G). Then P_{0} is well–placed in P (with respect to W) if (∗) holds for (P_{0}, P).

A proper subgroup M of G is strongly p–embedded if p divides |M| and

M ∩ M^{g} is a p′–group for every x ∈ G \ M. Let P ∈ Syl_{p}(G) and F(P) be a set of subgroups of P. Then F(P) is a conjugacy

family (for P in G), if for every g ∈ G and non–empty subsets A, A^{g} ⊆ P there exist subgroups Q_{1},…, Q_{n} ∈ F(P) and p–elements g_{i} ∈ N_{G}(Q_{i}) such that:

(I) g = g_{1} ⋯ g_{n}b for suitable b ∈ N_{G}(P), and

(II) A^{g_{1}···g_{i}} ⊆ Q_{i} for i = 1,…, n.

Let T (P) be the set of all subgroups P_{0} of P such that (i) P_{0} is well–placed in P with respect to W, and (ii) N_{G}(P_{0})/P_{0} contains a strongly p–embedded subgroup. Note that P ∉ T (P) and N_{G}(P) acts on T (P) by conjugation. Note further that

T (P) ≠ ∅ since the maximal Sylow intersections contained in P are in T (P). We now define a graph on which G operates.

Let C = {P^{g} | g ∈ G} and R be the set of all pairs (P^{x}, P^{y}) such that there exist x̃ ∈ N_{G}(P)x and ỹ ∈ N_{G}(P)y with

(1) P ∩ P^{x̃^{ỹ^{−1}}} = Q ∈ T (P), and

(2) x̃ỹ^{−1} ∈ O^{p′} (N_{G}(Q)).

Then R is a symmetric relation on C, and G operates by conjugation on the graph Γ = (C, R). In particular, the stabilizer of the vertex P^{g} in G is N_{G}(P^{g}). Next we show that Γ and G satisfy property (a) of Lemma 1.

Lemma 4. Let g ∈ G and D = P ∩ P^{g}. Then there exists a path joining P and P^{g} which is fixed by D.

Proof. Let (P, P^{g}) be a counterexample such that |D| is maximal. Set M = N_{G}(D) and P_{1} = N_{P} (D). Note that P ≠ D and thus P_{1} ≠ D.

Let C_{0} = {T ∈ C | D ≤ T } and Γ_{0} be the induced subgraph of Γ with vertex set C_{0}. Then Γ_{0} is not connected since (P, P^{g}) is a counterexample. Let C_{1} be the vertex set of the connected component of Γ_{0} which contains P. The maximality of |D| gives:

(∗) T ∈ C_{1} for every T ∈ C_{0} with T ∩ P ≠ D. By Lemma 3 there exists P̃ ∈ C_{0} such that D is well–placed in P̃ and P_{1} ≤ P̃. Hence

(∗) implies P̃ ∈ C_{1}, and we may assume that P̃ = P. With a symmetric argument we may assume that D is well placed in P^{g} and g ∈ O^{p′} (M).

Now let C_{0}^{∗} = {P^{m} | m ∈ M} and Γ_{0}^{∗} be the induced subgraph of Γ_{0} with vertex set C_{0}^{∗}, and let M_{0} be the stabilizer in M of the connected component of Γ_{0}^{∗} which contains P. Clearly P_{1} ≤ M_{0} and thus p divides |M_{0}/D|. Moreover, M_{0} is a proper subgroup of M since g ∈ M and (P, P^{g}) is a counterexample. Let x ∈ M \ M_{0} and D_{0} ∈ Syl_{p}(M_{0} ∩ M_{0}^{x}). Then P^{x} is not in the same connected component of Γ_{0}^{∗} as P. Hence, (∗) yields D = D_{0}, and M_{0} ∩M_{0}^{x}/D is a p′–group. It follows that D ∈ T (P) and (P, P^{g}) ∈ R, a contradiction.

Theorem 1. Let P ∈ Syl_{p}(G). Then T (P) is a conjugacy family for P in G.

Proof. Set B = N_{G}(P). Let A be a non–empty subset of G and h ∈ G. Suppose that A, A^{h} ⊆ P. We have to show:

There exist subgroups Q_{i} ∈ T (P) and p–elements g_{i} ∈ N_{G}(Q_{i}), i = 1,…, n, such that

(i) h = g_{1} ⋯ g_{n}b, b ∈ N_{G}(P), and (ii) A^{g_{1}···g_{i}} ⊆ Q_{i} for i = 1,…, n. We may assume that h ∉ B. Note that A ⊆ P ∩ P^{h^{−1}}. Hence, by Lemma 4 property

(a) of Lemma 1 is satisfyied. From the equivalent property (b) we conclude that there exist Q_{i} ∈ T (P) and y_{i} ∈ BO^{p′}(N_{G}(Q_{i})), i = 1,…, n such that (i) and (ii) hold. In particular y_{i} = b_{i}^{−1}z_{i} for suitable b_{i} ∈ B and z_{i} ∈ O^{p′} (N_{G}(Q_{i})). Set b̃_{i} = b_{1} ⋯ b_{i}. Note that Qb̃_{i} ; i ∈ T (P). Now Lemma 2 shows that there exist g_{i} ∈ O^{p′} (N_{G}(Qb̃_{i} ; i)) satisfying (i) and (ii).

The next result indicates that the conjugacy family T (P) is in a certain sense minimal among the conjugacy families for P. For any conjugacy family F(P) set F(G) = ⋃_{g∈G} F(P)^{g}.

Theorem 2. Let F_{1}(P),…, F_{r}(P) be the distinct conjugacy families for P in G. Then ⋂_{i=1}^{r} F_{i}(G) = T (G).

Proof. Clearly ⋂_{i=1}^{r} F_{i}(G) ⊆ T (G) since by Theorem 1 T (P) is a conjugacy family. Thus, it suffices to show that for every Q ∈ T (P) and k ∈ {1,…, r} there exists x ∈ G such that Q^{x} ∈ F_{k}(P).

Assume that Q ∈ T (P) is a counterexample such that |Q| is maximal. Let M = N_{G}(Q) and \overline{M} = M/Q. Then there exists a strongly p–embedded subgroup \overline{M}_{0} in \overline{M} containing \overline{M ∩ P}. Pick h ∈ M \ M_{0}. Then P ∩ P^{h} = Q since N_{P ∩P^{h}}(Q) = Q; in particular Q, Q^{h^{−1}} ⊆ P. Since F_{k}(P) is a conjugacy family there exist Q_{i} ∈ F_{k}(P) and p-elements y_{i} ∈ N_{G}(Q_{i}), i = 1,…, n such that

(i) h^{−1} = y_{1} ⋯ y_{n}b, b ∈ N_{G}(P), and

(ii) Q^{y_{1}···y_{i}} ≤ Q_{i} ≤ P.

Let R_{0} be the set of all pairs (P^{x}, P^{y}) for which there exist x̃ ∈ N_{G}(P)x, ỹ ∈ N_{G}(P)y and K ∈ F_{k}(P) such that

(∗) x̃ỹ^{−1} ∈ O^{p′} (N_{G}(K)).

Let Λ(C, R_{0}) be the corresponding graph. Define P_{0} = P and P_{i} = P^{y_{i}^{−1}···y_{1}^{−1}} for i = 1,…, n. Then (i) implies that P_{n} = P^{h}, and (ii) gives

(iii) Q ≤ Q_{i+1}^{y_{i+1}^{−1}···y_{1}^{−1}} ≤ P_{i} ∩ P_{i+1} for i = 1,…, n − 1.

Moreover, (P_{i}, P_{i+1}) ∈ R_{0} since y_{i+1} ∈ O^{p′} (N_{G}(Q_{i+1})). It follows that (P_{0},…, P_{n}) is a path of Λ(C, R_{0}).

Assume first that P_{i}∩P_{i+1} ≠ Q for every i = 1,…, n−1. Then also N_{P_{i}∩P_{i+1}}(Q) ≠ Q. On the other hand, if T_{1} and T_{2} are p-subgroups of M such that Q is properly contained in T_{1} ∩ T_{2} and T_{1} ≤ M_{0}, then also T_{2} ≤ M_{0} since \overline{M}_{0} is strongly p-embedded in \overline{M}. It follows that P^{h} ∩ M ≤ M_{0} and h ∈ M_{0}, a contradiction.

Thus there exists j ∈ {1,…, n − 1} such that P_{j} ∩ P_{j+1} = Q. But now (iii) implies that Q^{y_{1}···y_{j+1}} = Q_{j+1} ∈ F_{k}(P).

Acknowledgement. I would like to thank Prof. G. Stroth for pointing out a mistake in the proof of Theorem 2.

Literature

[1] J.L. Alperin: Sylow intersections and fusion, J. Alg. 6 (1967), 222 – 241.

[2] J.L. Alperin, D. Gorenstein: Transfer and fusion in finite groups, J. Alg. 6 (1967), 242 – 255.

[3] D. Goldschmidt: A conjugacy family for finite groups, J. Alg. 160 (1970), 138 – 142.

Received 20.10.94