MINIMAL CHARACTERISTIC BISETS AND FINITE GROUPS REALIZING RUIZ-VIRUEL EXOTIC FUSION SYSTEMS

SEJONG PARK

Abstract. Continuing our previous work [2], we determine a minimal left characteristic biset X for every exotic fusion system F on the extraspecial group S of order 7^{3} and exponent 7 discovered by Ruiz and Viruel [6], and analyze the finite group G obtained from X by the method of [2] which realizes F as the full subcategory of the 7-fusion system of G. In particular, we obtain an upper bound for the exoticity index for F.

1. Introduction

In our previous work [2], we observed that every saturated fusion system F on a finite p-group S can be realized by a finite group G containing S as a subgroup, in the sense that F = F_{S}(G), i.e. the fusion system on S whose morphisms are the G-conjugation maps among subgroups of S. This observation lead us to propose the notion of the exoticity index e(F) of F as the minimum of the values log_{p} |S_{0} : S| where S_{0} is a Sylow p-subgroup of G containing S as G runs over all finite groups such that F = F_{S}(G). With this definition, F is exotic if and only if e(F) > 0, and e(F) measures how far the fusion system F is from being the fusion system of some finite group. In [2] the finite group G was constructed by using a finite S-S-biset X called a left characteristic biset of the fusion system F as G = Aut(_{1}X), i.e. the automorphism group of X viewed as a right S-set ignoring the left S-action (or equivalently, restricting the left S-action to the trivial subgroup, whence the subscript 1 on the left). Then we have

G ≅ S ≀ Σ_{e(X)} where e(X) = |X|/|S|.

The upper bound of the exoticity index e(F) of F obtained from this G is as follows:

e(F) ≤ (e(X) − 1) log_{p} |S| + ∑_{i≥1} ⌊\frac{e(X)}{p^{i}}⌋ ≤ e(X)(1 + log_{p} |S|).

As far as we know, no other method for constructing a finite group realizing a given saturated fusion system has been found.

In this paper, we determine minimal left characteristic bisets for some saturated fusion systems F on the extraspecial group S of order p^{3} and exponent p including all exotic fusion systems on S discovered by Ruiz and Viruel [6], and thereby obtain finite groups realizing F which are the smallest among those obtained by the method of [2] described in the previous paragraph.

Date: November 2010.

2000 Mathematics Subject Classification. 20C20.

Key words and phrases. fusion systems, exotic fusion systems, exoticity index.

Theorem 1.1. Let p be an odd prime and let S be the extraspecial group of order p^{3} and exponent p. Let F be a saturated fusion system on S such that all subgroups of S of order p^{2} are F-radical. Then there exists a unique minimal left characteristic biset X for F, and we have

e(X) = \frac{p^{5} − 1}{p − 1}|Out_{F}(S)|.

A more detailed version of this theorem is stated and proved in §5 as Theorem 5.2 and Remark 5.3.

The upper bound for the exoticity index of a Ruiz-Viruel exotic fusion system F given by the above theorem is quite large. An extraspecial group S of order p^{3} and exponent p affords exotic fusion systems only when p = 7, and in that case there are three of them with the order of the outer automorphism groups of S being 48, 72 and 96, respectively. The upper bound that we obtain for the case where |Out_{F}(S)| = 48 is 425744, for example.

A natural question that follows is whether we can cut down the group G to get a smaller upper bound for the exoticity index. By Alperin’s fusion theorem, an obvious candidate is the subgroup H of G generated by the normalizers of the F-essential subgroups of S. After close analysis of some elements of the normalizers of F-essential subgroups of S, we find that even H is quite large, though this gives us a better understanding of the fusion action of G on S.

Theorem 1.2. Let p be an odd prime and let S be an extraspecial group of order p^{3} and exponent p. Let F be a saturated fusion system on S such that all subgroups of S of order p^{2} are F-radical. Let X be the minimal left characteristic biset X for F given by Theorem 1.1. Let G = Aut(_{1}X) ≅ S ≀ Σ_{e(X)} and let H be the subgroup of G generated by the normalizers of the F-essential subgroups of S. Then the image of H in Σ_{e(X)} is a transitive subgroup of Σ_{e(X)}.

This theorem is proved in §6 as Theorem 6.4.

Organization of the paper: In §2, we recall the definition of characteristic bisets for fusion systems and fix basic notations. We also compare notational conventions in the literature about bisets concerning fusion systems in Remark 2.6. In §3, we introduce the notion of layers of bisets and review some basic results about the number of fixed points which will be used in the subsequent sections. §4 contains a general result which determines the top two layers of a characteristic biset of any saturated fusion system. Using this result and detailed information of the saturated fusion systems on an extraspecial group S of order p^{3} and exponent p in [6], we determine minimal left characteristic bisets for saturated fusion systems F on S such that all subgroups of S of order p^{2} are F-radical in §5. A more detailed version of Theorem 1.1 is stated and proved in this section as Theorem 5.2 and Remark 5.3. In §6, we analyze some elements of the normalizers of F-essential subgroups of S and obtain Theorem 1.2 as a consequence. Finally in the Appendix A, we determine coefficients of characteristic idempotents as an easy application of results in §4 and §5.

Acknowledgments: The author would like to express his special thanks to Ronald Solomon and Justin Lynd for very helpful discussions about the content of this paper and their warm hospitality whenever he visited The Ohio State University.

2. Characteristic bisets for fusion systems

First let us fix notations. We keep the notations in our previous paper [2] and expand on them as needed. Our notations are largely compatible with (and in fact many of them derive from) those in [4, 5], except that the directions of maps are inverted. See Remark 2.6 for more details.

Let S_{1}, S_{2} be groups. An S_{1}-S_{2}-biset is a set X with left S_{1}-action and right S_{2}-action such that (ux)v = u(xv) for x ∈ X, u ∈ S_{1}, v ∈ S_{2}. An S_{1}-S_{2}-biset X can be viewed as a (left) (S_{1} × S_{2})-set via (u, v) · x = uxv^{−1} for x ∈ X, u ∈ S_{1}, v ∈ S_{2} and vice versa. Throughout this paper, we will go freely back and forth between S_{1}-S_{2}-bisets and (S_{1} × S_{2})-sets using this correspondence. That way, we can exploit the advantages of both perspectives. On one hand, bisets can be composed: if S_{3} is another group and Y is an S_{2}-S_{3}-biset, the set

X ×_{S_{2}} Y = (X × Y)/ ∼,

where (xu, y) ∼ (x, uy) for x ∈ X, y ∈ Y, u ∈ S_{2}, is an S_{1}-S_{3}-biset by the left action of S_{1} on X and the right action of S_{3} on Y. On the other hand, X viewed as an (S_{1} × S_{2})-set has the fixed-point subset X^{H} = {x ∈ X | hx = x for all h ∈ H} by any subgroup H ≤ S_{1} × S_{2}, and the number of fixed-points |X^{H}| as H runs over all subgroups of S_{1} × S_{2} determine the isomorphism type of X by Burnside’s theorem.

For a subgroup Q of S_{1} and an injective group homomorphism φ: Q → S_{2}, let

S_{1} ×_{(Q,φ)} S_{2} = (S_{1} × S_{2})/ ∼

where (xu, y) ∼ (x, φ(u)y) for x ∈ S_{1}, y ∈ S_{2}, u ∈ Q, and let ⟨x, y⟩ be the ∼-equivalence class containing (x, y). One can view this set as an S_{1}-S_{2}-biset by u · ⟨x, y⟩ = ⟨ux, y⟩, ⟨x, y⟩ · v = ⟨x, yv⟩ for x, u ∈ S_{1}, y, v ∈ S_{2}. It is a transitive S_{1}-S_{2}-biset with free left S_{1}- action and free right S_{2}-action. In fact, every such S_{1}-S_{2}-biset is of this form. Viewed as an (S_{1} × S_{2})-set, it is isomorphic to

(S_{1} × S_{2})/Δ_{Q}^{φ}

where Δ_{Q}^{φ} = {(u, φ(u)): u ∈ Q}.

Two transitive S_{1}-S_{2}-bisets S_{1} ×_{(Q,φ)} S_{2} and S_{1} ×_{(R,ψ)} S_{2} where Q, R ≤ S_{1}, φ: Q → S_{2}, ψ: R → S_{2} are isomorphic if and only if Δ_{Q}^{φ} is conjugate to Δ_{R}^{ψ} in S_{1}×S_{2}, which in turn holds if and only if there exist x ∈ S_{1}, y ∈ S_{2} such that ^{x}Q := xQx^{−1} = R and ψ ∘ c_{x}|_{Q} = c_{y} ∘ φ where c_{x}: S_{1} → S_{1} denotes the conjugation map by x given by c_{x}(u) = xux^{−1} for u ∈ S_{1}, and c_{y} is defined analogously. If this is the case, we say that φ and ψ are S_{1}-S_{2}-conjugate, and write φ ∼_{(S_{1},S_{2})} ψ. We write [φ] for the S_{1}-S_{2}-conjugacy class containing φ. More generally, we say that φ is S_{1}-S_{2}-subconjugate to ψ and write φ ≾_{(S_{1},S_{2})} ψ if there exist x ∈ S_{1}, y ∈ S_{2} such that ^{x}Q ≤ R and ψ ∘ c_{x}|_{Q} = c_{y} ∘ φ

Finally, for an S_{1}-S_{2}-biset X, Q ≤ S_{1}, and an injective group homomorphism φ: Q → S_{2}, let _{φ}X be the Q-S_{2}-biset obtained from X where the left Q-action is induced by φ; let QX := _{(id_{Q})}X. Equivalently,

φX ≅ (Q ×_{(Q,φ)} S_{1}) ×_{S_{1}} X.

For R ≤ S_{2} and an injective group homomorphism ψ: R → S_{2}, S_{1}-R-bisets X_{ψ} and X_{R} are defined analogously.

Definition 2.1. Let F be a fusion system on a finite p-group S. A finite S-S-biset X is called a left characteristic biset for the fusion system F if it satisfies the following conditions:

(A) Every transitive subbiset of X is of the form S ×_{(Q,φ)} S where Q ≤ S and φ ∈

Hom_{F}(Q, S). (B) e(X) := |X|/|S| ≢ 0 mod p. (C) (left F-stability) _{Q}X ≅ _{φ}X as Q-S-bisets for every Q ≤ S and φ ∈ Hom_{F}(Q, S).

Similarly, X is called a right characteristic biset for F if it satisfies the conditions (A), (B), and

(C’) (right F-stability) X_{Q} ≅ X_{φ} as S-Q-bisets for every Q ≤ S and φ ∈ Hom_{F}(Q, S). If X satisfies the conditions (A), (B), (C) and (C’), then X is called a characteristic biset for F.

Throughout this paper, we will refer to the conditions in this definition by the symbols (A), (B), (C) and (C’).

The following result, due to Broto, Levi and Oliver, is fundamental for this paper.

Theorem 2.2 ([1, 5.5]). Every saturated fusion system has a characteristic biset.

Remark 2.3. [1, 5.5] only states the existence of a right characteristic biset. But in fact what is constructed in the proof of [1, 5.5] is a characteristic biset, as pointed out by Ragnarsson in the paragraph right before Proposition 4.4 in [4].

We define an object which will play a central role in this paper.

Definition 2.4. Let F be a saturated fusion system on a finite p-group S. A (left or right) characteristic biset X for F is called minimal if it has the smallest size among all possible (left or right) characteristic bisets for F.

Finally, we introduce the opposite operation, which is useful when comparing left and right stability.

Definition 2.5. Let S_{1} and S_{2} be groups. For an S_{1}-S_{2}-biset X, let X^{op} be the S_{2}-S_{1}-biset which has the same underlying set as X and such that v · x · u = u^{−1}xv^{−1} for x ∈ X, u ∈ S_{1}, v ∈ S_{2}.

If X is a left (resp. right) characteristic biset for a fusion system F, then X^{op} is a right (resp. left) characteristic biset for F. If S_{1}, S_{2} are groups, Q ≤ S_{1} and φ: Q → S_{2} is an injective group homomorphism, then

(S_{1} ×_{(Q,φ)} S_{2})^{op} ≅ S_{2} ×_{(φ(Q),φ^{−1})} S_{1}

See [5, §3.5] for a more discussion of this opposite operation.

Remark 2.6. Our use of the terminology ‘characteristic biset’ derives from Ragnarsson [4], and Ragnarsson and Stancu [5], whereas Puig [3, Ch. 21] calls them basic sets. But our convention in which maps defining transitive bisets are directed from left to right is close to Puig’s; Ragnarsson and Stancu use the opposite notations. More precisely, the correspondence between Ragnarsson and Stancu’s notation and ours is as follows:

[S_{1} ×_{(Q,φ)} S_{2}] = [φ(Q), φ^{−1}]_{S_{2}}^{S_{1}} = ([Q, φ]_{S_{1}}^{S_{2}})^{op} ,

where the square bracket denotes the isomorphism class of bisets. Accordingly, our left (resp. right) stabilitiy corresponds to Ragnarsson and Stancu’s right (resp. left) stability.

3. Layers of characteristic bisets and counting fixed-points

Definition 3.1. Let S be a finite group. Let X be an S-S-biset with free left and right action of S. For each r ≥ 0, let X_{r} denote the ‘r-th layer’ of X, i.e. the union of the transitive subbisets of X of the form

S ×_{(Q,φ)} S with |S : Q| = p^{r},

so that

X = X_{0} ⨿ X_{1} ⨿ X_{2} ⨿ · · · .

Let X_{≤r} = ∐_{0≤i≤r} X_{i}.

A very effective way of studying finite sets with action of a finite group is counting the number of fixed-points by subgroups. Indeed it is a standard fact due to Burnside that, for a finite group G, two finite G-sets X and Y are isomorphic if and only if |X^{H}| = |Y^{H}| for all H ≤ G.

First we make a simple, yet useful, observation.

Lemma 3.2. Let F be a saturated fusion system on a finite p-group S with a left or right characteristic biset X.

(1) For U ≤ S ×S, we have X^{U} ≠ ∅ if and only if there is a transitive subbiset S ×_{(Q,φ)} S of X such that U is conjugate in S × S to a subgroup of Δ_{Q}^{φ}. In particular, U = Δ_{R}^{ψ} for some R ≤ S and ψ ∈ Hom_{F}(R, S).

(2) |X^{Δ_{Q}^{φ}}| = |X_{≤r}^{Δ_{Q}^{φ}}| if |S: Q| = p^{r}.

Proof. In general, if G is a group and H, K are subgroups of G, we have (G/H)^{K} ≠ ∅ if and only if K is G-conjugate to a subgroup of H. The lemma follows immediately from this fact and that the family of subgroups of S ×S of the form Δ_{Q}^{φ} where Q ≤ S and φ ∈ Hom_{F}(Q, S) is invariant under conjugation in S × S. □

In the following lemma, we record what the stability conditions (C) and (C’) in Definition 2.1 mean in terms of the number of fixed-points. It is a slight variation of [5, 4.8], which reduces the number of equations to consider.

Lemma 3.3. Let F be a fusion system on a finite p-group S, and let X be a finite S-S-biset satisfying the condition (A). Then the left stability condition (C) is equivalent to the following condition:

|X^{Δ_{Q}^{φ}}| = |X^{Δ_{φ(Q)}^{id}}| for all Q ≤ S, φ ∈ Hom_{F}(Q, S);

the right stability condition (C’) is equivalent to the following condition:

|X^{Δ_{Q}^{φ}}| = |X^{Δ_{Q}^{id}}| for all Q ≤ S, φ ∈ Hom_{F}(Q, S).

Proof. We consider the condition (C), the case for (C’) being analogous. We have _{Q}X ≅ _{φ}X if and only if |(_{Q}X)^{H}| = |(_{φ}X)^{H}| for all H ≤ S × S. Since X satisfies the condition (A), Lemma 3.2 implies that both (_{Q}X)^{H} and (_{φ}X)^{H} will be empty unless H = Δ_{R}^{ψ} for some

R ≤ Q and ψ ∈ Hom_{F}(R, S). Now (_{Q}X)^{Δ_{R}^{ψ}} = X^{Δ_{R}^{ψ}}, while (_{φ}X)^{Δ_{R}^{ψ}} = X^{Δ_{φ(R)}^{ψ∘φ^{−1}}}. Thus

QX ≅ _{φ}X if and only if |X^{Δ_{R}^{ψ}}| = |X^{Δ_{φ(R)}^{ψ∘φ^{−1}}}| for all R ≤ Q, ψ ∈ Hom_{F}(R, S). The latter condition implies in particular (when R = Q and ψ = φ) |X^{Δ_{Q}^{φ}}| = |X^{Δ_{φ(Q)}^{id}}|. Thus the condition (C) implies that |X^{Δ_{Q}^{φ}}| = |X^{Δ_{φ(Q)}^{id}}| for all Q ≤ S, φ ∈ Hom_{F}(Q, S). Conversely, if |X^{Δ_{Q}^{φ}}| = |X^{Δ_{φ(Q)}^{id}}| for all Q ≤ S, φ ∈ Hom_{F}(Q, S), then the condition (C) holds, because

|X^{Δ_{R}^{ψ}}| = |X^{Δ_{ψ(R)}^{id}}| = |X^{Δ_{φ(R)}^{ψ∘φ^{−1}}}|

for all R ≤ Q ≤ S, ψ ∈ Hom_{F}(R, S), φ ∈ Hom_{F}(Q, S). □

We recall the formula for the number of fixed-points of transitive bisets.

Lemma 3.4 ([5, 3.10]). Let S be a finite group. Let Q, R ≤ S, and let φ: Q → S, ψ: R → S be injective group homomorphisms. Then

|((S × S)/Δ_{Q}^{φ})^{Δ_{R}^{ψ}}| = \frac{|N_{ψ,φ}|}{|Q|}|C_{S}(ψ(R))|

where

N_{ψ,φ} = {x ∈ S | ^{x}R ≤ Q, ∃y ∈ S: φ ∘ c_{x}|_{R} = c_{y} ∘ ψ}.

Note that N_{ψ,φ} = ∅ unless ψ ≾_{(S,S)} φ. Also, N_{φ,φ} = N_{φ}, the largest subgroup of N_{S}(Q) to which φ can be extended. In the special case where Q = S, Lemma 3.4 takes the following form.

Lemma 3.5. Let S be a finite group. Let R ≤ S, and let α: S → S, ψ: R → S be injective group homomorphisms. Then

|((S × S)/Δ_{S}^{α})^{Δ_{R}^{ψ}}| = \begin{cases}|C_{S}(ψ(R)|, & if ψ ≾_{(S,S)} α, \\ 0, & otherwise. \end{cases}

Proof. By Lemma 3.4, ((S × S)/Δ_{S}^{α})^{Δ_{R}^{ψ}} ≠ ∅ if and only if N_{ψ,α} ≠ ∅, i.e. if there are x, y ∈ S such that α ∘ c_{x}|_{R} = c_{y} ∘ ψ. Since α ∘ c_{x} = c_{α(x)} ∘ α, we have N_{ψ,α} ≠ ∅ if and only if N_{ψ,α} = S. □

4. Minimal characteristic bisets up to index p subgroups

In this section, we determine the coefficients of the top two layers X_{0} and X_{1} of a left or right characteristic biset X for a saturated fusion system F. It is well-known that X_{0} has the following form.

Proposition 4.1. Let X be a left or right characteristic biset for a saturated fusion system F on a finite p-group S. Then there is a positive integer c_{0} which is not divisible by p such that

X_{0} ≅ c_{0} ∐_{[α]∈Out_{F} (S)}(S × S)/Δ_{S}^{α}.

Proof. Write X_{0} ≅ ∐_{[α]∈Out_{F} (S)} c_{α}(S × S)/Δ_{S}^{α} where the c_{α} are nonnegative integers. We have |X|/|S| ≡ |X_{0}|/|S| ≡ ∑_{α} c_{α} mod p. Thus the condition (B) is equivalent to that ∑_{α} c_{α} ≢ 0 mod p. In particular, there is β ∈ Aut_{F}(S) such that c_{β} > 0. Lemma 3.3 implies that |X^{Δ_{S}^{α}}| = |X^{Δ_{S}^{β}}| for every α ∈ Aut_{F}(S). But |X^{Δ_{S}^{α}}| = |X_{0}^{Δ_{S}^{α}}| = c_{α}|Z(S)| by Lemma 3.2 and Lemma 3.4. Thus we have c_{α} = c_{β} =: c_{0} for every β ∈ Aut_{F}(S). Then |X|/|S| ≡ ∑_{α} c_{α} ≡ c_{0}|Out_{F}(S)| ≢ 0 mod p. It follows that c_{0} ≢ 0 mod p. □

To determine X_{1}, we need the following definitions.

Definition 4.2. Let F be a fusion system on a finite p-group S. Let Q ≤ R ≤ S and let φ ∈ Aut_{F}(Q, S). We say that φ is extendable (in F) to R if there is a morphism ψ ∈ Hom_{F}(R, S) such that φ = ψ|_{Q}. We say that φ is extendable if it is extendable to some subgroup of S properly containing Q; otherwise, we say that φ is nonextendable. Let

Hom_{F}^{e,R}(Q, S) = {φ ∈ Hom_{F}(Q, S) | φ is extendable to R},
Hom_{F}^{e}(Q, S) = {φ ∈ Hom_{F}(Q, S) | φ is extendable },
Hom_{F}^{n}(Q, S) = {φ ∈ Hom_{F}(Q, S) | φ is nonextendable }.

Note that Hom_{F}^{e}(Q, S) = ⋃_{Q<R≤S} Hom_{F}^{e,R}(Q, S). Also, if |S : Q| = p, then Hom_{F}^{e,S}(Q, S) = Hom_{F}^{e}(Q, S).

The following proposition is implicit in [5, 5.5].

Proposition 4.3. Let X be a left or right characteristic biset for a saturated fusion system F on a finite p-group S. Let φ ∈ Hom_{F}^{n}(Q, S) for some Q ≤ S. Then X has a transitive subbiset isomorphic to S ×_{(Q,φ)} S.

Proof. Assume that X is a left characteristic biset for F; the proof is identical when X is a right characteristic biset. By Lemma 3.3, we have |X^{Δ_{Q}^{φ}}| = |X^{Δ_{φ(Q)}^{id}}|. By Proposition 4.1 and Lemma 3.5, we have |X^{Δ_{φ(Q)}^{id}}| ≥ |((S × S)/Δ_{S}^{id})^{Δ_{φ(Q)}^{id}}| > 0. Thus we have |X^{Δ_{Q}^{φ}}| ≠ 0. Lemma 3.2 and the assumption that φ is nonextendable, it follows that X has a transitive subbiset isomorphic to S ×_{(Q,φ)} S. □

We denote by Hom_{F}(Q, S)/ ∼_{(S,S)} the set of ∼_{(S,S)}-equivalence classes of elements of Hom_{F}(Q, S). For instance, we have Out_{F}(S) = Aut_{F}(S)/ ∼_{(S,S)}. The following proposition is based on an important property of F-centric subgroups which can be found in [1, A.8]. Both the proposition and the proof are crucial for many results in this paper.

Proposition 4.4. Let F be a saturated fusion system on a finite p-group S. Let Q ≤ R ≤ S be normal subgroups of S which are F-centric. Then restriction to Q induces a bijection

Hom_{F}(R, S)/ ∼_{(S,S)}⟶^{≅} Hom_{F}^{e,R}(Q, S)/ ∼_{(S,S)} .

In particular, if |S : Q| = p, then we have

Out_{F}(S) ⟶^{≅} Hom_{F}^{e}(Q, S)/ ∼_{(S,S)}

Proof. The given map is well-defined because Q ⊴ S. It is clearly surjective. To show that it is injective, let φ, ψ ∈ Hom_{F}(R, S) and suppose φ|_{Q} ∼_{(S,S)} ψ|_{Q}. That means there exist x ∈ N_{S}(Q) and y ∈ S such that c_{y} ∘ ψ|_{Q} = φ ∘ c_{x}|_{Q}. Since Q is F-centric, we apply [1, A.8] to c_{y} ∘ ψ|_{R} and φ ∘ c_{x}|_{R} (the latter is well-defined because R ⊴ S) and get z ∈ Z(Q) such that c_{y} ∘ ψ = φ ∘ c_{x} ∘ c_{z}|_{R}. In other words, φ ∼_{(S,S)} ψ, showing that the given map is also injective. □

Now we determine X_{1} of a characteristic biset X for a saturated fusion system, which amounts to solving a part of the system of linear equations given in [4, 5.6].

Proposition 4.5. Let F be a saturated fusion system on a finite p-group S. Let {P_{i} | 1 ≤ i ≤ n} be the set of subgroups of S of index p. For each 1 ≤ i ≤ n, let {φ_{i,j}: P_{i} → S | 0 ≤ j ≤ m_{i}} be a set of representatives of ∼_{(S,S)}-equivalence classes of F-morphisms with domain P_{i} such that φ_{i,0} = ι_{P_{i}}, the inclusion map P_{i} ↪ S. Let X be a finite S-S-biset with

X_{0} ≅ c_{0} ∐_{[α]∈Out_{F} (S)}(S × S)/Δ_{S}^{α}

for some positive integer c_{0} which is not divisible by p.

(1) If X is a right characteristic biset for F, then there is an integer c_{1}^{(i)} ≥ 0 for each 1 ≤ i ≤ n such that

X_{1} ≅ ∐_{1≤i,j≤n φ_{i,j} extendable} c_{1}^{(i)}(S × S)/Δ_{P_{i}}^{φ_{i,j}} ⨿ ∐_{1≤i,j≤n φ_{i,j} nonextendable}(c_{0} + pc_{1}^{(i)})(S × S)/Δ_{P_{i}}^{φ_{i,j}}.

(2) If X is a left characteristic biset for F, then there is an integer c_{1}^{(i)} ≥ 0 for each 1 ≤ i ≤ n such that

X_{1} ≅ ∐_{1≤i,j≤n φ_{i,j} extendable} c_{1}^{(i)}(S × S)/Δ_{φ_{i,j}(P_{i})}^{φ_{i,j}^{−1}} ⨿ ∐_{1≤i,j≤n φ_{i,j} nonextendable}(c_{0} + pc_{1}^{(i)})(S × S)/Δ_{φ_{i,j}(P_{i})}^{φ_{i,j}^{−1}}.

Proof. We prove (1) and obtain (2) by taking opposite of (1). Suppose that X is a right characteristic biset for F. Write X_{1} ≅ ∐_{i,j} c_{1}^{(i,j)}(S ×S)/Δ_{P_{i}}^{φ_{i,j}} where the c_{1}^{(i,j)} are nonnegative

integers. For each pair (i, j), we have |X^{Δ_{P_{i}}^{φ_{i,j}}} | = |X^{Δ_{P_{j}}^{φ_{i,0}}}| by Lemma 3.3. Since |S : P_{i}| = p,

we have |X^{Δ_{P_{i}}^{φ_{i,j}}} | = |X_{0}^{Δ_{P_{i}}^{φ_{i,j}}} | + |X_{1}^{Δ_{P_{i}}^{φ_{i,j}}} | by Lemma 3.2, and

|X_{0}^{Δ_{P_{i}}^{φ_{i,j}}} | = c_{0}a_{i,j}|C_{S}(φ_{i,j}(P_{i}))|,
|X_{1}^{Δ_{P_{i}}^{φ_{i,j}}} | = c_{1}^{(i,j)}|((S × S)/Δ_{P_{i}}^{φ_{i,j}})^{Δ_{P_{i}}^{φ_{i,j}}} | = c_{1}^{(i,j)} \frac{|N_{φ_{i,j}}|}{|P_{i}|}|C_{S}(φ_{i,j}(P_{i}))|,

where

a_{i,j} = |{[α] ∈ Out_{F}(S) | α|_{P_{i}} ∼_{(S,S)} φ_{i,j}}|,

by Lemma 3.4 and Lemma 3.5. Thus we have

(c_{0}a_{i,j} + c_{1}^{(i,j)}|N_{φ_{i,j}}|/|P_{i}|) |C_{S}(φ_{i,j}(P_{i}))| = (c_{0}a_{i,0} + c_{1}^{(i,0)}|N_{φ_{i,0}}|/|P_{i}|) |C_{S}(P_{i})|.

Case 1: φ_{i,j} is extendable. Then φ_{i,j} = α|_{P_{i}} for some α ∈ Aut_{F}(S). Then C_{S}(φ_{i,j}(P_{i})) = C_{S}(α(P_{i})) = α(C_{S}(P_{i})), so |C_{S}(φ_{i,j}(P_{i}))| = |C_{S}(P_{i})|. Also N_{φ_{i,j}} = N_{φ_{i,0}} = S. Now if β ∈ Aut_{F}(S) and β|_{P_{i}} ∼_{(S,S)} φ_{i,j}, then α|_{P_{i}} ∼_{(S,S)} β|_{P_{i}}, so β^{−1} ∘ α|_{P_{i}} ∼_{(S,S)} id_{P_{i}}. This shows that a_{i,j} = a_{i,0}. Hence, we have c_{1}^{(i,j)} = c_{1}^{(i,0)} =: c_{1}^{(i)}.

Case 2: φ_{i,j} is not extendable. Then P_{i} and φ_{i,j}(P_{i}) are F-centric by Alperin’s fusion theorem. So |C_{S}(φ_{i,j}(P_{i}))| = |Z(φ_{i,j}(P_{i}))| = |φ_{i,j}(Z(P_{i}))| = |Z(P_{i})| = |C_{S}(P_{i})|. Also N_{φ_{i,j}} = P_{i} and N_{φ_{i,0}} = S. We have a_{i,j} = 0 since φ_{i,j} is nonextendable. On the other hand, suppose α|_{P_{i}} ∼_{(S,S)} ι_{P_{i}}, i.e. there is α ∈ Aut_{F}(S) such that α|_{P_{i}} = c_{x}|_{P_{i}} for some x ∈ S. By [1, A8], we have α = c_{x} ∘ c_{z} for some z ∈ Z(P_{i}), and so α ∼_{(S,S)} id_{S}. Thus a_{i,0} = 1. Therefore, we have c_{1}^{(i,j)} = c_{0} + pc_{1}^{(i)}. □

5. Minimal bisets for Ruiz-Viruel exotic systems

In this section, we determine minimal left characteristic bisets for Ruiz-Viruel exotic fusion systems [6]. In particular, we show that every Ruiz-Viruel exotic fusion system has a unique minimal left characteristic biset, and that in fact it is also a unique minimal right characteristic biset.

First we review some of the results on the Ruiz-Viruel exotic fusion systems and fix notations. Throughout this section, let p be an odd prime, and let S be an extraspecial p-group of order p^{3} and exponent p. Then S has center Z(S) = ⟨z⟩ of order p, and exactly p + 1 subgroups

V_{i} = ⟨z, u_{i}⟩ (0 ≤ i ≤ p)

of order p^{2}, which are elementary abelian and centric in S. Therefore, for any fusion system F on S, all the V_{i} are F-centric. Let us fix the ordered basis (z, u_{i}) for V_{i} as an 𝔽_{p}- vector space. Via this ordered basis we get an isomorphism Aut(V_{i}) ≅ GL_{2}(p). Then

U_{i} := Aut_{S}(V_{i}) ≅ \{(\begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix})\} =: U ≤ GL_{2}(p). Set some subgroups of GL_{2}(p) as

B := N_{GL_{2}(p)}(U) = \{(\begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix})\} , T := \{(\begin{matrix}∗ & 0 \\ 0 & ∗ \end{matrix})\} , R := \{(\begin{matrix}0 & ∗ \\ ∗ & 0 \end{matrix})\} ,

and denote by B_{i}, T_{i} and R_{i} their isomorphic images in Aut_{F}(V_{i}). Note that T ⊔ R is a set of representatives of U-U-double cosets in GL_{2}(p).

Let F be a saturated fusion system on S. By the extension axiom, we have

Aut_{F}(V_{i}) ∩ B_{i} = Aut_{F}^{e}(V_{i}),
Aut_{F}(V_{i}) − B_{i} = Aut_{F}^{n}(V_{i}),

for 0 ≤ i ≤ p. Note that Aut_{F}(V_{i}) is closed under left and right Aut_{S}(V_{i})-action in Aut(V_{i}). Thus Aut_{F}(V_{i}) ∩ T_{i} is a set of representatives of extendable F-automorphisms of V_{i}, and Aut_{F}(V_{i})∩R_{i} is a set of representatives of nonextendable F-automorphisms of V_{i}, both with respect to S-S-conjugacy.

The subgroup V_{i} is F-radical if and only if Aut_{F}(V_{i}) contains SL_{2}(p). In this case, we have

Aut_{F}(V_{i}) ≅ SL_{2}(p) : r_{i} for some r_{i}|(p − 1).

Then

Aut_{F}(V_{i}) ∩ T_{i} = \{(\begin{matrix}k & 0 \\ 0 & l \end{matrix})_{V_{i}} | k, l ∈ 𝔽_{p}^{×}, kl ∈ ⟨μ^{(p−1)/r_{i}}⟩\} ,

Aut_{F}(V_{i}) ∩ R_{i} = \{(\begin{matrix}0 & l \\ k & 0 \end{matrix})_{V_{i}} | k, l ∈ 𝔽_{p}^{×}, −kl ∈ ⟨μ^{(p−1)/r_{i}}⟩\} ,

where 𝔽_{p}^{×} = ⟨μ⟩. In particular, we have

|Aut_{F}(V_{i}) ∩ T_{i}| = |Aut_{F}(V_{i}) ∩ R_{i}| = (p − 1)r_{i}.

For later use, set Λ_{i}^{e} = {(k, l) | kl ∈ ⟨μ^{(p−1)/r_{i}}⟩}, Λ_{i}^{n} = {(k, l) | −kl ∈ ⟨μ^{(p−1)/r_{i}}⟩}.

We will consider saturated fusion systems F on S such that all V_{i} are F-radical. In particular, all exotic fusion systems on S have this property. We list all of them in the following table.

p Out_{F}(S) |V_{i}^{F}| r_{i} Group
3  D_{8}  2, 2 2, 2 ^{2}F_{4}(2)′
3 SD_{16}  4  2  J_{4}
5  4S_{4}  6  4  T h
7 D_{16} × 3 4, 4 2, 2
7 6^{2} : 2 6, 2 2, 6
7 SD_{32} × 3 8  2

Here |V_{i}^{F}| denotes the size of the F-conjugacy class of subgroups of S containing V_{i}. In the columns |V_{i}^{F}| and r_{i}, numbers corresponding to different F-conjugacy classes are separated by commas, and they are in the same order in these two columns. The column “Group” indicates finite groups realizing corresponding saturated fusion systems. An empty entry in this column means that the corresponding saturated fusion system is exotic. For example, when p = 7, the saturated fusion system on S with Out_{F}(S) ≅ 6^{2} : 2 has exactly two Fconjugacy classes of subgroups of order 7^{2}. One of them consists of 6 subgroups of order 7^{2} and their automorphism groups are isomorphic to SL_{2}(7) : 2; the other consists of 2 subgroups of order 7^{2} and their automorphism groups are isomorphic to SL_{2}(7) : 6 = GL_{2}(7). This saturated fusion system is exotic.

One observes that the numerical relation |Out_{F}(S)| = (p−1)|V_{i}^{F}|r_{i} is satisfied in all cases. This is not a coincidence, and indeed a special case of a more general phenomenon.

Lemma 5.1. Let F be a saturated fusion system on S. Suppose that V_{i} is F-radical for some 1 ≤ i ≤ p. Then we have

f := \frac{|Out_{F}(S)|}{p − 1} = |V_{i}^{F}|r_{i} = |\{[α] ∈ Out_{F}(S) | α(z) = z^{m}\}|

for any m ∈ 𝔽_{p}^{×}.

Proof. By Proposition 4.4, we have

|Out_{F}(S)| = |Hom_{F}^{e}(V_{i}, S)/ ∼_{(S,S)} |.

We claim that

|Hom_{F}^{e}(V_{i}, S)/ ∼_{(S,S)} | = |V_{i}^{F}| · |Aut_{F}^{e}(V_{i})/ ∼_{(S,S)} |.

Indeed, if V_{i} ≅_{F} V_{j}, then there is α ∈ Aut_{F}(S) such that α(V_{i}) = V_{j} by Alperin’s fusion theorem. Then composition with α induces a bijection

Aut_{F}^{e}(V_{i})/ ∼_{(S,S)}⟶^{≃} {φ ∈ Hom_{F}^{e}(V_{i}, S) | φ(V_{i}) = V_{j}}/ ∼_{(S,S)} .

Since this holds for every j with V_{i} ≅_{F} V_{j}, the claim follows. Since

|Aut_{F}^{e}(V_{i})/ ∼_{(S,S)} | = |Aut_{F}(V_{i}) ∩ T_{i}| = (p − 1)r_{i},

as we have observed previously in this section, the first equaltiy of the proposition follows.

Now consider the restriction map Aut_{F}(S) → Aut_{F}(Z(S)). Since inner automorphisms of S acts as the identity map on Z(S), this restriction map induces the group homomorphism

Out_{F}(S) → Aut_{F}(Z(S)), [α] ↦ α|_{Z(S)}.

For each m ∈ 𝔽_{p}^{×}, we have (\begin{matrix}m & 0 \\ 0 & m^{−1} \end{matrix})_{V_{i}} ∈ Aut_{F}^{e}(V_{i}) as Aut_{F}^{e}(V_{i}) contains SL_{2}(p), and hence

there exists α ∈ Aut_{F}(S) such that α(z) = z^{m}. For each pair m, n ∈ 𝔽_{p}^{×}, choose β ∈ Aut_{F}(S) such that β(z) = z^{nm^{−1}}. Then composition with β defines the bijection

{[α] ∈ Out_{F}(S) | α(z) = z^{m}} → {[α] ∈ Out_{F}(S) | α(z) = z^{n}}.

Thus we get the second equality of the proposition. □

Now suppose that F is a saturated fusion system on S such that all V_{i} are F-radical. Let us enumerate F-morphisms between subgroups of S of index p up to S-S-conjugacy. Suppose V_{i} ≅_{F} V_{j}. By Alperin’s fusion theorem, there is α ∈ Aut_{F}(S) such that α(V_{i}) = V_{j}. Then α(Z(S)) = Z(S) and α(u_{i}) ∈ u_{j}^{k}⟨z⟩ for some 0 < k < p. We normalize the notations for the sake of convenience as follows. Since Aut_{F}(V_{j}) ≥ SL_{2}(p), by composing α with some extendable F-automorphism of V_{j} and replacing u_{j} by a suitable power of u_{j}, we obtain α_{i,j} ∈ Aut_{F}(S) sending z to z and u_{i} to u_{j}. We do this one by one for each F-conjugacy class of the V_{i} in a compatible way. By composing elements of Aut_{F}(V_{i}) ∩ R_{i} with α_{i,j}, we obtain nonextendable F-isomorphisms

φ_{i,j}^{k,l}: V_{i} → V_{j}

sending z to u_{j}^{k} and u_{i} to z^{l} where (k, l) ∈ Λ_{i}^{n}. The set

{φ_{i,j}^{k,l} | V_{i} ≅_{F} V_{j}, (k, l) ∈ Λ_{i}^{n}}

is a set of representatives of nonextendable isomorphisms between index p subgroups of S with respect to S-S-conjugacy. Similarly we have a set

{ψ_{i,j}^{k,l} | V_{i} ≅_{F} V_{j}, (k, l) ∈ Λ_{i}^{e}}

of representatives of extendable isomorphisms between index p subgroups of S with respect to S-S-conjugacy, where

ψ_{i,j}^{k,l}: V_{i} → V_{j}

sends z to z^{k} and u_{i} to u_{j}^{l}.

Then, by Proposition 4.5, if X is a right characteristic biset for F we have

X_{0} ≅ c_{0} ∐_{[α]∈Out_{F} (S)}(S × S)/Δ_{S}^{α}, (X_{0})

X_{1} ≅ ∐_{\begin{matrix}V_{i}≅_{F} V_{j} \\ (k,l)∈Λ_{i}^{e} \end{matrix}} c_{1}^{(i)}(S × S)/Δ_{V_{i}}^{ψ_{i,j}^{k,l}} ⨿ ∐_{\begin{matrix}V_{i}≅_{F} V_{j} \\ (k,l)∈Λ_{i}^{n} \end{matrix}}(c_{0} + pc_{1}^{(i)})(S × S)/Δ_{V_{i}}^{φ_{i,j}^{k,l}}, (X_{1})

for some nonnegative integers c_{0} and c_{1}^{(i)} such that p ∤ c_{0}.

Now we consider F-maps between subgroups of order p. Since we are assuming that all V_{i} are F-radical, all elements of S of order p are F-conjugate. For elements ξ, ζ ∈ S of order p, let Δ_{ξ}^{ζ} := Δ_{⟨ξ⟩}^{φ} where φ: ⟨ξ⟩ → ⟨ζ⟩ sends ξ to ζ. Then the F-graph subgroups of S × S (i.e. subgroups of S × S of the form Δ_{Q}^{φ} for some Q ≤ S and φ ∈ Hom_{F}(Q, S)) of order p up to S × S-conjugacy are

Δ_{ξ}^{ζ} where ξ ∈ {z, u_{i}}_{0≤i≤p}, ζ ∈ {z^{m}, u_{j}^{m}}_{0≤j≤p,1≤m≤p−1}.

Thus

X_{2} ≅ ∐_{1≤m≤p−1} c_{2}^{(z,z^{m})}(S × S)/Δ_{z}^{z^{m}} ⨿ ∐_{\begin{matrix}0≤i≤p \\ 1≤m≤p−1 \end{matrix}} c_{2}^{(u_{i},z^{m})}(S × S)/Δ_{u_{i}}^{z^{m}} (X_{2})

⨿ ∐_{\begin{matrix}0≤j≤p \\ 1≤m≤p−1 \end{matrix}} c_{2}^{(z,u_{j}^{m})}(S × S)/Δ_{z^{m}}^{u_{j}^{m}} ⨿ ∐_{\begin{matrix}0≤i,j≤p \\ 1≤m≤p−1 \end{matrix}} c_{2}^{(u_{i},u_{j}^{m})}(S × S)/Δ_{u_{i}}^{u_{j}^{m}}

for some nonnegative integers c_{2}^{(z,z^{m})}, c_{2}^{(u_{i},z^{m})}, c_{2}^{(z,u_{j}^{m})} and c_{2}^{(u_{i},u_{j}^{m})}.

Now we can state the main result of this section precisely.

Theorem 5.2. Let p be an odd prime and let S be an extraspecial group of order p^{3} and exponent p. Let F be a saturated fusion system on S such that all subgroups V_{i} of S of order p^{2} are F-radical. Then there exists a unique minimal right characteristic biset X for F given as follows:

X_{0} ≅ ∐_{[α]∈Out_{F} (S)}(S × S)/Δ_{S}^{α},

X_{1} ≅ ∐_{\begin{matrix}V_{i}≅_{F} V_{j} \\ (k,l)∈Λ_{i}^{n} \end{matrix}}(S × S)/Δ_{V_{i}}^{φ_{i,j}^{k,l}},

X_{2} ≅ ∐_{\begin{matrix}V_{i}≅_{F} V_{j} \\ 1≤m≤p−1 \end{matrix}}(f − r_{i})(S × S)/Δ_{u_{i}}^{u_{j}^{m}} ⨿ ∐_{\begin{matrix}V_{i}≅_{F̸} V_{j} \\ 1≤m≤p−1 \end{matrix}} f(S × S)/Δ_{u_{i}}^{u_{j}^{m}}.

Moreover, X is a unique minimal left characteristic biset for F and hence a unique minimal characteristic biset for F.

Proof. The proof is done by explicitly computing all right characteristic bisets for F and showing that there is the smallest one among them. For this, let X be an arbitrary right characteristic biset for F. The top two layers X_{0} and X_{1} of X are given as in (X_{0}) and (X_{1}) by Proposition 4.5. Write X_{2} as in (X_{2}). The coefficients of X_{2} are completely determined by the system of equations

|X^{Δ_{ξ}^{ζ}}| = |X^{Δ_{ξ}^{ξ}}|, ξ ∈ {z, u_{i}}_{0≤i≤p}, ζ ∈ {z^{m}, u_{j}^{m}}_{0≤j≤p,1≤m≤p−1}.

To solve this system of equations, first compute the numbers of fixed points of Δ_{ξ}^{ζ} on transitive subbisets of X using Lemma 3.4 and Lemma 3.5:

|((S × S)/Δ_{S}^{α})^{Δ_{ξ}^{ζ}}| = \begin{cases}p^{3}, & if α(ξ) = ζ ∈ ⟨z⟩ \\ p^{2}, & if α(ξ) ∈ ζ⟨z⟩ ≠ ⟨z⟩ \\ 0, & otherwise, \end{cases}

|((S × S)/Δ_{V_{i}}^{ψ_{i,j}^{k,l}})^{Δ_{ξ}^{ζ}}| = \begin{cases}p^{4}, & if (ξ, ζ) = (z, z^{k}) \\ p^{3}, & if (ξ, ζ) = (u_{i}, u_{j}^{l}) \\ 0, & otherwise, \end{cases}

|((S × S)/Δ_{V_{i}}^{φ_{i,j}^{k,l}})^{Δ_{ξ}^{ζ}}| = \begin{cases}p^{3}, & if (ξ, ζ) = (z, u_{j}^{k}) or (ξ, ζ) = (u_{i}, z^{l}) \\ p^{2}, & if ξ = u_{i}, ζ ∈ ⟨u_{j}⟩ \\ 0, & otherwise, \end{cases}

|((S × S)/Δ_{ξ}^{ζ})^{Δ_{ξ}^{ζ}}| = \begin{cases}p^{5}, & if ξ, ζ ∈ ⟨z⟩ \\ p^{4}, & if ξ ∈ ⟨z⟩ ∌ ζ or ξ ∉ ⟨z⟩ ∋ ζ \\ p^{3}, & if ξ ∉ ⟨z⟩ ∌ ζ. \end{cases}

Adding up those numbers, we get the following result using Lemma 5.1:

|X_{0}^{Δ_{z}^{z^{m}}}| = p^{3}fc_{0},  |X_{0}^{Δ_{u_{i}}^{z^{m}}}| = |X_{0}^{Δ_{z}^{u^{m}_{j}}}| = 0,
|X_{0}^{Δ_{u_{i}}^{u^{m}_{j}}}| = c_{0}p^{2}|{[α] ∈ Out_{F}(S) | α(u_{i}) ∈ u_{j}^{m}⟨z⟩}|

= \begin{cases}p^{2}r_{i}c_{0}, & if V_{i} ≅_{F} V_{j} \\ 0, & otherwise, \end{cases} ,

|X_{1}^{Δ_{z}^{z^{m}}}| = p^{4}f ∑_{0≤i≤p} c_{1}^{(i)},  |X_{1}^{Δ_{u_{i}}^{z^{m}}}| = p^{3}fc_{0} + p^{4}fc_{1}^{(i)}

|X_{1}^{Δ_{z}^{u^{m}_{j}}}| = p^{3}fc_{0} + p^{4}r_{j} ∑_{V_{i}≅_{F} V_{j}} c_{1}^{(i)},

|X_{1}^{Δ_{u_{i}}^{u^{m}_{j}}}| = \begin{cases}p^{3}r_{i}c_{1}^{(i)} + p^{2}(p − 1)r_{i}(c_{0} + pc_{1}^{(i)}), & if V_{i} ≅_{F} V_{j} \\ 0, & otherwise. \end{cases}

The second equality for |X_{0}^{Δ_{u_{i}}^{u^{m}_{j}}}| needs more explanation. If there is α ∈ Aut_{F}(S) such that α(u_{i}) ∈ u_{j}^{m}⟨z⟩, then α(V_{i}) = V_{j}. Thus if V_{i} ≅_{F̸} V_{j}, then {α ∈ Out_{F}(S) | α(u_{i}) ∈ u_{j}^{m}⟨z⟩} = ∅. Suppose V_{i} ≅_{F} V_{j}. Then by the extension axiom, there is α ∈ Aut_{F}(S) such that α(V_{i}) = V_{j}. Then α(u_{i}) ∈ u_{j}^{l}⟨z⟩ for some l ≠ 0. Since Aut_{F}(V_{i}) ≥ SL_{2}(p), we may modify α so that α(u_{i}) = u_{j}^{m}. Then α(z) = z^{k} for some k and there are r_{i} choices for k. By [1, A8], each choice of k determines α up to S-S-conjugacy.

|X_{≤1}^{Δ_{z}^{z^{m}}}| = p^{3}fc_{0} + p^{4}f ∑_{0≤i≤p} c_{1}^{(i)},

|X_{≤1}^{Δ_{u_{i}}^{z^{m}}}| = p^{3}fc_{0} + p^{4}fc_{1}^{(i)},
|X_{≤1}^{Δ_{z}^{u^{m}_{j}}}| = p^{3}fc_{0} + p^{4}r_{j} ∑_{V_{i}≅_{F} V_{j}} c_{1}^{(i)},

|X_{≤1}^{Δ_{u_{i}}^{u^{m}_{j}}}| = \begin{cases}p^{3}r_{i}c_{0} + p^{4}r_{i}c_{1}^{(j)}, & if V_{i} ≅_{F} V_{j} \\ 0, & otherwise. \end{cases}

Now |X^{Δ_{ξ}^{ζ}}| = |X^{Δ_{ξ}^{ξ}}| and |X^{Δ_{ξ}^{ζ}}| = |X_{≤1}^{Δ_{ξ}^{ζ}}| + c_{2}^{(ξ,ζ)}|((S × S)/Δ_{ξ}^{ζ})^{Δ_{ξ}^{ζ}}|. Thus

c_{2}^{(ξ,ζ)}|((S × S)/Δ_{ξ}^{ζ})^{Δ_{ξ}^{ζ}}| = c_{2}^{(ξ,ξ)}|((S × S)/Δ_{ξ}^{ξ})^{Δ_{ξ}^{ξ}}| + (|X_{≤1}^{Δ_{ξ}^{ξ}}| − |X_{≤1}^{Δ_{ξ}^{ζ}}|).

Then

|X^{Δ_{z}^{z^{m}}}| = |X^{Δ_{z}^{z}}| : c_{2}^{(z,z^{m})} = c_{2}^{(z,z)} =: c_{2}^{(z)},
|X^{Δ_{u_{i}}^{u^{m}_{j}}}| = |X^{Δ_{u_{i}}^{u_{i}}}| : c_{2}^{(u_{i},u_{j}^{m})} = c_{2}^{(u_{i},u_{i})} =: c_{2}^{(u_{i})}(if V_{i} ≅_{F} V_{j}),
|X^{Δ_{u_{i}}^{u^{m}_{j}}}| = |X^{Δ_{u_{i}}^{u_{i}}}| : c_{2}^{(u_{i},u_{j}^{m})} = r_{i}c_{0} + pr_{i}c_{1}^{(i)} + c_{2}^{(u_{i})}(if V_{i} ≅_{F̸} V_{j}),
|X^{Δ_{z}^{u^{m}_{j}}}| = |X^{Δ_{z}^{z}}| : c_{2}^{(z,u_{j}^{m})} = pc_{2}^{(z)} + (f − r_{j}) ∑_{V_{i}≅_{F} V_{j}} c_{1}^{(i)} + f ∑_{V_{i}≅_{F̸} V_{j}} c_{1}^{(i)},

|X^{Δ_{u_{i}}^{z^{m}}}| = |X^{Δ_{u_{i}}^{u_{i}}}| : pc_{2}^{(u_{i},z^{m})} = c_{2}^{(u_{i})} − (f − r_{i})c_{0} − p(f − r_{i})c_{1}^{(i)}.

The only restriction on coefficients (other than p ∤ c_{0}) is that they must be nonnegative because X is a genuine biset:

p ∤ c_{0} ≥ 1, c_{1}^{(i)} ≥ 0, c_{2}^{(z)} ≥ 0, c_{2}^{(u_{i})} ≥ (f − r_{i})c_{0} + p(f − r_{i})c_{1}^{(i)}.

Thus X is minimal if and only if

c_{0} = 1, c_{1}^{(i)} = 0, c_{2}^{(z)} = 0, c_{2}^{(u_{i})} = (f − r_{i}),

and in this case

c_{2}^{(u_{i},u_{j}^{m})} = \begin{cases}f − r_{i}, & if V_{i} ≅_{F} V_{j} \\ f, & if V_{i} ≅_{F̸} V_{j}, \end{cases}
c_{2}^{(u_{i},z^{m})} = c_{2}^{(z,u_{j}^{m})} = 0.

The last statement follows from observing that the unique minimal right characteristic biset X is isomorphic to its opposite X^{op}. □

Remark 5.3. Let us look more closely at numerical relations in the above minimal characteristic biset X. Let e := e(X) = |X|/|S| and e_{i} := |X_{i}|/|S| (i = 0, 1, 2) so that e = e_{0} +e_{1} +e_{2}.

Let d_{i} = e_{i}/p^{i} (i = 0, 1, 2). In other words, d_{i} is the number of transitive subbisets appearing in X_{i}. Then, using Lemma 5.1, we see that

d_{0} = |Out_{F}(S)|,
d_{1} = ∑_{0≤i≤p} |V_{i}^{F}||Λ_{i}^{n}| = ∑_{0≤i≤p} |V_{i}^{F}|(p − 1)r_{i} = (p + 1)|Out_{F}(S)|,
d_{2} = ∑_{0≤i≤p}(f − r_{i})|V_{i}^{F}|(p − 1) + ∑_{0≤i≤p} f((p + 1) − |V_{i}^{F}|)(p − 1) = p(p + 1)|Out_{F}(S)|.

Thus

e = d_{0} + pd_{1} + p^{2}d_{2} = \frac{p^{5} − 1}{p − 1}|Out_{F}(S)|.

We summarize all these numbers in the following table.

p Out_{F}(S) |V_{i}^{F}| r_{i} f d_{0} d_{1} d_{2} e(X) Group
3  D_{8}  2, 2 2, 2 4 8 32 96  968  ^{2}F_{4}(2)′
3 SD_{16}  4  2 8 16 64 192 1936  J_{4}
5  4S_{4}  6  4 24 96 576 2880 74976  T h
7 D_{16} × 3 4, 4 2, 2 8 48 384 2688 134448 ≤ 425744
7 6^{2} : 2 6, 2 2, 6 12 72 576 4032 201672 ≤ 638620
7 SD_{32} × 3 8  2 16 96 768 5376 268896 ≤ 851496

Numbers in the column “Group” are the upper bounds of the exoticity indices obtained by minimal characteristic bisets.

6. Finite groups realizing Ruiz-Viruel exotic fusion systems

In this section, we analyze how the finite group G realizes fusion more closely, and as a consequence prove Theorem 1.2. For this, the following basic double coset formula concerning restriction of bisets will be useful.

Lemma 6.1 ([4, §2]). Let S be a group. Let Q, R ≤ S and let φ: Q → S, ψ: R → S be injective group homomorphisms. Then

ψ((S × S)/Δ_{Q}^{φ}) ≅ ∐_{t∈[φ(R)\S/Q]}(R × S)/Δ_{ψ^{−1}(^{t}Q)}^{φ∘c_{t}^{−1}∘ψ}

Let F be a saturated fusion system on a finite p-group S with a left characteristic biset X. Let G = Aut(_{1}X), i.e. the group of bijections of X preserving the right S-action, and define ι: S → G by ι(u)(x) = ux for u ∈ S, x ∈ X. By our previous work [2], we know that ι is injective and F ≅ F_{ι(S)}(G). More precisely, let R ≤ S and let ψ ∈ Hom_{F}(R, S). Then the left F-stability of X implies that _{R}X and _{ψ}X are isomorphic as R-S-bisets. Then any isomorphism of R-S-bisets g: _{R}X ⟶^{∼} _{ψ}X viewed as an element of G realizes ψ in the sense that ψ(u) = gug^{−1} for all u ∈ R.

In fact, we can be more precise about the element g ∈ G. Let us recall the notations in [2]. Write

X = ∐_{i=1}^{n} S ×_{(Q_{i},φ_{i})} S

where Q_{i} ≤ S, φ_{i} ∈ Hom_{F}(Q_{i}, S) for 1 ≤ i ≤ n. For each i, fix a set {t_{ij}}_{j∈J_{i}} of representatives of the left cosets of Q_{i} in S. Set J = ∐_{i=1}^{n} J_{i}. For each 1 ≤ i ≤ n, we have a decomposition of right S-sets

S ×_{(Q_{i},φ_{i})} S = ∐_{j∈J_{i}}⟨t_{ij}, S⟩,

where ⟨t_{ij}, S⟩ := {⟨t_{ij}, x⟩ | x ∈ S} is a regular right S-set for all j ∈ J_{i}. So we have

G := Aut(_{1}X) ≅ S ≀ Sym(J),

where Sym(J) denotes the symmetric group on the set J.

Let \overline{G} be the image of G in Sym(J) and use bar notation for images of subgroups or elements of G in Sym(J). Then each J_{i} is an \overline{S}-orbit with |J_{i}| = |S : Q_{i}|. In particular, we have |J_{i}| = 1 precisely when the corresponding F-morphism φ_{i}: Q_{i} → S is an automorphism of S. Suppose that the Q_{i} are indexed so that |Q_{i}| ≥ |Q_{i+1}| for all i. Then there are integers 1 ≤ n_{0} ≤ n_{1} ≤ n_{2} ≤ ⋯ ≤ n such that |S : Q_{i}| = p^{r} for all i with n_{r} ≤ i < n_{r+1}. With this notation, \overline{S}-orbits of J can be represented schematically as follows:

\overbrace{∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗}^{[φ_{i}](n_{0}≤i<n_{1})} \overbrace{(∗ ∗ ∗) ⋯ (∗ ∗ ∗)}^{[φ_{i}](n_{1}≤i<n_{2})} \overbrace{(∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗) ⋯ (∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗)}^{[φ_{i}](n_{2}≤i<n_{3})} ⋯

Each asterisk in the above diagram represents an element of the index set J. Left \overline{S}-orbits of the asterisks are denoted by brackets, except for the first eight asterisks, each of which is a single left \overline{S}-orbit. Each left \overline{S}-orbit corresponds to a transitive subbiset of X of the form S ×_{(Q_{i},φ_{i})} S, and indexed by the S-S-conjugacy class of F-morphisms [φ_{i}] determining it. In a left \overline{S}-orbit indexed by the S-S-conjugacy class of φ_{i}: Q_{i} → S, each asterisk is indexed by a left coset of Q_{i} in S.

Remark 6.2. With the above notation in mind, suppose R ≤ S, ψ ∈ Hom_{F}(R, S). Then RX ≅ _{ψ}X, and by Lemma 6.1,

ψ((S × S)/Δ_{Q_{i}}^{φ_{i}}) ≅ ∐_{t∈[ψ(R)\S/Q_{i}]}(R × S)/Δ_{ψ^{−1}(^{t}Q_{i})}^{φ_{i}∘c_{t}^{−1}∘ψ},

R((S × S)/Δ_{Q_{i}}^{φ_{i}}) ≅ ∐_{s∈[R\S/Q_{i}]}(R × S)/Δ_{^{s}Q_{i}}^{φ_{i}∘c_{s}^{−1}}.

So one can find a bijective correspondence μ between the above transitive subbisets of _{R}X and _{ψ}X and a permutation μ̃ ∈ Sym(J) which respects μ, and construct an isomorphism of R-S-bisets g: _{R}X ⟶^{∼} _{ψ}X which, viewed as an element of G, realizes ψ and such that \overline{g} = μ̃.

Our focus is the behavior of the permutations \overline{g} ∈ Sym(J) associated with elements g ∈ G realizing F-automorphisms of the F-essential subgroups of S. First we consider F-automorphisms of S.

Proposition 6.3. Let F be a saturated fusion system on a finite p-group S with a left characteristic biset X. Let G = Aut(_{1}X) and use the notations introduced above. Then the image of Aut_{F}(S) in Sym(J) is transitive on J^{(0)} := ∐{J_{i} | |J_{i}| = 1}.

Proof. J^{(0)} is the union of singleton left \overline{S}-orbits of J, each corresponding to an outer automorphism of S. By Remark 6.2, for any α ∈ Aut_{F}(S) there is g ∈ G which realizes α and such that \overline{g} ∈ Sym(J) sends the element of J^{(0)} corresponding to [id_{S}] to the element of J^{(0)} corresponding to [α] because

α((S × S)/Δ_{S}^{φ_{i}}) ≅ (S × S)/Δ_{S}^{φ_{i}∘α}

for Q_{i} = S. □

Now we can prove Theorem 1.2, which we restate here for the convenience of the reader.

Theorem 6.4. Let p be an odd prime and let S be an extraspecial group of order p^{3} and exponent p. Let F be a saturated fusion system on S such that all subgroups of S of order p^{2} are F-radical. Let X be the minimal left characteristic biset X for F given by Theorem 1.1. Let G = Aut(_{1}X) ≅ S ≀ Σ_{e(X)} and let H be the subgroup of G generated by the normalizers of the F-essential subgroups of S. Then the image of H in Σ_{e(X)} is a transitive subgroup of Σ_{e(X)}.

Proof. We keep the notations of this section and §5. We have

H = ⟨Aut_{F}(S), Aut_{F}(V_{i}) | 1 ≤ i ≤ n⟩.

The index set J of the symmetric group Σ_{e(X)} can be represented schematically as follows:

\overbrace{∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗}^{[α]} \overbrace{(∗ ∗ ∗) ⋯ (∗ ∗ ∗)}^{[φ_{0,j}^{k,l}]} ⋯ \overbrace{(∗ ∗ ∗) ⋯ (∗ ∗ ∗)}^{[φ_{p,j}^{k,l}]} \overbrace{(∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗) ⋯ (∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗)}^{[u_{0}→u_{j}^{m}]} ⋯

By Proposition 6.3, we know that the image of Aut_{F}(S) in Sym(J) is transitive on the union of singleton left \overline{S}-orbits of J corresponding to outer automorphisms of S.

Now let φ = φ_{0,0}^{r,s} ∈ Aut_{F}(V_{0}) for some (r, s) ∈ Λ_{0}^{n}. Recall that Z(S) = ⟨z⟩, V_{i} = ⟨z, u_{i}⟩ for 0 ≤ i ≤ p. Let v ∈ S such that [v, u_{0}] = z so that v^{t}u_{0}v^{−t} = z^{t}u_{0} for all t. Then we have sets of left coset representatives as follows:

[S/V_{0}] = {v^{t}}_{0≤t≤p−1}, [S/V_{i}] = {u_{0}^{t}}_{0≤t≤p−1}(i ≠ 0),
[S/⟨u_{0}⟩] = {v^{t}z^{w}}_{0≤t,w≤p−1}, [S/⟨u_{i}⟩] = {u_{0}^{t}z^{w}}_{0≤t,w≤p−1}(i ≠ 0).

We have _{V_{0}}X ≅ _{φ}X as V_{0}-S-bisets, and for each transitive subbiset of X, the two restrictions to V_{0} via the inclusion V_{0} ↪ S and via φ are given as follows:

Y  _{V_{0}}Y  _{φ}Y
(S × S)/Δ_{S}^{α}  (V_{0} × S)/Δ_{V_{0}}^{α|_{V_{0}}}  (V_{0} × S)/Δ_{V_{0}}^{α∘φ}
(S × S)/Δ_{V_{0}}^{φ_{0,j}^{k,l}}  ∐_{t=0}^{p−1}(V_{0} × S)/Δ_{V_{0}}^{φ_{0,j}^{k,l}∘c_{v^{t}}} ∐_{t=0}^{p−1}(V_{0} × S)/Δ_{V_{0}}^{φ_{0,j}^{k,l}∘c_{v^{t}}∘φ}
(S × S)/Δ_{V_{i}}^{φ_{i,j}^{k,l}}(i ≠ 0)  (V_{0} × S)/Δ_{⟨z⟩}^{u_{j}^{rk}}  (V_{0} × S)/Δ_{⟨u_{0}⟩}^{u_{j}^{rl}}
(S × S)/Δ_{⟨u_{0}⟩}^{u_{j}^{m}}  ∐_{t=0}^{p−1}(V_{0} × S)/Δ_{⟨z^{t}u_{0}⟩}^{u_{j}^{m}} ∐_{t=0}^{p−1}(V_{0} × S)/Δ_{⟨z^{r^{−1}}u_{0}^{s^{−1}t}⟩}^{u_{j}^{m}}
(S × S)/Δ_{⟨u_{i}⟩}^{u_{j}^{m}}(i ≠ 0)  V_{0} × S  V_{0} × S

By comparing the size, we see that the transitive V_{0}-S-subbisets in the first two rows of the above table are isomorphic to each other, and those in the third and fourth rows are isomorphic to each other. Among the transitive subbisets in the first two rows, we can determine isomorphic pairs by the extendability of the associated maps. In the column _{V_{0}}Y, the maps α|_{V_{0}} are extendable, while φ_{0,j}^{k,l} ∘ c_{v^{t}} are nonextendable. On the other hand, in the column _{φ}Y, the maps α ∘ φ are nonextendable, while the maps φ_{0,j}^{k,l} ∘ c_{v^{t}} ∘ φ are extendable if t = 0 and nonextentable if t ≠ 0. Thus

(V_{0} × S)/Δ_{V_{0}}^{α|_{V_{0}}} ≅ (V_{0} × S)/Δ_{V_{0}}^{φ_{0,j}^{k,l}∘φ},
(V_{0} × S)/Δ_{V_{0}}^{α∘φ} ≅ (V_{0} × S)/Δ_{V_{0}}^{φ_{0,j}^{k,l}},
(V_{0} × S)/Δ_{V_{0}}^{φ_{0,j}^{k,l}∘c_{v^{t}}} ≅ (V_{0} × S)/Δ_{V_{0}}^{φ_{0,j}^{k,l}∘c_{v^{t′}} ∘φ} (t, t′ ≠ 0).

Similarly,

(V_{0} × S)/Δ_{⟨z⟩}^{u_{j}^{rk}} ≅ (V_{0} × S)/Δ_{⟨z^{r^{−1}}⟩}^{u_{j}^{m}},
(V_{0} × S)/Δ_{⟨u_{0}⟩}^{u_{j}^{m}} ≅ (V_{0} × S)/Δ_{⟨u_{0}⟩}^{u_{j}^{rl}},
(V_{0} × S)/Δ_{⟨z^{t}u_{0}⟩}^{u_{j}^{m}} ≅ (V_{0} × S)/Δ_{⟨z^{r^{−1}}u_{0}^{s^{−1}t′}⟩}^{u_{j}^{m}} (t, t′ ≠ 0).

Note that the numerical relations observed in Remark 5.3 guarantees that the numbers of transitive V_{0}-S-subbisets in the above list of isomorphisms do match up. This shows that for each φ ∈ Aut_{F}(V_{0}), there is an element g ∈ N_{G}(V_{0}) which realizes φ and such that \overline{g} ∈ Sym(J) respects the above list of isomorphisms of transitive V_{0}-S-bisets. Schematically, \overline{g} can be represented as follows:

\overbrace{• • • • • • • •}^{[α]} \overbrace{(∘ ∗ ∗) ⋯ (∘ ∗ ∗)}^{[φ_{0,j}^{k,l}]} | ⋯ \overbrace{(• • •) ⋯ (• • •)}^{[φ_{p,j}^{k,l}]} \overbrace{(∘ ∘ ∘ ∗ ∗ ∗ ∗ ∗ ∗) ⋯ (∘ ∘ ∘ ∗ ∗ ∗ ∗ ∗ ∗)}^{[u_{0}→u_{j}^{m}]} | ⋯

\overbrace{∘ ∘ ∘ ∘ ∘ ∘ ∘ ∘}^{[α]} \overbrace{(• ∗ ∗) ⋯ (• ∗ ∗)}^{[φ_{0,j}^{k,l}]} | ⋯ \overbrace{(∘ ∘ ∘) ⋯ (∘ ∘ ∘)}^{[φ_{p,j}^{k,l}]} \overbrace{(• • • ∗ ∗ ∗ ∗ ∗ ∗) ⋯ (• • • ∗ ∗ ∗ ∗ ∗ ∗)}^{[u_{0}→u_{j}^{m}]} | ⋯

where \overline{g} sends solid dots, circles and asterisks in the first row to solid dots, circles and asterisks in the second row, respectively, inside each compartment divided by vertical lines. In particular, the images of Aut_{F}(V_{i})(1 ≤ i ≤ n) in Sym(J) merge left \overline{S}-orbits of J of different sizes into a single \overline{H}-orbit. Together with the transitivity of the image of Aut_{F}(S) in Sym(J) on the singleton left S-orbits, this shows that \overline{H} is transitive on the whole index set J. □

Appendix A. Coefficients of characteristic idempotents

In this section, we determine some coefficients of characteristic idempotents. The reason why we include this section in the paper is two-fold. First, as characteristic idempotents are determined by imposing only one additional idempotency condition [4, 5.5, 5.6] on virtual characteristic bisets with coefficients in ℤ_{(p)}, this can be done without much effort using Proposition 4.5 and Theorem 5.2. More importantly, the information contained in this section can be useful for testing conjectures on the behavior of characteristic idempotents.

We follow Ragnarsson’s notation [4, 5] in this section. Let F be a saturated fusion system on a finite p-group S. Let ω = ω_{F} be the characteristic idempotent of F. As for characteristic bisets, let

ω = ω_{0} + ω_{1} + ⋯,

where

ω_{r} = ∑_{\begin{matrix}P ≤S \\ |S:P |=p^{r} \end{matrix}} ∑_{[P,φ]} c_{[P,φ]}(ω)[P, φ]_{S}^{S}.

(c_{[P,φ]}(ω) denotes the coefficient of the basis element [P, φ] for ω.) Then the idempotency condition tells us that

∑_{[φ]∈Hom_{F} (P,S)/∼_{(S,S)}} c_{[P,φ]} = \begin{cases}1, & if P = S \\ 0, & if P < S \end{cases}

Using this, it is easy to see that

ω_{0} = c_{0} ∑_{[α]∈Out_{F} (S)}[S, α]_{S}^{S},

where c_{0} = \frac{1}{|Out_{F} (S)|}.

Proposition A.1. Let F be a saturated fusion system on a finite p-group S. Let {P_{i} | 1 ≤ i ≤ n} be the set of subgroups of S of index p. For each 1 ≤ i ≤ n, let {φ_{i,j}: P_{i} → S | 0 ≤ j ≤ m_{i}} be a set of representatives of ∼_{(S,S)}-equivalence classes of F-morphisms with domain P_{i} such that φ_{i,0} = ι_{P_{i}}, the inclusion map P_{i} ↪ S. Let

d_{i}^{(e)} = |{1 ≤ j ≤ m_{i} | φ_{i,j} is extendable}|,
d_{i}^{(n)} = |{1 ≤ j ≤ m_{i} | φ_{i,j} is nonextendable}|.

Then

ω_{1} = ∑_{i,j s.t. φ_{i,j} extendable} c_{i}^{(e)}[P_{i}, φ_{i,j}]_{S}^{S} + ∑_{i,j s.t. φ_{i,j} nonextendable} c_{i}^{(n)}[P_{i}, φ_{i,j}]_{S}^{S},

where

c_{i}^{(e)} = −\frac{d_{i}^{(n)}}{d_{i}^{(e)} + pd_{i}^{(n)}} c_{0}, c_{i}^{(n)} = \frac{d_{i}^{(e)}}{d_{i}^{(e)} + pd_{i}^{(n)}} c_{0}.

Proof. By Proposition 4.5, we have

ω_{1} = ∑_{i,j s.t. φ_{i,j} extendable} c_{i}^{(e)}[P_{i}, φ_{i,j}] + ∑_{i,j s.t. φ_{i,j} nonextendable} c_{i}^{(n)}[P_{i}, φ_{i,j}]

where c_{i}^{(e)}, c_{i}^{(n)} ∈ ℤ_{(p)} such that

c_{i}^{(n)} = c_{0} + pc_{i}^{(e)}.

The coefficients of ω_{1} are determined by the idempotency condition:

d_{i}^{(e)}c_{i}^{(e)} + d_{i}^{(n)}c_{i}^{(n)} = 0 (1 ≤ i ≤ n).

Solving these equations for c_{i}^{(e)}, we get

c_{i}^{(e)} = −\frac{d_{i}^{(n)}}{d_{i}^{(e)} + pd_{i}^{(n)}} c_{0}.

Then

c_{i}^{(n)} = \frac{d_{i}^{(e)}}{d_{i}^{(e)} + pd_{i}^{(n)}} c_{0}.  □

For the next proposition, we use the notations of §5. Proposition A.2. Let p be an odd prime and let S be an extraspecial group of order p^{3} and exponent p. Let F be a saturated fusion system on S such that all subgroups of S of order p^{2} are F-radical. Then

ω_{1} = − \frac{c_{0}}{1 + p} ∑_{\begin{matrix}V_{i}≅_{F} V_{j} \\ (k,l)∈Λ_{i}^{e} \end{matrix}}[V_{i}, ψ_{i,j}^{k,l}]_{S}^{S} + \frac{c_{0}}{1 + p} ∑_{\begin{matrix}V_{i}≅_{F} V_{j} \\ (k,l)∈Λ_{i}^{n} \end{matrix}}[V_{i}, φ_{i,j}^{k,l}]_{S}^{S},
ω_{2} =\frac{p}{p^{3} − 1} ∑_{1≤m≤p−1}[z, z^{m}] − \frac{p}{(p + 1)(p^{3} − 1)} ∑_{\begin{matrix}0≤i≤p \\ 1≤m≤p−1 \end{matrix}}[u_{i}, z^{m}]

− \frac{p}{(p + 1)(p^{3} − 1)} ∑_{\begin{matrix}0≤j≤p \\ 1≤m≤p−1 \end{matrix}}[z^{m}, u_{j}^{m}] + ∑_{0≤i≤p} (\frac{1}{p^{3} − 1} − \frac{r_{i}c_{0}}{p + 1}) ∑_{\begin{matrix}V_{i}≅_{F} V_{j} \\ 1≤m≤p−1 \end{matrix}}[u_{i}, u_{j}^{m}]

+ \frac{1}{p^{3} − 1} ∑_{0≤i≤p} ∑_{\begin{matrix}V_{i}≅_{F̸} V_{j} \\ 1≤m≤p−1 \end{matrix}}[u_{i}, u_{j}^{m}].

where [ξ, ζ] := [⟨ξ⟩, φ] with φ: ⟨ξ⟩ → ⟨ζ⟩ such that φ(ξ) = ζ. Proof. By Proposition A.1,

ω_{1} = ∑_{\begin{matrix}V_{i}≅_{F} V_{j} \\ (k,l)∈Λ_{i}^{e} \end{matrix}} c_{i}^{(e)}[V_{i}, ψ_{i,j}^{k,l}]_{S}^{S} + ∑_{\begin{matrix}V_{i}≅_{F} V_{j} \\ (k,l)∈Λ_{i}^{n} \end{matrix}} c_{i}^{(n)}[V_{i}, φ_{i,j}^{k,l}]_{S}^{S},

where

c_{i}^{(e)} = −\frac{d_{i}^{(n)}}{d_{i}^{(e)} + pd_{i}^{(n)}} c_{0}, c_{i}^{(n)} = \frac{d_{i}^{(e)}}{d_{i}^{(e)} + pd_{i}^{(n)}} c_{0}.

So we only need to compute d_{i}^{(e)}, d_{i}^{(e)} to determine ω_{1}. By Lemma 5.1, we have

d_{i}^{(e)} = |\{ψ_{i,j}^{k,l} | V_{i} ≅_{F} V_{j}, (k, l) ∈ Λ_{i}^{e}\}| = |V_{i}^{F}|(p − 1)r_{i} = \frac{1}{c_{0}} ,
d_{i}^{(n)} = |\{φ_{i,j}^{k,l} | V_{i} ≅_{F} V_{j}, (k, l) ∈ Λ_{i}^{n}\}| = |V_{i}^{F}|(p − 1)r_{i} = \frac{1}{c_{0}} .

So

c_{i}^{(e)} = −\frac{c_{0}}{1 + p} =: c_{1}, c_{i}^{(n)} = \frac{c_{0}}{1 + p}.

Now we determine ω_{2}. By the proof of Theorem 5.2,

ω_{2} = ∑_{1≤m≤p−1} c_{2}^{(z,z^{m})}[z, z^{m}] + ∑_{\begin{matrix}0≤i≤p \\ 1≤m≤p−1 \end{matrix}} c_{2}^{(u_{i},z^{m})}[u_{i}, z^{m}]

+ ∑_{\begin{matrix}0≤j≤p \\ 1≤m≤p−1 \end{matrix}} c_{2}^{(z,u_{j}^{m})}[z, u_{j}^{m}] + ∑_{\begin{matrix}0≤i,j≤p \\ 1≤m≤p−1 \end{matrix}} c_{2}^{(u_{i},u_{j}^{m})}[u_{i}, u_{j}^{m}],

where

c_{2}^{(z,z^{m})} = c_{2}^{(z,z)} =: c_{2}^{(z)},

c_{2}^{(u_{i},u_{j}^{m})} = \begin{cases}c_{2}^{(u_{i},u_{i})} =: c_{2}^{(u_{i})}, & if V_{i} ≅_{F} V_{j} \\ c_{2}^{(u_{i})} + r_{i}c_{0} + pr_{i}c_{1}, & if V_{i} ≅_{F̸} V_{j}, \end{cases}

c_{2}^{(z,u_{j}^{m})} = pc_{2}^{(z)} + (f − r_{j}) ∑_{V_{i}≅_{F} V_{j}} c_{1} + f ∑_{V_{i}≅_{F̸} V_{j}} c_{1},

c_{2}^{(u_{i},z^{m})} = \frac{1}{p}c_{2}^{(u_{i})} − \frac{f − r_{i}}{p} c_{0} − (f − r_{i})c_{1}.

The coefficients of ω_{2} are determined by the idempotency condition:

0 = ∑_{1≤m≤p−1} c_{2}^{(z,z^{m})} + ∑_{\begin{matrix}0≤j≤p \\ 1≤m≤p−1 \end{matrix}} c_{2}^{(z,u_{j}^{m})},

0 = ∑_{1≤m≤p−1} c_{2}^{(u_{i},z^{m})} + ∑_{\begin{matrix}0≤j≤p \\ 1≤m≤p−1 \end{matrix}} c_{2}^{(u_{i},u_{j}^{m})} (0 ≤ i ≤ p).

Solving these equations, we get

c_{2}^{(z)} = \frac{p}{p^{3} − 1}, c_{2}^{(u_{i})} = \frac{1}{p^{3} − 1} − \frac{r_{i}}{p + 1}c_{0}.

Then

c_{2}^{(u_{i},u_{j}^{m})} = \frac{1}{p^{3} − 1}  (if V_{i} ≅_{F̸} V_{j}),
c_{2}^{(z,u_{j}^{m})} = c_{2}^{(u_{i},z^{m})} = −\frac{p}{(p + 1)(p^{3} − 1)}.  □

References

[1] Carles Broto, Ran Levi, and Bob Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003), no. 4, 779–856 (electronic).

[2] Sejong Park, Realizing a fusion system by a single finite group, Arch. Math. (Basel) 94 (2010), 405–410.

[3] Lluís Puig, Frobenius categories versus brauer blocks, Progress in Mathematics, vol. 274, Birkhäuser, Basel, 2009, The Grothendieck group of the Frobenius cateogry of a Brauer block.

[4] Kári Ragnarsson, Classifying spectra of saturated fusion systems, Algebr. Geom. Topol. 6 (2006), 195–252 (electronic).

[5] Kári Ragnarsson and Radu Stancu, Saturated fusion systems as idempotents in the double Burnside ring, preprint.

[6] Albert Ruiz and Antonio Viruel, The classification of p-local finite groups over the extraspecial group of order p^{3} and exponent p, Math. Z. 248 (2004), no. 1, 45–65.

Institute of Mathematics, University of Aberdeen, AB24 3UE, UK

E-mail address: s.park@abdn.ac.uk