Saturated Fusion Systems

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Saturated Fusion Systems

Author:

Oliver G¨ afvert, oliverg@kth.se Supervisor:

Tilman Bauer

SA104X - Degree Project in Engineering Physics, First Level Department of Mathematics

Royal Institute of Technology (KTH)

May 20, 2014

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Abstract

A fusion system is a category on a finite p-group, P , which encodes

”conjugacy” relations among the subgroups of P . In this thesis fusion systems of finite groups and ways to prove saturation of abstract fusion systems is investigated. First an introduction to fusion systems of finite groups and the notion of abstract fusion systems is given. Theorems of Burnside and Frobenius regarding fusion systems of finite groups are considered and proven. Alperins fusion theorem formulated for finite groups is considered and used in the proofs. It is proved that all fusion systems of finite groups are saturated.

An investigation in simpler ways of proving that a fusion system is saturated is done. First Alperins fusion theorem formulated for abstract fusion systems is considered which says that, a fusion system, denoted by F , is generated by the automorphism groups of some special subgroups.

Further investigation is done in how this set of special groups, that gen- erates F , can be used to check if F is saturated. A theorem of Craven, though originally stated by Puig, is then considered and proven. The theorem says that is suffices to check that the conjugacy classes of, so called, F -centric subgroups are saturated in order to check that the fusion system F is saturated. Also a theorem of [5] is considered and proven. The theorem says that an even smaller set of conjugacy classes than the set of F -centric subgroups is needed to check saturation.

Section 1-2 are written together with Karl Amundsson and Eric Ahlqvist.

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1 Introduction

The study of fusion systems is an active field of mathematics with applications in many areas including topology, representation theory and finite group theory. The axiomatic foundations of fusion systems started out with L. Puig in the early 1990’s, but the starting point of the theory reach back to classical theorems of Burnside and Frobenius with arguments of fusion of p-elements in finite groups [6]. In [11] Puig writes that one can view the Frobenius normal p-complement criterion as a ”conceptual origin” of the notion of the fusion system. It is currently being investigated if the theory of fusion systems can be used to simplify the classification of finite simple groups [4].

A fusion system on a finite group G is a category on a Sylow p-group P for some prime p dividing |G|. The objects of the category are the subgroups of P and the morphisms are all injective maps between subgroups of P which are induced by conjugation with elements in G. By looking at the local structure of the fusion system of a group one can get information about the group G.

Another way of looking at fusion systems is considering abstract fusion systems that are defined on some p-group P without requiring a larger group G.

With this definition it is possible to look at what properties of the fusion system determine if there exists a finite group G containing the fusion system. It is still being investigated what properties of the fusion system determine properties of the group G.

A special case of fusion systems are the saturated fusion systems and one can show that all finite groups have saturated fusion systems but not all saturated fusion systems belong to some finite group. This makes the study of saturated fusion systems very interesting since it might simplify classification of finite simple groups.

Two books were released on the subject in 2011. One by David Craven [6]

and one by Michael Aschbacher, Radha Kessar and Bob Oliver [5]. These have been crucial for this thesis.

We begin with some preliminary definitions.

Definition 1.1. Let p be a prime. A p-group is a group whose order is a power of p.

Definition 1.2. Let G be a group of order pⁿm, where p is a prime and p - m.

Then we say that a subgroup of G is a Sylow p-subgroup if its order is pⁿ. Theorem 1.3 (Sylow’s theorem). Let G be a group of order pⁿm, where p is a prime and p - m. Then

1. G has subgroups of order 1, p, p², · · · , pⁿ. 2. All Sylow p-subgroups of G are conjugate.

3. A subgroup of order p^k, 0 ≤ k ≤ n, is contained in some Sylow p-subgroup.

4. The number of subgroups of order p^k, 0 ≤ k ≤ n, is congruent to 1 modulo p.

5. The number of Sylow p-subgroups equals |G : N_G(P )|, where N_G(P ) is the normalizer of P ∈ Syl_p(G). In particular, |Syl_p(G)| divides m.

A proof can be found in [7].

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2 Fusion Systems

This section aims to give an introduction to fusion systems and hopefully also serve as a motivation as to why they are interesting to study. We begin with a definition of what is meant by fusion and for elements to be fused in some finite group.

Note that if x, y and g are elements in some group G, we write^xg = xgx⁻¹ and g^x = x⁻¹gx and the map cy : G → G denotes conjugation by the element y, i.e., cy(g) =^yg.

Definition 2.1. Let G be a finite group and let H ≤ K ≤ G be subgroups of G.

1. Let g, h ∈ H and suppose that g and h are not conjugate in H. If g and h are conjugate by an element in K, then g and h are said to be fused in K. Similarly, two subgroups are said to be fused if they are conjugate by an element in K.

2. The subgroup K is said to control weak fusion in H with respect to G if, whenever g, h ∈ H are fused in G, they are fused in K.

3. The subgroup K is said to control G-fusion in H, if whenever two subgroups A and B are conjugate via a conjugation map ϕ_g : A → B for some g ∈ G, then there is some k ∈ K such that ϕ_g and ϕ_k agree on A, i.e., if ∀a ∈ A, ϕ_g(a) = ϕ_k(a).

Definition 2.2. Let G be a finite group and P a Sylow p-subgroup of G. The fusion system of G on P is the category F_P(G), whose objects are all subgroups of P and whose morphisms are

Hom_F_P_(G)(Q, R) = HomG(Q, R), Q, R ≤ P,

i.e., the set of all group homomorphism from Q to R induced by conjugation with elements in G. The composition of morphisms is the composition of group homomorphisms.

Definition 2.3. Let F_P(G) be the fusion system of G on P and let Q be a subgroup of P . Then we define the automorphism group Aut_P(Q) by Aut_P(Q) = Hom_P(Q, Q).

Remark. Aut_P(Q) is isomorphic to N_P(Q)/C_P(Q), which can be seen by ap- plying the first isomorphism theorem on the natural homomorphism N_P(Q) → AutP(Q) defined by x 7→ cx.

Definition 2.4. Let G be a finite group and P a Sylow p-subgroup of G. Let F = F_P(G) be the fusion system of G on P and let Q, R be any two subgroups of P . We say that Q and R are F -conjugate if there is a morphism φ : Q → P in F such that φ(Q) = R, i.e., if there is a g ∈ G such that gQg⁻¹ = R.

One may also say that two F -conjugate subgroups are F -isomorphic.

Definition 2.5. Let G be a finite group and P a Sylow p-subgroup of G. Let F = FP(G) be the fusion system of G on P and Q a subgroup of P . The F - conjugacy class containing Q is the class of subgroups of P that are F -conjugate to Q.

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Definition 2.6. Let G be a finite group and P a Sylow p-subgroup of G. Let F = F_P(G) be the fusion system of G on P . The skeleton of F is the category Fsc, whose objects are representatives for the F -conjugacy classes. For any two objects A, B in Fsc we put Hom_F_sc(A, B) = Hom_F(A, B).

In category theory, one would say that Fsc is equivalent to F . Example 2.1. Let G be the symmetric group S4 and let

P = {1, (12), (34), (12)(34), (13)(24), (14)(23), (1324), (1423)}.

Then P ∈ Syl2(G) and P ∼= D8. In Figure 1 we see the subgroup lattice of P .

Figure 1: Fusion system on S4

Conjugation by (13)(24) is a morphism in F = FP(G) which maps {1, (12)}

and {1, (34)} onto each other and we say that

{1, (12)} and {1, (34)} are F -conjugate.

We also have that the groups {1, (12)(34)}, {1, (13)(24)} and {1, (14)(23)} are F -conjugate. These isomorphisms are induced by conjugation by (123) and (132). Hence we see that {1, (12)(34)} and {1, (13)(24)} are fused in G but not in P and equally for {1, (12)(34)} and {1, (14)(23)}.

The skeleton Fsc of the fusion system FP(G) is shown in Figure 2.

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Figure 2: The skeleton of the fusion system on S₄

The number on the edge between two subgroups Q and R is the size of Hom_F(Q, R). Note that Figure 2 does not cover all information of the fusion system since it does not tell what Hom_F(Q, R) are explicitly.

We have that Aut_F(Q) acts on Hom_F(Q, R) on the right and that Aut_F(R) acts on Hom_F(Q, R) on the left. For example, Let Q = {1, (12)(34)} ∼= Z/2Z and R = {1, (12)(34), (13)(24), (14)(23)} ∼= V4∼= Z/2Z⊕Z/2Z. Then AutF(R) ∼= S3 acts on Hom_F(Q, R) from the left as S3 on the set of three letters while Aut_F(Q) = 1 acts trivially from the right on Hom_F(Q, R).

Denote 1, (12)(34), (13)(24), (14)(23) by a, b, c, d respectively. Then Hom_F(R, P ) have as domain, {a, b, c, d} since a is always fixed under HomF(R, P ), we may denote the elements of HomF(R, P ) as cycles by α =IdR, β = (bc), γ = (bd), δ = (cd), = (bcd), ζ = (bdc).

Now consider the group AutF(P ) with action restricted to R. For any ψ ∈ Aut_F(P ) there are only two possibilities for ψ

R. Either ψ

R = Id_R or ψ R

swaps (13)(24) and (14)(23). Hence the left action of Aut_F(P ) on Hom_F(R, P ) will induce only two maps, the identity map and the following

α = Id 7→ (cd) = δ and δ = (cd) 7→ Id = α β = (bc) 7→ (bdc) = ζ and ζ = (bdc) 7→ (bc) = β γ = (bd) 7→ (bcd) = and = (bcd) 7→ (bd) = γ

This map can be written in cycle form as (αδ)(βζ)(γ). Hence we conclude that Aut_F(P ) will act on Hom_F(R, P ) as h(12)(34)(56)i acts on the set {1, 2, 3, 4, 5, 6}.

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Note that if Q is not F -isomorphic to any other subgroup in F , then if α ∈ Hom_F(Q, R) is the inclusion map of Q into R then Hom_F(Q, R) = α ◦ Aut_F(Q) and we have |Hom_F(Q, R)| = |Aut_F(Q)|.

The next theorem is a classical theorem of Burnside which gives information on the fusion in FP(G) when the p-group P is abelian.

Theorem 2.7 (Burnside). Let G be a finite group and let P be a Sylow-p- subgroup of G. If P is abelian, then FP(G) = FP(NG(P )).

Proof. Let Q, R ≤ P and let ϕ : Q → R be a morphism in FP(G) such that ϕ(q) = ^xq = xqx⁻¹ for some x ∈ G. Since P is abelian, everything in P centralizes^xQ ≤ P . Also, if xpx⁻¹∈^xP and xqx⁻¹∈^xQ, then

xpx⁻¹xqx⁻¹= xpqx⁻¹= xqpx⁻¹= xqx⁻¹xpx⁻¹

so that ^xP centralizes ^xQ. Thus, both P and ^xP are Sylow-p-subgroups of C_G(^xQ) and hence we can find a c ∈ C_G(^xQ) such that P =^cxP . This means that cx ∈ N_G(P ) and since ^cxu = c(xux⁻¹)c⁻¹ = xux⁻¹= ϕ(u) for u ∈ Q, we are done.

Definition 2.8. Let p be a prime and let G be a finite group. Then we say that a subgroup of G is a p⁰-group if its order is coprime to p.

Theorem 2.9. Any two normal p⁰-subgroups of a finite group G generate a normal p⁰-subgroup.

Proof. Let H and K be two normal p⁰-subgroups of the finite group G. Then, since H (or K) is normal, the generated subgroup is just HK. But by the second isomorphism theorem, we know that |HK| = |K||H|/|H ∩ K| and so the order of |HK| cannot possibly be divisible by p. Thus HK is a p⁰-group. To see that it is normal, take any g ∈ HK and write it as g = hk, where h ∈ H and k ∈ K. If x ∈ G, then

xgx⁻¹ = xhkx⁻¹= xhx⁻¹xkx⁻¹ ∈ HK, since H and K are normal.

Remark. The above implies that the subgroup generated by all the normal p⁰- subgroups is itself a normal p⁰-subgroup, that is, G has a unique maximal normal p⁰-subgroup. This maximal normal p⁰-subgroup will be denoted by Op⁰(G).

Definition 2.10. Let G be a finite group and let P be a Sylow-p-subgroup of G. Then G is said to be p-nilpotent if P has a normal complement K; that is, K is a normal subgroup of G such that G = KP and K ∩ P = 1. That is G ∼= K o P . Note that K = Op⁰(G).

K is called the normal p-complement of P in G.

Lemma 2.11. Let G be a group and suppose H and K are normal subgroups of G such that their orders are relatively prime. Then hk = kh for all h ∈ H, k ∈ K.

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Proof. First note that H ∩ K = 1 since if x ∈ H ∩ K, the order of x divides both H and K. But the orders are coprime, so it must be that x = 1. Take any h ∈ H and k ∈ K. Then we get

[h, k] = h kh⁻¹k⁻¹

| {z }

∈H

= hkh⁻¹

| {z }

∈K

k⁻¹∈ P ∩ K = 1,

so that hk = kh.

Next is another classical theorem, known as; Frobenius’ normal p-complement theorem. This gives a criterion for when P controls G-fusion in P , i.e., when every morphism in F_P(G) is induced by conjugation in P . The last implication of this theorem will be proved in section 4.

Theorem 2.12 (Frobenius). Let G be a finite group and let P be a Sylow p-subgroup of G. The following are equivalent:

1. G is p-nilpotent.

2. NG(Q) is p-nilpotent for any non-trivial Q ≤ P . 3. We have FP(G) = FP(P ).

4. For any Q ≤ P , the group AutG(Q) = NG(Q)/CG(Q) is a p-group.

Proof. 1) =⇒ 2): Since G is p-nilpotent, we can write G = KP , where K E G, P ∈ Syl^p(G) and K ∩ P = 1. As P ∼= G/K is a p-group, we see that K = {x ∈ G : gcd(|x|, p) = 1}. It follows that for any H ≤ G, H ∩ K = {x ∈ H : gcd(|x|, p) = 1} = Op⁰(H). Hence H = Op⁰(H) (P ∩ H), and so H is p-nilpotent. Thus, every subgroup of a p-nilpotent subgroup is p-nilpotent and so in particular, NG(Q) is p-nilpotent for any subgroup Q of P .

1) =⇒ 3): Let Q, R be subgrous of P and ϕ ∈ HomG(Q, R). Then ϕ = cx

for some x ∈ G. We want to show that cx(u) = xux⁻¹ = pup⁻¹ = cp(u) for some p ∈ P and every u ∈ Q. Since G = KP , we can write x = yz where y ∈ K and z ∈ P . But then

[y, zuz⁻¹] = y

|{z}

∈K

zuz⁻¹y⁻¹zu⁻¹z⁻¹

| {z }

∈K

= xux⁻¹

| {z }

∈P

zu⁻¹z⁻¹

| {z }

∈P

∈ P ∩ K = 1,

so that xux⁻¹= zuz⁻¹.

3) =⇒ 4): This follows from the fact that

Aut_G(Q) = Hom_G(Q, Q) = Hom_P(Q, Q) = N_P(Q)/C_P(Q) is a p-group.

2) =⇒ 4):

Let Q be a non-trivial subgroup of P . Then we can write NG(Q) = K1o P1. Since CG(Q) is a subgroup of NG(Q), CG(Q) is also p-nilpotent by the first part of the proof. Thus we can write CG(Q) = K2 o P2. It follows that

|NG(Q)|/|CG(Q)| = ^|K_|K¹^||P¹^|

2||P2|, so we are done if we can show that |K1| = |K2|.

Obviously, |K₂| ≤ |K1|. Now, Q and K1 are both normal in N_G(Q) and have coprime orders, so that by the lemma, K₁ commutes with every element in Q.

Thus K₁≤ C_G(Q) = K₂P₂, and so K₁≤ K₂and finally |K₁| = |K₂|.

The fact that 4) implies 1) will be proved in section 4.

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2.1 Transfer and the Focal Subgroup Theorem

To simplify the proof of the last part of Frobenius’ normal p-complement theorem we want to use Alperin’s fusion theorem. For this we need a property of the so called focal subgroup which was first introduced by Higman [9]. To prove the focal subgroup theorem we use the transfer homomorphism. If G is a group, H a subgroup of G and A is any abelian group, the transfer is a way of extending a homomorphism φ : H → A to a homomorphism τ : G → A.

Definition 2.13. Let G be a finite group and H ≤ G. Let φ : H → A be a homomorphism of H into an abelian group A. Let X be a set of right coset representatives for H in G and let I be the index set of X. For each g ∈ G, xi ∈ X we have that xig ∈ Hxj for a unique xj ∈ X. Define σg : I → I by σg(i) = j. Then xigx⁻¹_j = hi,g ∈ H where hi,g depends on i and g. Define τ : G → A by

τ (g) =Y

i∈I

φ(hi,g) =Y

i∈I

φ(xigx⁻¹_σ

g(i)) We say that τ is the transfer of G into A via φ.

Theorem 2.14. Let G, H and τ be chosen as in definition 3.1. Then we have the following:

1. The transfer τ is a homomorphism of G into A.

2. τ is independent on the choice of coset representatives of H in G.

For a proof of this theorem, see [8].

The next theorem will be necessary in our proof of the focal subgroup theorem.

Theorem 2.15. Let τ be the transfer of G into an abelian group A via H ≤ G and the homomorphism φ : H → A. For any g ∈ G, ∃{x1, ..., xt} ⊆ G with t and xi depending on g, with the following properties:

1. x_ig^rⁱx⁻¹_i ∈ H for some positive integers r_i, 0 ≤ i ≤ t.

2. Pt

i=1ri= n = |G : H|

3. τ (g) = φ(Q

i∈Ix_ig^rⁱx⁻¹_i )

In the proof of this theorem we follow the proof in Daniel Gorenstein [8].

Proof. Let y_i be coset representatives of H in G, 0 ≤ i ≤ n. Let σ_g ∈ S_n be defined by yig ∈ Hy_σ_g_(i). Decompose σg into disjoint cycles and reorder the yi

such that the decomposition assumes the form:

(12...r1)(r1+ 1...r1+ r2)(r1+ r2+ 1...r1+ r2+ r3)....(...r1+ r2+ ... + rt).

Then the i:th cycle has length ri, 0 ≤ i ≤ t and hence

t

X

i=1

ri = n = |G : H|

Hence (2) is proved.

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Now let x₁, ..., x_t be coset representatives for the cosets labeled 1, r₁ + 1, r₁+ r₂+ 1, ..., r₁+ r₂ + ... + r_t−1 + 1 respectively. Then by definition of σg, xig^jis a coset representative of H in G corresponding to the (j + 1)th coset of the ith cycle of σg. Hence

R := {xig^j|1 ≤ i ≤ t, 0 ≤ j ≤ ri− 1}

form a complete set of coset representatives for H in G. But then xig^rⁱ ∈ Hxi

by definition of ri and hence xig^rⁱx⁻¹_i ∈ H which proves (1).

Now we prove (3). Lets use R as coset representatives and compute τ (g).

Let yk = xig^j (k depend on i and j) and consider ykg = hk,gyk⁰_g, hk,g ∈ H. If j < ri− 1, then ykg = xig^j+1= yk⁰_g. Since xig^j+1is a coset representative in R and the xig^j+1∈ Hyk⁰_g if and only if yk⁰_g = xig^j+1, by definition of hk,g.

This implies that hk,g = 1 whenever j < ri− 1. Hence τ (g) is the product of those φ(hk,g) which corresponds to the elements yk = xig^rⁱ⁻¹ for which we have

y_kg = x_ig^rⁱ∈ Hxi⇒ ykg = (x_ig^rⁱx⁻¹_i )x_i, (x_ig^rⁱx⁻¹_i ) ∈ H.

Hence x_i= y_k⁰

g and we get h_k,g = x_ig^rⁱx⁻¹_i for each y_k. Therefore we get that τ (g) = φ

t

Y

i=1

xig^rⁱx⁻¹_i

!

and since φ is a homomorphism we get (3).

Theorem 2.16 (The Focal Subgroup Theorem, [9]). Let G be a finite group, P ∈ Syl_p(G) and let G⁰ be the commutator subgroup of G. Then

P ∩ G⁰= h[x, g] = x⁻¹x^g: x ∈ P, g ∈ G, x^g∈ P i

= hx⁻¹φ(x) : x ∈ P, φ ∈ Hom_F_G(P )(hxi, P )i

Note that ones the first equality is proved the second one is trivial, since it is just a matter of translation into the setting of fusion systems.

Proof. Let P^∗ = hx⁻¹x^g : x ∈ P, g ∈ G, x^g ∈ P i. We want to show that P^∗ = P ∩ G⁰. Since x⁻¹x^g = [x, g], we obviously have P^∗ ≤ P ∩ G⁰ and since P⁰≤ P^∗, P/P^∗is abelian.

Let φ : P → P/P^∗ be the natural homomorphism and let τ : G → P/P^∗be the transfer of G into P/P^∗ relative to P and φ.

Claim: If G/ ker τ ∼= P/P^∗, then P^∗= P ∩ G⁰.

pf. Let K = ker τ . G⁰≤ K as G/K is abelian. Also, G/K is a p-group and hence G = P K ⇒ G/K ∼= P/(P ∩ K) and P ∩ G⁰≤ G⁰≤ K ⇒ |P ∩ G⁰| ≤ |P^∗|.

Thus, since P^∗ ≤ P ∩ G⁰ we have P^∗= P ∩ G⁰.

Now we use the same notation as in Theorem 3.2. Let x ∈ P and choose elements xi∈ G and integers ri, 0 ≤ i ≤ t such that

τ (x) = φ

t

Y

i=1

xix^rⁱx⁻¹_i

!

≡

t

Y

i=1

xix^rⁱx⁻¹_i (mod P^∗).

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Since P/P^∗ is abelian, we have

τ (x) = φ

t

Y

i=1

x^rⁱx^−rⁱxix^rⁱx⁻¹_i

!

≡

t

Y

i=1

x^rⁱ

! _t Y

i=1

x^rⁱxix^rⁱx⁻¹_i

!

(mod P^∗).

But since x^−rⁱxix^rⁱx⁻¹_i = [x^rⁱ, xi] ∈ P⁰≤ P^∗ we have that

τ (x) ≡

t

Y

i=1

x^rⁱ(mod P^∗) ≡ x^{P r}ⁱ(mod P^∗) ≡ xⁿ(mod P^∗).

But since |G : P | = n, we get that gcd(p, n)=1 and hence, if x /∈ P^∗, then τ (x) /∈ P^∗. Thus τ maps P onto P/P^∗ and hence it also maps G onto P/P^∗ and we have

τ (G) = P/P^∗⇒ G/ker τ ∼= P/P^∗.

2.2 Proof of Frobenius 4⇒1

The goal of this section is to prove that (4) implies (1) in Frobenius’ normal p-complement theorem. To do this we use Alperin’s fusion theorem which is a strong result about conjugation in finite groups. The theorem was first stated and proved by Alperin in [3] in 1967.

Definition 2.17. Let P and Q be Sylow p-subgroups of G. R = P ∩ Q is called the tame intersection of P and Q in G if both N_P(R) and N_Q(R) are Sylow p-subgroups of N_G(R).

We will later see that in a fusion system F_P(G) of a finite group G on a Sylow p-subgroup P , a tame intersection of P with any Q ∈ Syl_p(G) will be so-called fully F -normailized which is an important property for a group in a fusion system.

Example 2.2. Every Sylow p-subgroup is a tame intersection, as we see from Q = Q ∩ Q.

Theorem 2.18 (Alperins Fusion Theorem, [3]). Let G be a finite group and P ∈ Syl_p(G). Let A, A^g⊆ P , for some g ∈ G.

Then there exists elements x1, x2, ..., xn, subgroups Q1, Q2, ..., Qn ∈ Syl_p(G) and an y ∈ NG(P ) such that

1. g = x₁x₂...x_ny,

2. P ∩ Q_i is a tame intersection, 0 ≤ i ≤ n, 3. x_i is a p-element of N_G(P ∩ Q_i), 0 ≤ i ≤ n,

4. A ⊆ P ∩ Q₁and A^x¹^x²^...xⁱ ⊆ P ∩ Q_i+1, 0 ≤ i ≤ n − 1.

Proposition 2.19 (Frobenius 4 ⇒ 3). Suppose for any Q ≤ P , the group AutG(Q) = NG(Q)/CG(Q) is a p-group. Then FP(G) = FP(P ).

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Proof. We want to show that for any morphism in F_P(G) induced by conjugation with some g ∈ G, g can be written as a product of elements x₁, x₂. . . , x_n, y ∈ P . From Theorem 2.18 we know we can find Qi∈ Syl_p(G) such that P ∩ Qi is a tame intersection and such that xi∈ NG(P ∩ Qi), for 0 ≤ i ≤ n. So if we can show that xi∈ P for 1 ≤ i ≤ n and y ∈ P , we are done.

Since P ∩ Qi is a tame intersection we have that NP(P ∩ Qi) ∈ Syl_p(NG(P ∩ Qi)), thus NP(P ∩ Qi)CG(P ∩ Qi)/CG(P ∩ Qi) ∈ Syl_p(NG(P ∩ Qi)/CG(P ∩ Qi)).

Now by assumption, AutG(P ∩ Qi) = NG(P ∩ Qi)/CG(P ∩ Qi) is a p-group

⇒ NG(P ∩ Qi) = NP(P ∩ Qi)CG(P ∩ Qi) and since NP(P ∩ Qi) ∩ CG(P ∩ Qi) = CP(P ∩ Qi) we have that

NP(P ∩ Qi)CG(P ∩ Qi) C_G(P ∩ Q_i)

∼= NP(P ∩ Qi) N_P(P ∩ Q_i) ∩ C_G(P ∩ Q_i)

∼= AutP(P ∩ Q_i) (1)

Hence each for 1 ≤ i ≤ n, xi∈ P .

Now we use the same argument again to show that y ∈ P . P is a tame intersection with itself and hence NP(P ) ∈ Syl_p(NG(P )). But NG(P ) was assumed to be a p-group and hence NG(P ) = NP(P ) which implies that y ∈ P .

Now to the proof of that (4) implies (1), in Frobenius Theorem. We will denote by O^p(G), the smallest normal subgroup of G such that G/O^p(G) is a p-group.

Proposition 2.20. Let G be a group, H a normal subgroup of G and K a characteristic subgroup of H. Then K is normal in G.

Proof. Let H ≤ G and K char H. For any g ∈ G, we have that gHg⁻¹ = H and hence g induces a map cg ∈ Aut(H). Since K is characteristic in H we have that gHg⁻¹ = cg(H) = H and K is fixed by any automorphism of H, thus K is normal in G.

Proposition 2.21 (Frobenius 4 ⇒ 1). Suppose for any Q ≤ P , the group AutG(Q) = NG(Q)/CG(Q) is a p-group. Then G is p-nilpotent.

Proof. We will proceed by induction on G. The proof will be in two steps. First we will show that if G has a proper normal subgroup H such that G/H is a p-group, then the statement is true. The second step is to show that G actually contains such an H.

So first assume that there is an H C G such that G/H is a p-group. Let Q be a proper p-subgroup of H. By assumption N_G(Q)/C_G(Q) is a p-group. We have that H ∩ N_G(Q) = N_H(Q) and H ∩ C_G(Q) = C_H(Q) and hence N_H(Q)/C_H(Q) is a p-group. Hence, by induction H has a normal p-complement K = O_p⁰(H).

By Theorem 2.9, K is the unique maximal normal p’-subgroup and hence K is characteristic in H. Thus, since H C G, we have that K C G, by Proposition 2.20. But both G/H and H/K are p-groups and hence, so is G/K. Hence K is a normal p-complement in G and the induction is complete.

Now we prove the second step. From Proposition 2.19 and the focal subgroup theorem we have that

P ∩ G⁰= hx⁻¹x^g|x, x^g∈ P, g ∈ Gi = hx⁻¹x^g|x, x^g, g ∈ P i = P⁰ (2)

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and we have that P⁰ < P since P is a p-group. Now consider the group G⁰O^p(G) E G. Let φ : P → P/P⁰ be the natural homomorphism and let τ be the transfer from G into P/P⁰ via φ. Then we have that

G/ ker τ ∼= P/P⁰ 6= 1. (3)

This means that G⁰ ≤ ker τ since P/P⁰ is abelian and O^p(G) ≤ ker τ since P/P⁰ is a p-group. Thus G⁰O^p(G) ≤ ker τ and hence G⁰O^p(G) is a proper normal subgroup of G. Obviously G/(G⁰O^p(G)) is a p-group as G/O^p(G) is by definition of O^p(G). So if we put H = G⁰O^p(G) in part one, we are done.

2.3 Abstract Fusion Systems

In this section we will introduce a more general definition of a fusion system.

Instead of defining it on a finite group G we will define a fusion system on a p-group P directly, without requiring that P is a subgroup of some larger group G. We will also loosen the requirement of every morphism being induced by conjugation. Instead we are satisfied if every morphism, induced by conjugation in P , is in the fusion system.

It is however hard to work with the definition of an abstract fusion system alone and hence we introduce the notion of a saturated fusion system. We will prove that for any finite group G with a Sylow p-subgroup P , F_P(G) is saturated.

Since we may construct a fusion system on any finite group but not every fusion system is realisable on a finite group, the concept of an abstract fusion system implies that the class of abstract fusion systems is bigger than the class of finite groups. This observation gives a prospect that the theory of fusion systems could help in simplifying the the classification of finite simple groups.

The fusion systems that are not realisable on any finite group G, are called exotic fusion systems.

We begin with the definition of an abstract fusion system.

Definition 2.22. Let P be a finite p-group. A fusion system on P is a category F , whose objects are all subgroups of P and whose morphisms Hom_F(Q, R) are sets of injective homomorphisms having the following three properties:

1. For each g ∈ P such that^gQ ≤ R, cg: Q → R defined by cg(x) =^gx is in HomF(Q, R).

2. For each φ ∈ Hom_F(Q, R), the induced isomorphism Q → φ(Q) and its inverse lies in Hom_F(Q, φ(Q)) and Hom_F(φ(Q), Q) respectively.

3. Composition of morphisms in F is the usual composition of group homomorphisms.

The axioms are inspired from G-conjugacy and Definition 2.2. The first axioms guarantees agreement with Definition 2.2 in the case one has an under- lying group. The second and third axioms are there to make F -conjugacy into an equivalence relation, as it is for G-conjugacy.

Example 2.3. Let P be a p-group and F the fusion system on P such that, for any subgroups Q, R of P , Hom_F(Q, R) consists of all injective homomorphisms from Q to R. Then F is called the universal fusion system on P .

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Consider the fusion system on D₈ in Example 2.1. For the Klein 4-group V = {id, (12), (34), (12)(34)} we have Aut_F(V ) < Aut(V ) and hence, this is not the universal fusion system on D8. However, if we let G = A6 an construct the fusion system on D8we see that FD₈(G) is the universal fusion system on D8as in this fusion system, both Klein 4-groups have their full automorphism group.

To define a saturated fusion system we need the following definitions.

Definition 2.23. Let F be a fusion system on a finite p-group P . A subgroup Q of P is said to be fully F -automized if AutP(Q) ∼= NP(Q)/CP(Q) is a Sylow p-subgroup of AutF(Q).

Definition 2.24. Let F be a fusion system on a finite p-group P and let Q ≤ P . For any φ : Q → P in F we set

N_φ= {y ∈ N_P(Q)| ∃ z ∈ N_P(φ(Q)) such that φ(^yu) =^zφ(u), ∀u ∈ Q} (4) Remark. Note that QC_P(Q) E NφE NP(Q).

Remark. Nφ is the preimage in NP(Q) of AutP(Q) ∩^φAutP(φ(Q)).

Definition 2.25. Let F be a fusion system on a finite p-group P . A subgroup Q of P is said to be receptive if every morphism φ, whose image is Q, is extensible to Nφ.

To get insight into what these definitions mean we look at the following proposition.

Proposition 2.26. Let F be a fusion system on a finite group G and let P ≤ G such that P ∈ Syl_p(G). If Q ≤ P is such that NP(Q) ∈ Syl_p(NG(Q)) then Q is receptive.

Proof. Suppose that φ : R → Q is an isomorphism in F . Now

N_φ= {x ∈ N_P(R)| ∃ y ∈ N_P(Q) such that φ(xrx⁻¹) = yφ(r)y⁻¹, ∀r ∈ R}, (5) thus c_x−1◦ φ⁻¹◦ cy◦ φ centralizes R and cy◦ φ ◦ c_x−1◦ φ⁻¹ centralizes φ(R) = Q ⇒ φ ◦ cx = φ⁰ ◦ φ, for some φ⁰ that is induced by conjugation with some element g ∈ CG(Q). Thus

φ(Nφ) ≤ NP(Q)CG(Q) (6)

and since Nφ is a p-group and NP(Q) ∈ Syl_p(NG(Q)) ⇒ there exists a ψ in F induced by some c ∈ CG(Q) such that ψ(φ(Nφ)) ≤ NP(Q). Thus we can define θ = ψ ◦ φ such that θ : N_φ→ NP(Q), and the proof is done.

Definition 2.27. Let F be a fusion system on a finite p-group P . A subgroup Q of P is said to be fully F -centralized if |C_P(R)| ≤ |C_P(Q)| for any R ≤ P F -conjugate to Q. Q is called fully F -normalized if |N_P(R)| ≤ |N_P(Q)| for any R ≤ P F -conjugate to Q.

Note that if a subgroup is fully F -normalized and fully F -centralized it must be fully F -automized since |NP(Q)| = |CP(Q)||AutP(Q)|, for some subgroup Q.

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Example 2.4. Take G = S₄ and let P = h(1243), (12)i then P ∈ Syl_p(G) and P ∼= D8. Now look at the subgroup Q = {id, (12)(34), (13)(24), (14)(23)}

which is a subgroup of P and which is isomorphic to the Klein-4-group. Since Q is normal in G and conjugation by (123) permutes (12)(34), (13)(24) and (14)(23) transitively, we know that H1= {id, (12)(34)}, H2= {id, (13)(24)} and H3= {id, (14)(23)} make a conjugacy class in G. We note that H1C P while H2 and H3 are not, hence H1 is fully F -normalized. Furthermore, NP(H1) = P ∈ Syl_p(NG(H1)) and thus, by Proposition 2.26, H1 is receptive.

Proposition 2.28. Let F be a fusion system on a finite group G and let P ≤ G such that P ∈ Syl_p(G). Let Q ≤ P

1. Q is fully F -centralized if and only if CP(Q) ∈ Syl_p(CG(Q)) 2. Q is fully F -normalized if and only if N_P(Q) ∈ Syl_p(N_G(Q))

Proof. First we prove 1. Let S ∈ Syl_p(C_G(Q)) such that C_P(Q) ≤ S. By Sylow’s theorem there is a g ∈ G such that (QS)^g ≤ P and we have that Q ∼= Q^g. For any y ∈ S^g, gyg⁻¹ ∈ C_G(Q) which implies that (gyg⁻¹)z(gy⁻¹g⁻¹) = z ⇔ y(g⁻¹zg)y⁻¹= g⁻¹zg for all z ∈ Q. Hence S ≤ CG(Q^g) ∩ P = CP(Q^g) and we conclude that |CP(Q)| ≤ |S| ≤ |CP(Q^g)|. From here it is easy to see that Q is fully F -centralized if and only if |CP(Q)| = |S|.

To prove 2, just use the same argument for normalizers insted of centralizers.

We will now define what is meant by a saturated fusion system. The axioms for fusion systems are quite hard to work with alone whereas the concept saturation solves a lot of these problems.

Definition 2.29. Let F be a fusion system on a finite p-group P . We say that F is saturated if every F -conjugacy class of subgroups of P contains a subgroup that is both receptive and fully F -automized.

Theorem 2.30. Let G be a finite group and let P be a Sylow p-subgroup of G. The fusion system FP(G) is saturated.

Proof. We want to show that every FP(G)-conjugacy class of P contains a fully F -automized and receptive subgroup. Take any R ≤ P , then there is some Q ≤ P F -conjugate to R such that Q is fully F -normalized. Hence, by Proposition 2.28, NP(Q) ∈ Syl_p(NG(Q)) and then by Proposition 2.26, Q is receptive. Now notice that NP(Q)CG(Q)/CG(Q) ∈ Syl_p(NG(Q)/CG(Q)) and AutG(Q) ∼= NG(Q)/CG(Q) and

NP(Q)CG(Q) C_G(Q)

∼= NP(Q)

N_P(Q) ∩ C_G(Q) =NP(Q) C_P(Q)

∼= AutP(Q). (7)

Hence AutP(Q) is a Sylow p-group of AutG(Q) = Aut_F_P_(G)(Q), which shows that Q is fully automized.

Theorem 2.30 implies that the class of saturated fusion systems is at least as big as the class of finite groups.

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Proposition 2.31. Let F be a fusion system on a finite p-group P and let Q, R be F -conjugate subgroups of P such that R is fully F -automized . There exists an isomorphism ψ : Q → R in F such that Nψ= NP(Q), i.e. if R is receptive then ψ extends to a morphism from NP(Q) to NP(R) in F .

Proof. Since R is fully F -automized AutP(R) is a Sylow p-subgroup of AutF(R).

If φ : Q → R is an isomorphism in F , then φ ◦ AutP(Q) ◦ φ⁻¹ is a p-subgroup of AutF(R) and hence there is an α ∈ AutF(R) such that α ◦ φ ◦ AutP(Q) ◦ φ⁻¹◦ α⁻¹ ≤ AutP(R). Now let ψ = α ◦ φ. Now this says exactly that for any x ∈ N_P(Q) there is an y ∈ N_P(R) such that ψ ◦ c_x◦ψ⁻¹= c_y ⇐⇒ ψ ◦cx= c_y◦ψ as ψ is injective. But then N_P(R) satisfies the definition of N_ψ and we get N_ψ= N_P(Q).

We denote by cQ, the map NP(Q) → AutP(Q) defined by cQ(g) = cg for g ∈ NP(Q).

Proposition 2.32. Let F be a saturated fusion system on a finite p-group P . Let Q and R be F -conjugate subgroups of P and let φ : Q → R be a morphism in F . Suppose φ extends to a morphism ¯φ : S → P for some S ≤ NP(Q).

Then the image of ¯φ is contained within NP(R) and we have that cQ(S) ≤ Aut_P(Q) ∩^φ⁻¹Aut_P(R).

Proof. Let x ∈ S. For all g ∈ Q we have that ^xg ∈ Q and thus ^φ^¯(x ¯φ(g)x⁻¹) = φ(xgx¯ ⁻¹) ∈ R ⇒ ¯φ(x) ∈ NP(R). Thus the image of ¯φ is in NP(R). Also, we have that cR( ¯φ(x)) ∈ AutP(R), therefore^φ(cQ(S)) = cR( ¯φ(S)) ≤ AutP(R).

And since c_Q(S) ≤ Aut_P(Q) we have that c_Q(S) ≤ Aut_P(Q) ∩^φ⁻¹Aut_P(R).

Proposition 2.33. Let F be a saturated fusion system on a finite p-group P . Q ≤ P is receptive if and only if it is fully F -centralized.

Proof. Suppose a subgroup Q ≤ P is receptive and not fully F -centralized.

There exists a fully F -centralized subgroup R of P which is F -conjugate to Q.

Now define φ : R → Q to be an isomorphism in F . Since Q is receptive there is a morphism ψ : Nφ → NP(Q) such that ψ

_R = φ. Now RCP(R) ≤ Nφ and ψ(R) = Q. Now suppose that x ∈ ψ(CP(R)). Any q ∈ Q may be written as ψ(r) for some r ∈ R. Hence ψ⁻¹(x)r(ψ⁻¹(x))⁻¹= r ⇔ xψ(r)x⁻¹= ψ(r) which implies that x ∈ C_P(Q). Hence ψ(C_P(R)) ≤ C_P(Q) ⇒ |C_P(R)| ≤ |C_P(Q)|, which is a contradiction.

Conversely, suppose Q is fully F -centralized but not receptive. Since F is saturated there exists some R ≤ P , F -conjugate to Q and such that R is fully F -automized and receptive. Then by Proposition 2.31 there is an isomorphism ψ : Q → R that can be extended to a morphism ¯ψ : NP(Q) → NP(R). Hence

ψAutP(Q) ≤ AutP(R). Now suppose that S ≤ P is F -conjugate to Q and that φ : S → Q is a morphism in F . Then

AutP(S) ∩^φ⁻¹AutP(Q) ≤ AutP(S) ∩^φ⁻¹^ψ⁻¹AutP(R) (8) hence we must have that N_φ ≤ N_ψφ. Since ψ has an extension ¯ψ : N_P(Q) → N_P(R) and θ = ψφ has an extension ¯θ : N_φ → N_P(R) we need to show that θ(N¯ φ) ≤ ¯ψ(NP(Q)). Now, by part 1, both Q and R are fully F -centralized and hence we have that ¯ψ : CP(Q) → CP(R) is an isomorphism so by Proposition 2.32 we have that ¯ψ(NP(Q)) is the full preimage in NP(R) of^ψAutP(Q). Also

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θ(N¯ _φ) is the preimage in N_P(R) of c_R(¯θ(N_φ)) =^θ(c_S(N_φ)) ≤^ψAut_P(Q) since

φ(c_S(N_φ)) ≤ Aut_P(Q) and thus ¯θ(N_φ) ≤ ¯ψ(N_P(Q)) so that ¯ψ⁻¹|θ(N¯ _φ)◦ ¯θ is a map from Nφ to NP(Q) extending φ = ψ⁻¹θ.

Theorem 2.34. Let F be a saturated fusion system on a finite p-group P , and let Q ≤ P . Then Q is fully F -normalized if and only if Q is fully F -automized and receptive.

Proof. Suppose that Q is fully F -automized and receptive. Then we have from Proposition 2.33 that Q is fully F -centralized. Thus, since Q is both fully F - centralized and fully F -automized we have from |NP(Q)| = |CP(Q)||AutP(Q)|

that Q is fully F -normalized.

Now suppose that Q is fully F -normalized. Then, since F is saturated, we can find an R ≤ P such that R is fully F -automized and receptive. Now from Proposition 2.33 we know that R is fully F -centralized since it is receptive.

Thus, from the arguments above, we have that R is fully F -normalized. This implies that |NP(R)| = |NP(Q)|, hence

|NP(Q)| = |AutP(Q)||CP(Q)| = |AutP(R)||CP(R)| = |NP(R)| (9) Now since R is fully F -automized and fully F -centralized we must have that

|C_P(R)| ≥ |C_P(Q)| and |Aut_P(R)| ≥ |Aut_P(Q)| hence we must have equality for both atomizers and centralizers. And thus Q is fully F -automized and receptive, since, from Proposition 2.33, fully F -centralized implies receptive.

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3 Proving Saturation of Fusion Systems

We have shown that all fusion systems on finite groups have saturated fusion systems but the converse is not true. There are saturated fusion systems that are not isomorphic to a fusion system on some finite group. These fusion systems are called exotic fusion systems.

When dealing with abstract fusion systems it is generally only saturated fusion systems that are studied. Hence a nice way of showing that a fusion system is saturated is desirable. It is generally difficult to show that the fusion system is saturated directly from the definition. In the following sections we will study how proving that a fusion system is saturated can be simplified and also how to construct saturated fusion systems using those methods.

To get a better understanding of what might be controlling saturation in a fusion system we will start with looking at F -essential subgroups and Alperin’s fusion theorem formulated for fusion systems.

3.1 Centric and Essential Subgroups

First we begin with a definition of what it means for a group to have a strongly p-embedded subgroup.

Definition 3.1. Let G be a finite group and p a prime dividing |G|, and let M < G. M is strongly p-embedded in G, if M contains a Sylow p-subgroup of G and M ∩^gM is a p⁰−group for all g ∈ G \ M .

Remark. If G has a strongly p-embedded subgroups and P, Q ∈ Syl_p(G) then P ∩ Q = 1.

This is definition is needed for the definition of an F -essential subgroup.

Also we need to define what it means for a subgroup to be F -centric.

Definition 3.2. Let F be a fusion system on a finite p-group P . A subgroup Q is F -centric if for every subgroup R, F -conjugate to Q, CP(R) ≤ R.

To get a better understanding of this definition we look at the following proposition.

Proposition 3.3. Let F be a saturated fusion system on a finite p-group P . If Q ≤ P is fully F -centralized then QC_P(Q) is F -centric.

Proof. Suppose Q is fully F -centralized and let ˜Q = QC_P(Q). Since Q ≤ ˜Q and C_P(Q) ≤ ˜Q, C_P( ˜Q) centralizes both Q and C_P(Q) and hence C_P( ˜Q) ≤ CP(Q) ∩ CP(CP(Q)) ≤ ˜Q.

Let φ : ˜Q → R be an F -isomorphism. If φ(a) ∈ φ(CP(Q)), then ∀φ(q) ∈ φ(Q), φ(a)φ(q)φ(a)⁻¹ = φ(q) ⇒ φ(a) ∈ CP(φ(Q)). Hence we have that R = φ(QCP(Q)) = φ(Q)φ(CP(Q)) ≤ φ(Q)CP(φ(Q)).

Now, as Q is fully centralized, |CP(Q)| ≤ |CP(φ(Q))| and hence CP(Q) ≤ R, by the same argument that CP( ˜Q) ≤ ˜Q. Thus Q i F -centric.

Definition 3.4. Let F be a fusion system on a finite p-group P . A subgroup Q is F -essential if Q is F -centric and OutF(Q) ∼= AutF(Q)/Inn(Q) contains a strongly p-embedded subgroup.

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Example 3.1. Let P = D₈ then P = hab | |a| = 4, |b| = 2 and ab = ba⁻¹i and

|P | = 8. Denote T₀= ha², abi and T₁= ha², bi. Then T₀ and T₁are isomorphic to C2× C2. We see that both T0 and T1 have order 4 and since we can find elements in P \ Ti that are in AutP(Ti), e.g b ∈ P for i = 0 and a ∈ P for i = 1, we must have that both T0 and T1are F -centric.

Suppose now that F is a saturated fusion system on P . We have that both T0 and T1 must be fully F -normalized since there are no subgroups of P that are conjugate to Ti, for i ∈ {0, 1} and T0 and T1 are not conjugate to each other. Thus, since F is saturated, we must also have, from Theorem 2.34, that they are fully F -automized and receptive. Then AutP(Ti) ∈ Syl_p(AutF(Ti)) and thus AutP(Ti)/Inn(Ti) ∼= OutP(Ti) ∈ Syl_p(OutF(Ti)) since OutF(Ti) ∼= AutF(Ti)/Inn(Ti). Now also we notice that Inn(Ti) is in fact trivial since Ti

is abelian so that Aut_F(T_i) ∼= OutF(T_i). We also notice that |Aut(T_i)| = 6 and Aut(T_i) ∼= S3 so that |Aut_F(T_i)| ∈ {2, 6}. Also S₃ contains a strongly 2- embedded subgroup since it contains 2-groups of order 2 that are all isomorphic and do not intersect, i.e if Q ∈ Syl₂(S₃) then Q∩^gQ = 1 for any g ∈ S₃\Q, and 1 is a 2⁰-element. However, if |Aut_F(T_i)| = 2 then clearly Aut_F(Q) cannot contain a strongly 2-embedded subgroup. With this we see that if |Aut_F(Ti)| = 6 then Ti is F -essential.

Now we look at P itself. We see that |Aut(P )| = 8 which is a 2-group and can thus not contain a strongly 2-embedded subgroup. Thus P can not be F -essential. So take the cyclic group C4 = hai then |Aut(C4)| = 2 and hence Out_F(C4) can not contain a strongly 2-embedded subgroup and C4 cannot be F -essential. We now look at the subgroups of P of order 2 and we see that none of them are F -centric and thus the only possible F -essential subgroups of a saturated fusion system on P are the subgroups T0 and T1.

We now look at Alperin’s fusion theorem formulated in the setting of abstract fusion systems.

Theorem 3.5 (Alperin’s Fusion Theorem). Let F be a saturated fusion system on a finite p-group P and let Q denote the set of fully F -normalized essential subgroups of P . Let Q, R ≤ P be F -conjugate and let φ ∈ Hom(R, Q). Then there exists

1. a sequence of F -conjugate subgroups Q = Q0, Q1, . . . , Qn+1= R, 2. a sequence S1, S2, . . . , Sn∈ Q such that Qi−1, Qi≤ Si,

3. a sequence of automorphisms φi ∈ Hom(Si, Si) such that φi(Qi−1) = Qi, and

4. an automorphism ψ ∈ Hom(P, P ) such that ψ(Qn) = Qn+1and such that (φ1φ2. . . φnψ)|Q = φ.

A detailed proof can be found in [2].

Example 3.2. Suppose P = D₈ and let T₀, T₁ be defined as in Example 3.1.

The following holds for any fusion system over P :

1. Since P is fully F -automized and |Aut(P )| = 8 and P is a 2-group, we must have that Aut_F(P ) = AutP(P ).

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2. If R = hai and Q = P ⇒ Hom_F(R, Q) = Hom_P(R, Q) = Hom(R, Q).

3. The subgroups T₀ and T₁ can not be F -conjugate since they contain elements of different cycle type.

4. Let R = ha²i and let Q ≤ P be any subgroup of order 2. Since R C P ⇒ R is fully F -normalized and hence fully automized and receptive. So if Q is F -conjugate to R, by Lemma 3.11, ∃ a map ψ : Ti → NP(R), where Ti= NP(Q), that restricts to a map φ : Q → R.

5. By (1-4) F is completely determined by AutF(T0) and AutF(T1). For each i, AutP(Ti) ≤ AutF(Ti) ≤ Aut(Ti), and thus AutF(Ti) has order 6 or 2. From Example 3.1 we know that Ti is F -essential if and only if

|AutF(T_i)| = 6 and these are the only F -essential groups in P .

Thus there are only four saturated fusion systems on P . Denote these fusion systems F_ij where i = 0 if |Aut_F(T₀)| = 2 and i = 1 if |Aut_F(T₀)| = 6 and similarly for j and |Aut_F(T₁)|. Then F₀₀ is isomorphic to the fusion system on the Sylow 2-group of D₈, i.e. D₈ itself. We also have that F₁₀ ∼= F01 and they are isomorphic to the fusion system on the Sylow 2-group of S4. F11 is isomorphic to the fusion system on the Sylow 2-group of A6. Note that there can be no exotic fusion systems on P .

Corollary 3.6. Let F be a fusion system on a finite p-group P . Then any morphism in F can be written as a finite composition of restrictions of automorphisms of F -essential subgroups of P and automorphisms of P .

3.2 The Surjectivity Property

In this section we will introduce a new property that a subgroup of P can have, namely the surjectivity property. This property makes being fully F -automized and Q-receptive into a single condition.

Definition 3.7. Let F be a fusion system on a finite p-group P . Let Q, R ≤ P such that Q and R are F -conjugate. We say that Q is R-receptive if for any isomorphism φ : R → Q in F , φ can be extended to a map ψ : Nφ→ NP(Q).

Remark. If Q is R-receptive for all R ≤ P that are F -conjugate to Q then Q is receptive.

Proposition 3.8. Let F be a fusion system on a finite p-group P . If Q ≤ P is such that Q is Q-receptive and that for any R ≤ P , F -conjugate to Q, there is a map φ : R → Q that extends to some ψ : N_P(R) → N_P(Q) such that ψ|_R= φ.

Then Q is receptive.

Proof. Take any α : R → Q in F . Now take φ : R → Q such that φ can be extended to some ψ : NP(R) → NP(Q). Then the map θ = α ◦ ψ⁻¹ is an automorphism of Q and thus is has an extension ¯θ : Nθ → NP(Q) since Q is Q−receptive. Now Nαis the preimage of AutP(R)∩φ⁻¹◦AutP(Q)◦φ in R and ψ(Nα) induces the automorphism group ψ ◦ AutP(R) ◦ ψ⁻¹∩ ψ ◦ φ⁻¹◦ AutP(Q) ◦ φ ◦ ψ⁻¹ of Q. Since ψ ◦ AutP(R) ◦ ψ⁻¹≤ AutP(Q) we see that this intersection is contained in AutP(Q) ∩ θ ◦ AutP(Q) ◦ θ⁻¹ so that ψ(Nα) ≤ Nθ. Hence there is an extension of α to ¯α : Nα→ NP(Q), proving that Q is receptive.

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We now gather the property of being fully F -automized and Q-receptive into one property, the surjectivity property.

Definition 3.9. Let Q be a subgroup of P .

1. If R be a subgroup of P such that Q ≤ R ≤ N_P(Q), then by Aut(Q ≤ R) it is meant the automorphisms of R in F that restricts to automorphisms of Q.

2. Q has the surjectivity property of F if whenever QC_P(Q) ≤ R ≤ N_P(Q) there is a map φ : Aut_F(Q ≤ R) → N_Aut_F_(Q)(Aut_R(Q)) such that φ is surjective.

Remark. If Q has the surjectivity property then QCP(Q) ≤ R ≤ NP(Q) and any map φ ∈ Aut_F(Q) that normalizes AutR(Q) extends to some ψ ∈ Aut_F(R).

Example 3.3. Let P = D₈ and let F be a saturated fusion system on P . Suppose P = hab | |a| = 4, |b| = 2 and ab = ba⁻¹i and let Q = ha², abi. Suppose

|Aut_F(Q)| = 6 and Aut_F(Q) ∼= S3 as in Example 3.1. Now N_P(Q) = P and Aut_F(P ) = AutP(P ) thus Aut_F(Q ≤ P ) = AutP(Q ≤ P ) = AutP(Q) = hbi.

Now we need to check if the morphisms φ, φ⁻¹ ∈ Aut_F(Q) \ AutP(Q) are in N_Aut_F_(Q)(AutP(Q)). We know that Aut_F(Q) ∼= S3 and that |AutP(Q)| = 2, then AutP(Q) is isomorphic to a 2-group of S3 and the normalizer of any 2- group in S3is the 2-group itself, thus N_Aut_F_(Q)(AutP(Q)) = Aut_F(Q ≤ R) and hence Q has the surjectivity property.

Definition 3.10. Let F be a fusion system on a finite p-group P and let Q ≤ P , then F is inductively saturated with respect to Q if all F -conjugacy classes properly containing Q have a fully F -automized and receptive member.

Also, F is inductively saturated with respect to Q for some F -conjugacy class Q if F is inductively saturated with respect to some Q ∈ Q. An F -conjugacy class Q is said to be saturated if it contains a fully F -automized and receptive member.

Lemma 3.11. Let F be a saturated fusion system on a finite p-group P . If Q ≤ P is fully F -normalized then

1. if R ≤ P is F -conjugate to Q there is a ψ : R → Q that extends to a morphism ¯ψ : NP(R) → NP(Q).

2. Q has the surjectivity property.

Proof. Since F is saturated and Q is fully F -normalized we know from Theorem 2.34 that Q is fully F -automized and receptive. Suppose R ≤ P is F -conjugate to Q. Since Q is receptive we know, from the definition, that any morphism φ : R → Q in F extends to a morphism ¯φ : Nφ → NP(Q). Now since Q is fully F -automized we know from Proposition 2.31 that there is a morphism ψ : R → Q in F such that N_ψ = N_P(R), hence ψ extends to a morphism ψ : N¯ _P(R) → N_P(Q) since Q is receptive.

Now to prove (2) we have from above that Q is receptive. Hence any morphism φ : Q → Q in F extends to a morphism ¯φ : N_φ → N_P(Q) in F . Let QCP(Q) ≤ R ≤ NP(Q). We now want to show that any α ∈ Aut_F(Q) that normalizes AutR(Q) extends to a morphism ¯ψ ∈ Aut_F(R), then Q has the surjectivity property by definition. Suppose that α ∈ Aut_F(Q) normalize AutR(Q).

Saturated Fusion Systems