Fusion Systems on Finite Groups and Alperin's Theorem

Fusion Systems On Finite Groups and Alperin’s Theorem

Author: Eric Ahlqvist, [email protected]

Supervisor: Tilman Bauer

SA104X - Degree Project in Engineering Physics, First Level Department of Mathematics Royal Institute of Technology (KTH) May 21, 2014

1 Abstract Let G be a group and P a Sylow p-subgroup of G.A fusion system of G on P , denoted by FP (G), is the category with objects; subgroups of P , and morphisms induced by conjugation in G. This thesis gives a brief introduction to the theory fusion systems. Two classical theorems of Burnside and Frobenius are stated and proved. These theorems may be seen as a starting point of the theory of fusion systems, even though the axiomatic foundation is due to Puig in the early 1990’s. An abstract fusion system F on a p-group P is defined and the notion of a saturated fusion system is discussed. It turns out that the fusion system of any finite group is saturated, but the converse; that a saturated fusion system is realizable on a finite group, is not always true. Two versions of Alperin’s fusion theorem are stated and proved. The first one is the classical formulation of Alperin and the second one, due to Puig, a version stated in the language of fusion systems. The differences between these two are investigated. The fusion system F of GL2(3) on the Sylow 2-subgroup isomorphic to SD16 is determined and the subgroups generating F are found.

Section 1-5 is written together with Karl Amundsson and Oliver G¨afvert.

Contents

1 Introduction 3

2 Fusion systems 4

3 Transfer and the focal subgroup theorem 8

4 Proof of Frobenius (4)⇒(1) 11

5 Abstract fusion systems 12

6 Alperins fusion theorem 16

7 Centric, radical and essential subgroups 19

8 Alperin’s theorem for fusion systems 22

9 A fusion system on GL2(3) 27

A Some properties of ﬁnite groups 30

2 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 [5]. 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 [2]. 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 [5] and one by Michael Aschbacher, Radha Kessar and Bob Oliver [4]. 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 pk, 0 ≤ k ≤ n, is contained in some Sylow p-subgroup. 4. The number of subgroups of order pk, 0 ≤ k ≤ n, is congruent to 1 modulo p.

5. The number of Sylow p-subgroups equals |G : NG(P )|, where NG(P ) is the normalizer of P ∈ Sylp(G). In particular, |Sylp(G)| divides m. A proof can be found in [6].

3 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 x −1 and g = x gx and the map cy : G → G denotes conjugation by the element y y, i.e., cy(g) = g. 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 ϕg : A → B for some g ∈ G, then there is some k ∈ K such that ϕg and ϕk agree on A, i.e., if ∀a ∈ A, ϕg(a) = ϕk(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 FP (G), whose objects are all subgroups of P and whose morphisms are

HomFP (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 FP (G) be the fusion system of G on P and let Q be a subgroup of P . Then we deﬁne the automorphism group AutP (Q) by AutP (Q) = HomP (Q, Q).

Remark. AutP (Q) is isomorphic to NP (Q)/CP (Q), which can be seen by ap- plying the first isomorphism theorem on the natural homomorphism NP (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 = FP (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-isomorphic 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-isomorphic subgroups are F-conjugate. 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-isomorphic to Q.

4 Deﬁnition 2.6. 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 . 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 HomFsc (A, B) = HomF (A, B).

In category theory, one would say that Fsc is equivalent to F.

Example 2.7. 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-isomorphic.

We also have that the groups {1, (12)(34)}, {1, (13)(24)} and {1, (14)(23)} are F-isomorphic. 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.

5 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 HomF (Q, R). Note that Figure 2 does not cover all information of the fusion system since it does not tell what HomF (Q, R) are explicitly. We have that AutF (Q) acts on HomF (Q, R) on the right and that AutF (R) ∼ acts on HomF (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 HomF (Q, R) from the left as S3 on the set of three letters while AutF (Q) = 1 acts trivially from the right on HomF (Q, R). Denote 1, (12)(34), (13)(24), (14)(23) by a, b, c, d respectively. Then HomF (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 ∈

AutF (P ) there are only two possibilities for ψ R. Either ψ R = IdR or ψ R swaps (13)(24) and (14)(23). Hence the left action of AutF (P ) on HomF (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 AutF (P ) will act on HomF (R,P ) as h(12)(34)(56)i acts on the set {1, 2, 3, 4, 5, 6}.

6 Note that if Q is not F-isomorphic to any other subgroup in F, then if α ∈ HomF (Q, R) is the inclusion map of Q into R then HomF (Q, R) = α ◦ AutF (Q) and we have |HomF (Q, R)| = |AutF (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 2.8 (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 x x cx CG( Q) and hence we can find a c ∈ CG( Q) such that P = P . This means cx −1 −1 −1 that cx ∈ NG(P ) and since u = c(xux )c = xux = ϕ(u) for u ∈ Q, we are done. Definition 2.9. 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 2.10. 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 0 0 p -subgroup. This maximal normal p -subgroup will be denoted by Op0 (G). Definition 2.11. 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. 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 FP (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:

7 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 ∈ Sylp(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 −1 −1 for some x ∈ G. We want to show that cx(u) = xux = pup = 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 by Lemma A.1

[y, zuz−1] = y zuz−1y−1zu−1z−1 = xux−1 zu−1z−1 ∈ P ∩ K = 1, |{z} | {z } | {z } | {z } ∈K ∈K ∈P ∈P so that xux−1 = zuz−1. 3) =⇒ 4): This follows from the fact that

AutG(Q) = HomG(Q, Q) = HomP (Q, Q) = NP (Q)/CP (Q) is a p-group. 2) =⇒ 4): Let Q be a non-trivial subgroup of P . Then we can write NG(Q) = K1 oP1. 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 |K1||P1| |NG(Q)|/|CG(Q)| = , so we are done if we can show that |K1| = |K2|. |K2||P2| Obviously, |K2| ≤ |K1|. Now, Q and K1 are both normal in NG(Q) and have coprime orders, so that by the lemma, K1 commutes with every element in Q. Thus K1 ≤ CG(Q) = K2P2, and so K1 ≤ K2 and ﬁnally |K1| = |K2|. The fact that 4) implies 1) will be proved in section 4.

3 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 3.1. 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

8 −1 σg(i) = j. Then xigxj = hi,g ∈ H where hi,g depends on i and g. Deﬁne τ : G → A by Y Y −1 τ(g) = φ(hi,g) = φ(xigx ) σg (i) i∈I i∈I We say that τ is the transfer of G into A via φ.

Theorem 3.2. 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 3.3. 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:

ri −1 1. xig xi ∈ H for some positive integers ri, 0 ≤ i ≤ t. Pt 2. i=1 ri = n = |G : H|

Q ri −1 3. τ(g) = φ( i∈I xig xi ) In the proof of this theorem we follow the proof in Daniel Gorenstein [8].

Proof. Let yi be coset representatives of H in G, 0 ≤ i ≤ n. Let σg ∈ Sn be deﬁned by yig ∈ Hyσg (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 ri = n = |G : H| i=1

Hence (2) is proved.

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

j R := {xig |1 ≤ i ≤ t, 0 ≤ j ≤ ri − 1}

ri form a complete set of coset representatives for H in G. But then xig ∈ Hxi ri −1 by deﬁnition of ri and hence xig xi ∈ H which proves (1). Now we prove (3). Lets use R as coset representatives and compute τ(g). j Let y = x g (k depend on i and j) and consider y g = h y 0 , h ∈ H. If k i k k,g kg k,g

9 j+1 j+1 j < r − 1, then y g = x g = y 0 . Since x g is a coset representative in R i k i kg i j+1 j+1 and the x g ∈ Hy 0 if and only if y 0 = x g , by definition of h . i kg kg i k,g This implies that hk,g = 1 whenever j < ri − 1. Hence τ(g) is the product ri−1 of those φ(hk,g) which corresponds to the elements yk = xig for which we have ri ri −1 ri −1 ykg = xig ∈ Hxi ⇒ ykg = (xig xi )xi, (xig xi ) ∈ H. r −1 Hence x = y 0 and we get h = x g i x for each y . Therefore we get that i kg k,g i i k t ! Y ri −1 τ(g) = φ xig xi i=1 and since φ is a homomorphism we get (3). Theorem 3.4 (The Focal Subgroup Theorem, [9]). Let G be a finite group, 0 P ∈ Sylp(G) and let G be the commutator subgroup of G. Then P ∩ G0 = h[x, g] = x−1xg : x ∈ P, g ∈ G, xg ∈ P i −1 = hx φ(x): x ∈ P, φ ∈ HomFG(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−1xg : x ∈ P, g ∈ G, xg ∈ P i. We want to show that P ∗ = P ∩ G0. Since x−1xg = [x, g], we obviously have P ∗ ≤ P ∩ G0 and since P 0 ≤ 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 = PK ⇒ 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 t ! t Y ri −1 Y ri −1 ∗ τ(x) = φ xix xi ≡ xix xi (mod P ). i=1 i=1 Since P/P ∗ is abelian, we have t ! t ! t ! Y ri −ri ri −1 Y ri Y ri ri −1 ∗ τ(x) = φ x x xix xi ≡ x x xix xi (mod P ). i=1 i=1 i=1

−ri ri −1 ri 0 ∗ But since x xix xi = [x , xi] ∈ P ≤ P we have that t Y P τ(x) ≡ xri (mod P ∗) ≡ x ri (mod P ∗) ≡ xn(mod P ∗). i=1 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 ∗.

10 4 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 [1] in 1967. Definition 4.1. 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 NP (R) and NQ(R) are Sylow p-subgroups of NG(R).

We will later see that in a fusion system FP (G) of a ﬁnite group G on a Sylow p-subgroup P , a tame intersection of P with any Q ∈ Sylp(G) will be so-called fully F-normailized which is an important property of a group in a fusion system. Example 4.2. Every Sylow p-subgroup is a tame intersection, as we see from Q = Q ∩ Q. Theorem 4.3 (Alperins Fusion Theorem, [1]). Let G be a ﬁnite group and g P ∈ Sylp(G). Let A, A ⊆ P , for some g ∈ G. Then there exists elements x1, x2, ..., xn, subgroups Q1,Q2, ..., Qn ∈ Sylp(G) and an y ∈ NG(P ) such that

1. g = x1x2...xny,

2. P ∩ Qi is a tame intersection, 0 ≤ i ≤ n,

3. xi is a p-element of NG(P ∩ Qi), 0 ≤ i ≤ n,

x1x2...xi 4. A ⊆ P ∩ Q1 and A ⊆ P ∩ Qi+1, 0 ≤ i ≤ n − 1. Proposition 4.4 (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 ).

Proof. We want to show that for any morphism in FP (G) induced by conjugation with some g ∈ G, g can be written as a product of elements x1, x2 . . . , xn, y ∈ P . From Theorem 4.3 we know we can ﬁnd Qi ∈ Sylp(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) ∈ Sylp(NG(P ∩ Qi)), thus NP (P ∩Qi)CG(P ∩Qi)/CG(P ∩Qi) ∈ Sylp(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) ∼ NP (P ∩ Qi) ∼ = = AutP (P ∩ Qi) (1) CG(P ∩ Qi) NP (P ∩ Qi) ∩ CG(P ∩ Qi)

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 ) ∈ Sylp(NG(P )). But NG(P ) was assumed to be a p-group and hence NG(P ) = NP (P ) which implies that y ∈ P .

11 Now to the proof of that (4) implies (1), in Frobenius Theorem. We will denote by Op(G), the smallest normal subgroup of G such that G/Op(G) is a p-group. Proposition 4.5 (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 CG such that G/H is a p-group. Let Q be a proper p-subgroup of H. By assumption NG(Q)/CG(Q) is a p-group. We have that H ∩ NG(Q) = NH (Q) and H ∩ CG(Q) = CH (Q) and hence NH (Q)/CH (Q) is a p-group. Hence, by induction H has a normal p-complement K = Op0 (H). By Theorem 2.10, 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 Lemma A.4. 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 4.4 and the focal subgroup theorem we have that

P ∩ G0 = hx−1xg|x, xg ∈ P, g ∈ Gi = hx−1xg|x, xg, g ∈ P i = P 0 (2) and we have that P 0 < P since P is a p-group. Now consider the group 0 p 0 G O (G) E G. Let φ : P → P/P be the natural homomorphism and let τ be the transfer from G into P/P 0 via φ. Then we have that

G/ ker τ =∼ P/P 0 6= 1. (3)

This means that G0 ≤ ker τ since P/P 0 is abelian and Op(G) ≤ ker τ since P/P 0 is a p-group. Thus G0Op(G) ≤ ker τ and hence G0Op(G) is a proper normal subgroup of G. Obviously G/(G0Op(G)) is a p-group as G/Op(G) is by deﬁnition of Op(G). So if we put H = G0Op(G) in part one, we are done.

5 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 , FP (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

12 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 5.1. 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 HomF (Q, R) are sets of injective homomorphisms having the following three properties:

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

2. For each φ ∈ HomF (Q, R), the induced isomorphism Q → φ(Q) and its inverse lies in HomF (Q, φ(Q)) and HomF (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 5.2. Let P be a p-group and F the fusion system on P such that, for any subgroups Q, R of P , HomF (Q, R) consists of all injective homomorphisms from Q to R. Then F is called the universal fusion system on P . Consider the fusion system on D8 in Example 2.7. For the Klein 4-group V = {id, (12), (34), (12)(34)} we have AutF (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 D8 we see that FD8 (G) is the universal fusion system on D8 as 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 5.3. 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 5.4. 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

y z Nφ = {y ∈ NP (Q)| ∃ z ∈ NP (φ(Q)) such that φ( u) = φ(u), ∀u ∈ Q} (4)

Note that QCP (Q) E Nφ E NP (Q). Definition 5.5. 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 5.6. Let F be a fusion system on a finite group G and let P ≤ G such that P ∈ Sylp(G). If Q ≤ P is such that NP (Q) ∈ Sylp(NG(Q)) then Q is receptive.

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

−1 −1 Nφ = {x ∈ NP (R)| ∃ y ∈ NP (Q) such that φ(xrx ) = yφ(r)y , ∀r ∈ R}, (5) −1 −1 thus cx−1 ◦ φ ◦ cy ◦ φ centralizes R and cy ◦ φ ◦ cx−1 ◦ φ centralizes φ(R) = 0 0 Q ⇒ φ ◦ cx = φ ◦ φ, for some φ 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) ∈ Sylp(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 : Nφ → NP (Q), and the proof is done. Definition 5.7. 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 |CP (R)| ≤ |CP (Q)| for any R ≤ P F-isomorphic to Q. Q is called fully F-normalized if |NP (R)| ≤ |NP (Q)| for any R ≤ P F-isomorphic 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.

Example 5.8. Take G = S4 and let P = h(1243), (12)i then P ∈ Sylp(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 H1 C P while H2 and H3 are not, hence H1 is fully F-normalized. Furthermore, NP (H1) = P ∈ Sylp(NG(H1)) and thus, by Proposition 5.6, H1 is receptive. Proposition 5.9. Let F be a fusion system on a ﬁnite group G and let P ≤ G such that P ∈ Sylp(G). Let Q ≤ P

1. Q is fully F-centralized if and only if CP (Q) ∈ Sylp(CG(Q))

2. Q is fully F-normalized if and only if NP (Q) ∈ Sylp(NG(Q))

Proof. First we prove 1. Let S ∈ Sylp(CG(Q)) such that CP (Q) ≤ S. By Sylow’s theorem there is a g ∈ G such that (QS)g ≤ P and we have that Q =∼ Qg. For g −1 −1 −1 −1 any y ∈ S , gyg ∈ CG(Q) which implies that (gyg )z(gy g ) = z ⇔ −1 −1 −1 g g y(g zg)y = g zg for all z ∈ Q. Hence S ≤ CG(Q ) ∩ P = CP (Q ) and we g conclude that |CP (Q)| ≤ |S| ≤ |CP (Q )|. 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 5.10. 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.

14 Theorem 5.11. 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 5.9, NP (Q) ∈ Sylp(NG(Q)) and then by Proposition 5.6, Q is receptive. Now notice that NP (Q)CG(Q)/CG(Q) ∈ Sylp(NG(Q)/CG(Q)) and ∼ AutG(Q) = NG(Q)/CG(Q) and

NP (Q)CG(Q) ∼ NP (Q) NP (Q) ∼ = = = AutP (Q). (7) CG(Q) NP (Q) ∩ CG(Q) CP (Q)

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

Theorem 5.11 implies that the class of saturated fusion systems is at least as big as the class of ﬁnite groups. Proposition 5.12. Let F be a fusion system on a ﬁnite p-group P and let Q, R be F-isomorphic 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). −1 If φ : Q → R is an isomorphism in F, then φ ◦ AutP (Q) ◦ φ is a p-subgroup of AutF (R) and hence there is an α ∈ AutF (R) such that α ◦ φ ◦ AutP (Q) ◦ −1 −1 φ ◦ α ≤ AutP (R). Now let ψ = α ◦ φ. Now this says exactly that for any −1 x ∈ NP (Q) there is an y ∈ NP (R) such that ψ◦cx ◦ψ = cy ⇐⇒ ψ◦cx = cy ◦ψ as ψ is injective. But then NP (R) satisﬁes the deﬁnition of Nψ and we get Nψ = NP (Q).

We denote by cQ, the map NP (Q) → AutP (Q) defined by cQ(g) = cg for g ∈ NP (Q). Proposition 5.13. Let F be a saturated fusion system on a finite p-group P . Let Q and R be F-isomorphic 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) ≤ φ−1 AutP (Q) ∩ AutP (R). Proof. Let x ∈ S. For all g ∈ Q we have that xg ∈ Q and thus φ¯(xφ¯(g)x−1) = ¯ −1 ¯ ¯ φ(xgx ) ∈ R ⇒ φ(x) ∈ NP (R). Thus the image of φ is in NP (R). Also, ¯ φ ¯ we have that cR(φ(x)) ∈ AutP (R), therefore (cQ(S)) = cR(φ(S)) ≤ AutP (R). φ−1 And since cQ(S) ≤ AutP (Q) we have that cQ(S) ≤ AutP (Q)∩ AutP (R). Proposition 5.14. 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

15 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 ∈ CP (Q). Hence ψ(CP (R)) ≤ CP (Q) ⇒ |CP (R)| ≤ |CP (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 5.12 there is an isomorphism ¯ ψ : Q → R that can be extended to a morphism ψ : NP (Q) → NP (R). Hence ψ AutP (Q) ≤ AutP (R). Now suppose that S ≤ P is F-conjugate to Q and that φ : S → Q is a morphism in F. Then

φ−1 φ−1ψ−1 AutP (S) ∩ AutP (Q) ≤ AutP (S) ∩ AutP (R) (8) ¯ hence we must have that Nφ ≤ Nψφ. Since ψ has an extension ψ : NP (Q) → ¯ NP (R) and θ = ψφ has an extension θ : Nφ → NP (R) we need to show that ¯ ¯ θ(Nφ) ≤ 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 ¯ ψ 5.13 we have that ψ(NP (Q)) is the full preimage in NP (R) of AutP (Q). Also ¯ ¯ θ ψ θ(Nφ) is the preimage in NP (R) of cR(θ(Nφ)) = (cS(Nφ)) ≤ AutP (Q) since φ −1 (c (N )) ≤ Aut (Q) and thus θ¯(N ) ≤ ψ¯(N (Q)) so that ψ¯ |¯ ◦ θ¯ is a S φ P φ P θ(Nφ) −1 map from Nφ to NP (Q) extending φ = ψ θ. Theorem 5.15. 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 5.14 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 , F-isomorphic to Q, such that R is fully F-automized and receptive. Now from Proposition 5.14 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 |CP (R)| ≥ |CP (Q)| and |AutP (R)| ≥ |AutP (Q)| as |AutF (R)| = |AutF (Q)|. Hence we must have equality for both atomizers and centralizers. And thus Q is fully F-automized and receptive, since, from Proposition 5.14, fully F- centralized implies receptive.

6 Alperins fusion theorem

In this chapter we will look att Alperin’s fusion theorem [1], which tells that conjugation in a ﬁnite group can be carried out in a series of conjugations by elements with certain properties. In chapter 9 we will look at a reformulation

16 of Alperin’s fusion theorem which is more suitable for fusion systems. We will also bring up some examples to show what consequences this has on the fusion systems on ﬁnite groups. We start with some deﬁnitions and then a series of lemmas which we will use to prove Alperin’s fusion theorem

Deﬁnition 6.1. Let P be Sylow p-subgroups of G. For R,Q ∈ Sylp(G) write R ∼ Q if there exists Sylow p-subgroups Q1,Q2,...,Qn and elements x1, x2, . . . , xn s.t.

1. P ∩ Qi is a tame intersection, 1 ≤ i ≤ n,

2. xi is a p-element in NG(P ∩ Qi), 1 ≤ i ≤ n,

x1x2...xi 3. P ∩ R ≤ P ∩ Q1 and (P ∩ R) ≤ P ∩ Qi+1, 1 ≤ i ≤ n − 1

x 4. R = Q where x = x1x2...xn we say R ∼ Q via x and that the set {Qi, xi : 1 ≤ i ≤ n} accomplish R ∼ Q. Lemma 6.2. The relation ∼ is transitive.

Proof. If {Ri, yi : 1 ≤ i ≤ m} and {Qi, xi : 1 ≤ i ≤ n} accomplish S ∼ R and R ∼ Q respectively, then R1,...,Rm,Q1,...,Qn accomplish S ∼ Q. Lemma 6.3. If S ∼ P via x, Qx ∼ P and P ∩ Q ≤ P ∩ S, then Q ∼ P . Proof. By Lemma 6.2 it is enough to show that Q ∼ Qx. Claim: If {Si, xi : 1 ≤ i ≤ n} accomplish S ∼ P then {Si, xi : 1 ≤ i ≤ n} also accomplish Q ∼ Qx. pf: (1),(2) and (4) in Deﬁnition 6.1 are trivial and (3) is clear as P ∩ Q ≤ P ∩ S ≤ P ∩ Si since S ∼ P .

Lemma 6.4. Assume R,Q ∈ Sylp(G) with R ∼ P and P ∩Q < R∩Q. Assume further, for all S ∈ Sylp(G) with |S ∩ P | > |Q ∩ P |, that S ∼ P . Then Q ∼ P . Proof. By assumption ∃x ∈ G such that R ∼ P via x. Now P ∩Qx = Rx ∩Qx = (R ∩ Q)x, so |P ∩ Qx| = |R ∩ Q| > |P ∩ Q|. Hence Qx ∼ P and by lemma 6.3 Q ∼ P . Lemma 6.5. Assume P and Q be Sylow p-subgroups of G, P ∩ Q is a tame intersection, and S ∼ P , ∀S ∈ Sylp(G) such that |P ∩ S| > |P ∩ Q|. Then Q ∼ P .

Proof. By Lemma 6.2 we may assume that Q 6= P . Thus P ∩ Q < P0 = NP (P ∩ Q) ∈ Sylp(NG(P ∩ Q)) and NQ(P ∩ Q) ∈ Sylp(NG(P ∩ Q)). x Hence there is an x ∈ NG(P ∩ Q) such that NQ(P ∩ Q) = NP (P ∩ Q) and thus Q ∼ Qx is accomplished by {Q,x}. x x Furthermore P ∩ Q < P0 ≤ P ∩ Q , so by hypothesis Q ∼ P . Hence by Lemma 6.2 Q ∼ Qx ∼ P ⇒ Q ∼ P .

Lemma 6.6. If Q, P ∈ Sylp(G) then Q ∼ P .

17 Proof. Proceed by induction on |P : P ∩ Q| =: n. If n = 1 ⇒ P ∩ Q = P ⇒ Q = P and Q ∼ P . Now assume Q 6= P and that S ∼ P, ∀S ∈ Sylp(G) such that |S∩P | > |Q∩P |. Take S ∈ Sylp(NG(P ∩ Q)) containing NP (P ∩ Q), and let R ∈ Sylp(G) such that S ≤ R ⇒ P ∩ R ≥ P ∩ S ≥ Np(P ∩ Q) > P ∩ Q ⇒ R ∼ P via some x ∈ G, by induction. Thus, since P ∩ R ≥ P ∩ Q and R ∼ P via some x ∈ G we need only show that Qx ∼ P to establish Q ∼ P . First, (P ∩ Q)x ≤ Sx ≤ P so that P ∩ Qx ≥ P ∩ (P ∩ Q)x = (P ∩ Q)x. However, if |P ∩Qx| > |P ∩Q| then Qx ∼ P by induction. Thus we may assume |P ∩ Qx| = |P ∩ Q| ⇒ P ∩ Qx = (P ∩ Q)x. x x Claim: NP (P ∩ Q ) ∈ Sylp(NG(P ∩ Q )). x x To prove this take S ∈ Sylp(NG(P ∩ Q)) ⇒ S ∈ Sylp(NG(P ∩ Q )) but x x x x NG(P ∩ Q) = NG((P ∩ Q) ) = NG(P ∩ Q ) ⇒ S ∈ Sylp(NG(P ∩ Q )). x x x x x However S ≤ R = P so that S ≤ NP (P ∩ Q ) but NP (P ∩ Q ) is a p- x x subgroup of NG(P ∩ Q ) containing a Sylow p-subgroup of NG(P ∩ Q ) thus x x NP (P ∩ Q ) ∈ Sylp(NG(P ∩ Q )). x x Let T ∈ Sylp(NG(P ∩ Q )) containing NQx (P ∩ Q ) and let U ∈ Sylp(G) such that T ≤ U.

Claim: It is enough to show that U ∼ P to complete the proof. x x x x x Since P ∩ Q < Q as |P ∩ Q | = |P ∩ Q| so that U ∩ Q ≥ NQx (P ∩ Q ) > P ∩ Qx ⇒ if we show U ∼ P then by Lemma 6.4 Qx ∼ P and we are done. P ∩ U ≥ P ∩ T ≥ P ∩ Qx so if P ∩ U > P ∩ Qx we are done by induction. x x So assume P ∩ U = P ∩ Q . T = NU (P ∩ Q ) by choice of T and U ⇒ x since P ∩ Q = P ∩ Q we have NU (P ∩ U) ∈ Sylp(NG(P ∩ U)) ⇒ P ∩ U is a tame intersection. Finally, |P ∩ U| = |P ∩ Q| so that S ∼ P, ∀S ∈ Sylp(G) with |P ∩ S| > |P ∩ U| and by Lemma 6.6, U ∼ P .

Now the proof of Alperin’s fusion theorem will be easy. Theorem 6.7 (Alperins Fusion Theorem, [1]). Let G be a ﬁnite group and g P ∈ Sylp(g). Let A, A ⊆ P , for some g ∈ G. Then there exists elements x1, x2, . . . , xn, subgroups Q1,Q2,...,Qn ∈ Sylp(G) and an y ∈ NG(P ) such that

1. g = x1x2 . . . xny,

2. P ∩ Qi is a tame intersection, 0 ≤ i ≤ n,

3. xi is a p-element of NG(P ∩ Qi), 0 ≤ i ≤ n,

x1x2...xi 4. A ⊆ P ∩ Q1 and and A ⊆ P ∩ Qi + 1, 0 ≤ i ≤ n − 1.

−1 Proof. A ⊆ P ⇒ Ag ⊆ P g ⇒ A ⊆ P g ∩ P . By Lemma 6, there is an g−1 x ∈ G such that P ∼ P via x. Let {Qi, xi : 1 ≤ i ≤ n} accomplish g−1 g−1x −1 P ∼ P ⇒ P = P which implies that y = x g ∈ NG(P ). (2) and (3) are clear and for (1) and (4) we have

−1 (1) g = x(x g) = x1x2 . . . xny. −1 −1 −1 (4) A ⊆ P ∩ P g ≤ P ∩ Q, since P g ∼ P and Ax1...xn ⊆ (P ∩ P g )x1...xn ≤ P ∩ Qi+1.

18 Alperin’s theorem deals with the collection of pairs (Hi,Ti) where H = P ∩Qi is a tame intersection for some Qi ∈ Sylp(G) and Ti consists of all p-elements of NG(H). This observation lead naturally to the following definition. Definition 6.8. Let G be a finite group and P a Sylow p-subgroup of G. 1. A family is a collection of subgroups of P . We denote a family by C. 2. A family C is a conjugation family if whenever A, Ag are subgroups of P for some g ∈ G, then there are elements Hi ∈ C, elements xi ∈ NG(Hi), x1x2...xi for 1 ≤ i ≤ n, and y ∈ NG(P ) such that g = x1x2 . . . xny and A ∈ Hi+1 for 0 ≤ i ≤ n − 1. 3. A family C is a weak conjugation family if whenever A, Ag are subgroups of P for some g ∈ G, then there elements Hi ∈ C, elements xi ∈ NG(Hi), g x1x2...xny x1x2...xi for 1 ≤ i ≤ n, and y ∈ NG(P ) such that A = A and A ∈ Hi+1 for 0 ≤ i ≤ n − 1. Alperin’s theorem clearly implies that if C is the family consisting of all H ≤ P such that H = P ∩ Q is a tame intersection for some Q ∈ Sylp(G), then C is a conjugation family. Note that in a weak conjugation family we do not require the elements g and x1x2 . . . xny to be equal, only that they induce the same map in HomG(A, P ). We see that weak conjugation families are the right concept for fusion systems, as we do not care about the particular element, but only the conjugation map it induces. An equivalent definition of a weak conjugation family, which is better suited for fusion systems, is the following one. Definition 6.9. Let P be a p-group and F a saturated fusion system on P . A weak conjugation family C is a collection of subgroups of P such that F = hAutF (Q): Q ∈ Ci, i.e. every morphisms in F can be written as a finite composition of automorphisms in F, of subgroups in C.

Alperin showed in [1], that the family CA, consisting of subgroups H of P where H = P ∩ Q is a tame intersection for some Q ∈ Sylp(G) and such that CG(H) ≤ H, is a weak conjugation family. This will be investigated further in the next section.

7 Centric, radical and essential subgroups

The aim of Section 8 is to prove Alperin’s fusion theorem for fusion systems. To describe the set of subgroups which controls conjugation in a fusion system we need the concept of centric, radical and essential subgroups. Goldschmidt showed in [7], as a refinement of Alperin’s fusion theorem, that if F is a saturated fusion system on a p-group P , then the class of essential subgroups of P determines the fusion system. Puig showed the class of essential subgroups is also the smallest class which will determine the fusion system. As Linckelmann writes in [10]; ”The class of essential subgroups are essential”. We denote by Op(G), the largest normal p-subgroup of G. Definition 7.1. Let G be a finite group and p a prime dividing |G|. A proper subgroup M is strongly p-embedded in G, if M contains a Sylow p-subgroup P of G and M ∩ P g = 1 for any g ∈ G \ M.

19 Example 7.2. Let G = S4 and P a Sylow 2-subgroup of G. G has a normal subgroup N of order 22 and hence G can not have a strongly p-embedded subgroup, since for any M ≤ G, M ∩ M g contains N for all g ∈ G. In fact we have the following:

Remark. If G contains a strongly p-embedded subgroup, then Op(G) = 1. We will prove this in the proof of Proposition 7.8. Deﬁnition 7.3. Let F be a fusion system on a ﬁnite p-group P . A subgroup Q is F-centric if for every subgroup R, F-isomorphic to Q, CP (R) ≤ R.

With this deﬁnition it is relevant to look back at the family CA, described at the end of Section 6. Proposition 7.4. Let G be a ﬁnite group, P a Sylow p-subgroup of G and F = FP (G). If H is a subgroup of P such that H = P ∩Q is a tame intersection for some Q ∈ Sylp(G) and such that CG(H) ≤ H, then H is fully F-normalized and F-centric.

Proof. Since NP (H) ∈ Sylp(NG(H)), H is fully F-normalized by Proposition 5.9. We have that CP (H) ≤ CG(H) ≤ H. So we want to show that for any g g g g g ∈ G, CP ( H) ≤ H. If H = H then this is obvious so suppose H 6= H. g −1 −1 −1 Let x be any element in CG( H). Then ∀h ∈ H, xghg x = ghg and −1 g g hence g xg ∈ CG(H). This implies that x ∈ CG(H) and hence CP ( H) ≤ g g g CG( H) ≤ CG(H) ≤ H. From this result, it is clear that Alperin’s theorem implies that the collection of all fully normalized, centric subgroups determines the hole fusion system. This is however not the smallest class of subgroups which determines the fusion system. Lemma 7.5. Let F be a saturated fusion system on a ﬁnite p-group P . If Q ≤ P is fully F-centralized then QCP (Q) is F-centric.

Proof. Suppose Q is fully F-centralized and let Q˜ = QCP (Q). Since Q ≤ Q˜ and CP (Q) ≤ Q˜, CP (Q˜) centralizes both Q and CP (Q) and hence CP (Q˜) ≤ CP (Q) ∩ CP (CP (Q)) ≤ Q˜. Let φ : Q˜ → R be an F-isomorphism. If φ(a) ∈ φ(CP (Q)), then ∀φ(q) ∈ −1 φ(Q), φ(a)φ(q)φ(a) = 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 7.6. Let F be a fusion system on a finite p-group P . A subgroup Q is F-radical if Op(AutF (Q)) = Inn(Q). Note in particular that if F is a saturated fusion system on a p-group P , then P is F-radical as Inn(P ) = AutP (P ) ∈ Sylp(AutF (P )). Definition 7.7. 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.

20 Here it is good to make the following observation. Proposition 7.8. Let F be a fusion system on a ﬁnite p-group P . If Q is an F-essential subgroup of P , then Q is F-radical.

Proof. Suppose Q ≤ P is F-essential and let R = OutF (Q). Then R contains a strongly p-embedded subgroup M, i.e. M < R contains a Sylow p-subgroup

S of R such that p 6 |M ∩ Sg| for all g ∈ R \ M. But then R can not contain a non-trivial normal p-group since if it did, this group would be contained in every Sylow p-subgroup of R and hence also in M ∩Sg which contradicts the fact that g p 6 |M ∩ S |. Hence Op(R) = 1 which implies that Op(AutF (Q)/Inn(Q)) = 1. But this is the same as saying that Op(AutF (Q)) = Inn(Q). As we will see in Section 8, the family of fully F-normalized, F-essential subgroups is a weak conjugation family. Hence by Proposition 7.8 and Propo- sition 7.4, we can make the following observation. If C is a family of subgroups H of P , such that for every H in C, H = P ∩ Q is a tame intersection for some Q ∈ Sylp(G), CG(H) ≤ H and NG(H)/H contains no non-trivial normal p-subgroup, then C is a weak conjugation family. Next Proposition will be necessary in the proof of Alperin’s fusion theorem for fusion systems. Furthermore, it gives a picture of what it means for a subgroup to be strongly p-embedded. Proposition 7.9. Let G be a ﬁnite group and let p be a prime dividing |G|. Let X be the partially ordered set of non-trivial p-subgroups of G, where the partial order is given by inclusion of one subgroup into the next. Let Ap(G) be the undirected graph corresponding to X in the natural way. Then G contains a strongly p-embedded subgroup if and only if Ap(G) is disconnected. Proof. Suppose that G has a strongly p-embedded subgroup M and P a Sylow p-subgroup of G such that P ≤ M. Let g ∈ G \ M. If P and P g are in diﬀerent components in Ap(G) then of course, Ap(G) is disconnected. Hence suppose, g to get a contradiction, that P and P are in the same component in Ap(G). Denote by S, the set of all Sylow p-subgroups of G and let SM be the set of all Sylow p-subgroups of M. Let the distance on S be the function

d : S × S → N, (R1,R2) 7→ d(R1,R2) deﬁned as follows; for any R1,R2 ∈ S, if K1 ≤ K2 ≥ · · · ≥ Kk is a shortest path from R1 to R2, then d(R1,R2) = k. 0 g 0 g Now choose R ∈ SM such that for any R ∈ SM , d(R,P ) ≤ d(R ,P ). Let Q1 ≤ Q2 ≥ · · · ≥ Qn be a path of minimal length connecting Q1 ≤ R g g and Qn ≤ P , i.e. d(R,P )=n. Since R is a Sylow p-subgroup of M, there is an h ∈ M such that R = P h. Hence R ∩ P g = 1, since if it is not, then −1 −1 1 6= P ∩ P gh ≤ M ∩ P gh which is a contradiction since gh−1 ∈/ M. This implies that n ≥ 3. Now Q2 is contained in some Sylow p-subgroup of G and hence there is an m m m ∈ G such that Q2 ≤ P . If m ∈ G \ M, then since Q1 ≤ M ∩ P , this contradicts that M is strongly p-embedded. Hence m ∈ M. But then P m ∈ S g and Q3 ≤ Q4 ≥ · · · ≥ Qn is a shorter path then the one from R to P . Then g m g m d(R,P ) > d(P ,P ), which is a contradiction since P ∈ SM and hence P g and P must be in diﬀerent components in Ap(G).

21 Conversely, suppose that Ap(G) is disconnected and let P be a Sylow p- subgroup of G. Let M be the set of all g ∈ G such that P g is in the same component of Ap(G) as P . Then M is a proper subgroup of G containing P . x By deﬁnition of M, P and P are in diﬀerent components of Ap(G) and hence P ∩ P g = 1 for any g ∈ G \ M. Now suppose, to get a contradiction, that x ∈ G \ M and M ∩ P x contains a p-group Q. Then there is a Sylow p-subgroup S of M containing Q and hence, for some y ∈ M, Sy = P . Also Syx = P x which means that Q ≤ Syx and we get that Q ≤ S ∩ Syx. But this implies that Qy ≤ Sy ∩ Syxy = P ∩ P xy. Since M is a group, y−1 ∈ M and since x∈ / M we have that xy∈ / M. Hence P ∩ P xy ≥ Qy 6= 1 is a contradiction and we conclude that M is strongly p-embedded in G.

8 Alperin’s theorem for fusion systems

In this section we will prove a version of Alperin’s fusion theorem more suitable for abstact fusion systems. The formulation given here is due to Puig. Instead of conjugation by an element g we will consider an F-isomorphism θ and prove that it can always be written as a composition series of F-automorphisms of subgroups of fully F-normalized essential subgroups of P . All F-essential subgroups are F-centric radical and hence Alperin’s fusion theorem implies that a fusion system is determined by its subcategory of F- centric radical subgroups [10]. We will begin by a proposition followed by some lemmas that we will will use in the proof of the theorem. Definition 8.1. If M is a set of morphisms and φ is a morphism we write φ−1 ◦ M ◦ φ = M φ. The following proposition makes it clear that if F is a fusion system on a finite p-group P and Q ≤ P is F-essential, then so is every subgroup F- isomorphic to Q. Proposition 8.2. Let F be a fusion system on a finite p-group P . If E is an F-essential subgroup of P and θ ∈ AutF (P ), then θ(E) is an F-essential subgroup of P . Proof. Since θ(E) is F-isomorphic to E, θ(E) is F-centric as E is. So we want to show that OutF (θ(E)) contains a strongly p-embedded subgroup. We know that there is an M ∈ OutF (E) such that M contains a Sylow φ p-subgroup of OutF (E) and M ∩ M is a p’-group ∀φ ∈ OutF (E)\M. −1 Observe that if α ∈ M, then θ ◦ α ◦ θ ∈ OutF (θ(E)). Conjugating M ∩ M φ by θ is an isomorphism and hence (M ∩ M φ)θ = M θ ∩ M φ◦θ which is then a p’-group. But we have that M φ◦θ = θ−1 ◦φ−1 ◦M ◦φ◦θ = −1 (M θ)θ◦φ◦θ . θ So we want to show that ∀β ∈ OutF (θ(E))\M , ∃φβ ∈ OutF (E)\M such −1 θ that β = θ ◦ φβ ◦ θ. But this is obvious since, if β ∈ OutF (θ(E))\M , then −1 −1 −1 θ ◦ β ◦ θ ∈/ M and θ ◦ β ◦ θ ∈ OutF (E), and hence we put φβ = θ ◦ β ◦ θ . θ θ β θ Thus M ∩ (M ) is a p’-group ∀β ∈ OutF (θ(E))\M and hence θ(E) is F-essential.

22 Now we will prove a few lemmas that will be necessary in the proof of Alperin’s theorem for fusion systems. Lemma 8.3. Let F be a fusion system on a ﬁnite p-group P and let Q, R be subgroups of P . Let φ : Q → R be an isomorphism in F such that R is fully F-normalized. Then there is an isomorphism ψ : Q → R in F such that Nψ = NP (Q), i.e. ψ extends to a morphism from NP (Q) to P in F. Proof. From Theorem 5.15, we see that R is fully F-automized and receptive. Hence by Proposition 5.12, there is an isomorphism ψ : Q → R in F such that Nψ = NP (Q). Proposition 8.4. Let P be a ﬁnite group and Q a subgroup of P . If R and S are both subgroups of NP (Q), containing QCP (Q) and such that AutR(Q) = AutS(Q), then R = S.

Proof. Let x ∈ S. Since AutR(Q) = AutS(Q), there is an y ∈ R such that −1 −1 −1 xqx = yqy , ∀q ∈ Q. Hence x y ∈ CP (Q) which is a subgroup of both R and S. But since x ∈ S and y−1 ∈ R we have that x ∈ R and y ∈ S, and thus S = R. Lemma 8.5. Let F be a fusion system on a ﬁnite p-group P . Let Q be a fully F-normalized subgroup of P and let α ∈ AutF (Q). Then R = Nα is the unique −1 subgroup of NP (Q) such that QCP (Q) ≤ R and AutR(Q) = AutP (Q) ∩ (α ◦ AutP (Q) ◦ α). Proof. The uniqueness follows from Proposition 8.4 To prove that R ≤ Nα, let r ∈ R. Then r ∈ NP (Q) and the conjugation −1 map cr : Q → Q is in AutR(Q), and hence in AutP (Q) ∩ (α ◦ AutP (Q) ◦ α) ⇒ −1 cr = α ◦ ca ◦ α for some ca ∈ AutP (Q). But then a ∈ NP (Q) and we have r −1 a q = α ( α(q)), ∀q ∈ Q. Hence r ∈ Nα and we have R ⊆ Nα. Now we prove that Nα ≤ R. Let a ∈ Nα, then ∃z ∈ NP (α(Q)) such that a z α( q) = α(q), ∀q ∈ Q. Hence ca ∈ AutR(Q) and thus there is an r ∈ R such r a −1 −1 that q = q, ∀q ∈ Q ⇒ r a ∈ CP (Q) ≤ R and hence rr a = a ∈ R ⇒ Nα ⊆ R and we thus have R = Nα.

Note that we do not use the property of the fusion system that HomP (A, B) ⊆ HomF (A, B), for any subgroups A, B of P . In [10] Linckelmann uses another definition that he calls a category on a finite p-group, which is similar to what we call a fusion system on a finite p-group except that it misses the property HomP (A, B) ⊆ HomF (A, B) for subgroups A, B of P . Lemma 8.6. Let F be a saturated fusion system on a finite p-group P and let Q ≤ P be a fully F-normalized. Then there is a unique subgroup R of NP (Q) such that QCP (Q) ≤ R and Op(AutF (Q)) = AutR(Q). Furthermore, every automorphism α ∈ AutF (Q) extends to a automorphism β ∈ AutF (R). Proof. The uniqueness follows from Proposition 8.4 −1 For every α ∈ AutF (Q) we have α ◦ AutR(Q) ◦ α = AutR(Q) and hence, by Lemma 8.5, R ⊆ Nα. Thus α extends to a morphism β : R → P . Hence, for any x ∈ R and any q ∈ Q we have α(xq) = β(xq) = β(x)α(q) which is equal to −1 saying α ◦ cx ◦ α = cβ(x). Since AutR(Q) is normal in AutF (Q) we have that −1 cβ(x) ∈ AutR(Q). Hence for every q ∈ Q we have cβ(x)(q) = β(x)qβ(x) =

23 −1 −1 rqr for some r ∈ R and we get that r β(x) ∈ CP (Q). Since CP (Q) is a subgroup of R we see that β(x) ∈ R and hence β ∈ AutF (R). Lemma 8.7. Let F be a saturated fusion system on a ﬁnite p-group P and let Q be a fully F-normalized subgroup of P . Let φ ∈ AutF (Q) such that AutQ(Q) −1 is a proper subgroup of AutP (Q) ∩ (φ ◦ AutP (Q) ◦ φ ). Then φ extends to a morphism ψ : R → P in F for some subgroup R of NP (Q) such that R properly contains Q.

Proof. Let R be as in Lemma 8.5. Since AutQ(Q) < AutR(Q), there exists an x ∈ R such that cx ∈ AutR(Q) but cx ∈/ AutQ(Q). Hence x∈ / Q and we have that Q is a proper subgroup of R. By Lemma 8.5 we have that R ⊆ Nφ and hence φ extends to a morphism ψ : R → P . Theorem 8.8 (Alperin’s fusion theorem). Let F be a saturated fusion system on a ﬁnite p-group P and let L be the set of fully F-normalized essential subgroups of P . If Q ≤ P and φ : Q → R is an isomorphism in F, then φ can be written as a ﬁnite composition of

1. F-automorphisms φ1, φ2, . . . , φn, such that Q = Q0, φ1(Q) = Q1, φ2(Q1) = Q2,..., φn(Qn−1) = Qn, and

2. an F-automorphism ψ of P such that ψ(Qn) = R.

Furthermore, for 1 ≤ i ≤ n, Qi−1,Qi ≤ Si and φi ∈ AutF (Si) for some Si ∈ L. Proof. Let M be the set of all isomorphisms in F that satisfy the claim of the theorem. Then M is closed under composition and if α : A → B is a morphism in M, so is α−1. We also have that the restriction of α to any subgroup H of A is in M. Let φ : Q → R be any isomorphism in F. We proceed by induction on the index of Q in P . If Q = P , there is nothing to prove so suppose that Q is a proper subgroup of P . We are going to use the fact that if Q is a proper subgroup of Nα then φ extends to Nφ and hence φ ∈ M by induction. Let ψ : R → T be an isomorphism in F such that T is fully F-normalized. By Lemma 8.3, ψ may be chosen so that it can be extended to NP (R) and hence ψ ∈ M by induction. Then, as ψ−1 ∈ M, we only have to prove that ψ ◦ φ ∈ M. We have that ψ ◦ φ : Q → T and since T is fully F-normalized, again by Lemma 8.3, there is an isomorphism β : Q → T in F, which extends to NP (Q). Hence, by induction, β ∈ M and it will suﬃce to show that ψ ◦ φ ◦ β−1 ∈ M. −1 Note that ψ ◦ φ ◦ β ∈ AutF (T ) and that T is fully F-normalized. Let −1 ξ = ψ ◦ α ◦ β . By Lemma 8.6 there is a unique subgroup S of NP (T ) containing TCP (T ) and every automorphism in AutF (T ) extends to a morphism in AutF (S). If S > T then we are done by induction, so assume that S = T . Then by Lemma 8.6 Op(AutF (T )) = AutS(T ) = AutT (T ) = Inn(T ) and TCP (T ) ≤ T implies CP (T ) ≤ T . Hence T is F-radical centric.

If T is essential, we are done so suppose it is not. Then by Proposition 7.9, the graph Ap(OutF (T )) is connected. Put A = AutP (T ) and B = ξ ◦AutP (T )◦ ξ−1. Since T is fully F-normalized, both A and B are Sylow p-subgroups of

24 AutF (T ). Since Ap(OutF (T )) is connected and OutF (T ) = AutF (T )/AutT (T ) there is a sequence

−1 AutP (T ) = S1,S2,...,Sn = ξ ◦ AutP (T ) ◦ ξ

such that AutT (T ) < Si ∩ Si+1 for 1 ≤ i ≤ n − 1. For 1 ≤ j ≤ n, choose −1 θj ∈ AutF (T ) such that θj ◦ AutP (T ) ◦ θj = Sj. Choose θ1 = IdR, then obviously θ ∈ M. Now we proceed by induction over j to show that θj ∈ M for 1 ≤ j < n. Let 1 ≤ j < n. Then we have that

−1 −1 AutT (T ) < Sj ∩ Sj+1 = (θj ◦ AutP (T ) ◦ θj ) ∩ (θj+1 ◦ AutP (T ) ◦ θj+1)

and hence, as AutT (T ) is normal in AutF (T ),

−1 −1 −1 AutT (T ) = θj ◦ AutT (T ) ◦ θj < AutP (T ) ∩ (θj ◦ θj+1 ◦ AutP (T ) ◦ θj+1 ◦ θj).

−1 But then, by Lemma 8.7 and induction, θj ◦ θj+1 ∈ M. Since θ1 ∈ M we get, −1 by induction over j, that θj ∈ M and as M is closed under compositions, so is θj+1. Finally, we have that

−1 −1 Sn = θn ◦ AutP (T ) ◦ θn = ξ ◦ AutP (T ) ◦ ξ

which means that ξ ◦ θn ∈ NAutF (T )(AutP (T )). Hence ξ ◦ θn extends to NP (T ) −1 and we have that ξ ◦ θn ∈ M. But θn ∈ M which means that θn ∈ M. Hence ξ ∈ M which implies that φ ∈ M. As a consequence of Theorem 8.8 we have the following result. Example 8.9. Let P be a dihedral, semidihedral or generalized quaternion group of order 2n ≥ 16. These groups have the following presentations.

2n−1 2 −1 D2n = hr, s : r = s = 1, sr = r si

2n−1 2 2n−2−1 SD2n = hr, s : r = s = 1, sr = r si

2n−1 4 −1 2n−2 2 Q2n = hr, s : r = s = 1, sr = r s, r = s i Let r, s be generators of P , such that |r| = 2n−1, |s| = 2 if P is semidihedral. Consider the order of the element ris.

i) If P is dihedral, then (ris)2 = risris = rir−iss = id for all i ∈ Z.

n−2 ii) If P is semidihedral, then (ris)2 = risris = rir2 −1sri−1s = ··· = n−2 n−2 n−2 ri(r2 −1)i = r2 i and hence (ris)4 = (r2 i)2 = id for all i ∈ Z. iii) If P is generalized quaternion, then (ris)2 = risris = hih−ixx = x2 and hence (ris)4 = id for all i ∈ Z. Hence we conclude that |ris| is either 2 or 4 for all i ∈ Z. Note also the following three facts. a) If P is dihedral, then every subgroup of P is either dihedral, cyclic or a ∼ n−1 Klein 4-group. This is easy to see since D2n = (Z/2 Z) oφ (Z/2Z), where φ(0) is the identity map and φ(1) is the inversion map. Observe ∼ that Z2 oφ Z2 = Z2 × Z2 = V4 as the inverse of 1 in Z/2Z is 1.

25 b) If P is generalized quaternion, then every subgroup of P is either cyclic or generalized quaternion. This follows since ∼ n−1 n−2 Q2n = ((Z/2 Z) oα (Z/4Z))/h(2 , 2)i where we deﬁne (a, b)(c, d) = (a+α(a)c, b+d) = (a+(−1)bc, b+d). Every n−1 subgroup of this quotient corresponds to a subgroup of (Z/2 Z) oα (Z/4Z) containing h(2n−1, 2)i and hence we conclude that all subgroups must be either cyclic or generalized quaternion. c) If P is semidihedral, then P has three maximal subgroups. One cyclic, one dihedral and one generalized quaternion. The maximal cyclic group is ob- n−2 viously hri. The maximal dihedral group is hr2, si as sr2 = r2(2 −1)s = r−2s = (r2)−1s which is the dihedral relation. Finally, it is easy to check that the maximal generalized quaternion group is hr2, rsi. Now 2-groups which are either dihedral or cyclic will have automorphism groups of order, a power of 2, and hence can not be F-essential. 2n−2 i ∼ i For every i ∈ Z, put Ei = hr , r si = V4 if |r s| = 2, and Ei = 2n−3 i ∼ i hr , r si = Q8 if |r s| = 4. Let F be any saturated fusion system on P . Then we have the following.

1. If φ ∈ Aut(P ) have odd order, then since hri =∼ Z/2n−1Z we must have ∼ that φ hri = Idhri and P/hri = hsi. Hence, by Lemma A.5, φ = IdP . Thus Aut(P ) is a 2-group, and hence so is AutF (P ). Note that this argument also holds for any subgroup Q of P , such that Q is either dihedral, semidihedral or generalized quaternion and |Q| ≥ 16. Now, since AutP (P ) is a Sylow 2-subgroup of AutF (P ), we must have that AutF (P ) = AutP (P ) = Inn(P ). 2. From a), b), c) and (1), one may conclude that the only subgroups of P whose automorphism groups are not 2-groups are the Ei and hence these are the only subgroups which could be F-essential. For example, if Q ≤ P is generalized quaternoin of order ≥ 16, then by (1), Aut(Q) is a 2-group and hence, so is AutF (Q). ∼ ∼ 3. For each i we have that Out(Ei) = S3 and OutP (Ei) = Z/2Z. Hence OutF (Ei) must be one of these groups. ∼ 4. If i ≡ j(mod 2), then Ei,Ej are P -conjugate, and hence OutF (Ei) = OutF (Ej). This is not hard to prove since if i ≡ j(mod 2), then j = 2m + i for some m ∈ Z, and hence, if P is dihedral or generalized quaternion, rm(ris)r−m = r2m+is. If P is semidihedral, then rm(ris)r−m = m+i (1−2n−2)m 2m+i−2n−2m m −m 2n−3 m −m r r s = r ∈ r (Ei)r and since r ∈ r (Ei)r 2n−3 2m 2m+i−2n−2m 2m+i m −m we get that (r ) (r ) = r ∈ r (Ei)r and hence m −m r (Ei)r = Ej. This implies that there are, up to conjugates, at most two diﬀerent F-essential subgroups. For each F-essential subgroup, there are only two possibilities for the outer automorphism group and hence we conclude that there are at most 4 distinct fusion systems on P . All these fusion systems are all realizable by ﬁnite groups [5, 4]. For example, ∼ n−2 n−1 if P = SD2n and q ≡ 2 − 1(mod 2 ), then an example of four groups

26 realizing distinct fusion systems on P are, SD2n , GL2(q), PSL3(q) and a speciﬁc 2 extension of PSL2(q ) by Z/2Z [4]. In Section 9, we will study the case where P is the semidihedral group of order 16. Then one fusion system is also realizable by GL2(3).

9 A fusion system on GL2(3)

Let G = GL2(3) and let P be the Sylow 2-subgroup of G generated by

0 1 0 1 2 1 1 1 , , and 2 0 1 0 1 1 2 1

2 8 Then P is isomorphic to the semidihedral group SD16 = hs, r : s = r = 0 1 1 2 1, sr = r3si. Here s = and r = . Let F = F (G). In Figure 1 0 1 1 P 3, we see the subgroup lattice of P .

Figure 3: Subgroup lattice of SD16

27 Let us for simplicity denote the subgroup of P , isomorphic to Q8, by Q8 etc.

F-automorphisms of Q8 ( ) 1 0 2 0 0 2 0 1 2 2 2 1 1 1 1 2 Q = , , , , , , , 8 0 1 0 2 1 0 2 0 2 1 1 1 1 2 2 2 and these matrices can be represented as 1, -1, i, -i, j, -j, k, -k respectively. ∼ ∼ By Proposition A.7, Aut(Q8) = S4 and hence |AutF (Q8)| 24. Inn(Q8) = Q8/Z(Q8) which has order 4. Conjugation by the matrices 1 1 1 2 1 0 1 0 2 1 1 0 1 1 1 2 0 1 , , , , , , , 0 1 0 1 1 1 2 1 0 1 1 2 0 2 0 2 2 2 induces 9 distinct automorphisms not in Inn(Q8) and hence we have found 13 ∼ automorphisms and since |AutF (Q8)| 24 we have that AutF (Q8) = S4.

F-automorphisms of D8 ( ) 1 0 2 0 2 0 1 0 0 1 0 2 0 1 0 2 D = , , , , , , , 8 0 1 0 2 0 1 0 2 1 0 2 0 2 0 1 0

∼ ∼ By Proposition A.6, Aut(D8) = D8 and Inn(D8) = Z/2Z ⊕ Z/2Z and hence we have |AutF (D8)| ∈ {4, 8}. Since |Inn(D8)| = 4 we need only find one map not in Inn(D8) to show that AutF (D8) = D8. Since every map in Inn(D8) fixes a 1 1 non-identity element and conjugation by fixes only the id matrix, we 1 2 have found a 5th map and hence AutF (D8) = D8.

F-automorphisms of Z/8Z ( ) 1 0 2 0 0 2 0 1 1 1 1 2 2 2 2 1 /8 = , , , , , , , Z Z 0 1 0 2 1 0 2 0 2 1 1 1 1 2 2 2

Every automorphism of Z8 is determined by how one of the generators are mapped and since there are 4 elements of order 8 in Z8 we have |AutF (Z8)| ≤ 4. 1 1 1 1 1 2 Conjugation by maps to . Suppose that conjugation 1 2 2 1 1 1 −1 a b 1 1 2 2 a b by a matrix maps to . Then if γ = det = c d 2 1 1 2 c d 1 ad−bc , we have a b 1 1 d −b ad − ac + 2bd − bc a2 − 2b2 γ = γ c d 2 1 −c a 2d2 − c2 ac − cb + ad − 2bd 2 2 But if this is equal to we get that 1 2 ad − ac + 2bd − bc = 2(ad − bc) (10) ac − bc + ad − 2bd = 2(ad − bc) (11) Adding (8) and (9) gives 2(ad − bc) = 4(ad − bc) ⇒ ad − bc = 0 which is a ∼ contradiction. Hence AutF (Z8) = S2.

28 F-automorphisms of SD16 ∼ |P | = 16 and |Z(P )| = 2. We have that AutF (P ) = NG(P )/CG(P ) and since |G| = 48 and Z(P ) ≤ CG(P ), we must have that |AutF (P )| ∈ {2, 3, 4, 6, 8, 12, 16, 24}. Every automorphism of D8 is induced by conjugation in P and hence |AutF (P )| is at least 8 ⇒ |AutF (P )| ∈ {8, 12, 16, 24}. Since FG(P ) is saturated, AutP (P ) ∈ Syl2(AutF (P )) and hence |AutF (P )|= 6 ∼ 16. Also, AutP (P ) = P/Z(P ) which has order 8. Hence 8 |AutF (P )| and we have |AutF (P )| ∈ {8, 24}. Now, all automorphisms of SD16 is determined by how the generators s and r are mapped. There are 5 elements of order 2 of which one is central and there are 4 elements of order 8. Hence |Aut(SD16)| = 4 · 4 = 16 and we have |AutF (P )| = 8.

The skeleton Fsc of the fusion system FP (G) is shown in Figure 4.

Figure 4: The skeleton of the fusion system on SD16 in GL2(3)

F-essential subgroups

Since Q8 is the only subgroup of P whose automorphism group is not a p- ∼ group, Q8 is the only potential candidate for being F-essential. AutF (Q8) =

29 NG(Q8)/CG(Q8) and |AutF (Q8)| = 24. Thus |CG(Q8)| = 2 ⇒ CG(Q8) = Z(G) ≤ Q8 and hence Q8 is F-centric.

∼ AutF (Q8) OutF (Q8) = Inn(Q8) and this group has order 6. Since Inn(Q8) has no element of order 4 it is not ∼ cyclic and hence Inn(Q8) = V4 and we get

∼ AutF (Q8) ∼ ∼ OutF (Q8) = = S4/V4 = S3 Inn(Q8)

We know that S3 has 3 Sylow 2-subgroups of which none are ﬁxed under conjugation by an element not in the Sylow 2-subgroup itself. Hence if we choose M to be one of these Sylow 2-subgroups we get that M is strongly 2-embedded in OutF (Q8). Hence Q8 is F-essential. Q8 is also fully F-normalized as it is not isomorphic to any other subgroup of P . Let K1,K2 denote the two subgroups isomorphic to the Klein-4-group. Since neither of these are included in Q8 which is the only F-essential subgroup, Alperin’s fusion theorem implies that any isomorphism from K1 to K2 must be a restriction of some ψ ∈ AutF (P ). 2 0 The only subgroup of order 2 contained in Q is Z(P ) = . Hence 8 0 2 any isomorphism α : Q → R, where |Q| = |R| = 2, must be a restriction to Q of some α ∈ AutF (P ). 1 1 2 1 If A = and B = , then Alperin’s theorem implies for 1 2 1 1 any isomorphism φ : A → B, there is a ψ1 ∈ AutF (Q8) and a ψ ∈ AutF (P ) such that φ = ψ ◦ φ1 . Note that ψ or φ1 or both, might be the identity φ1(A) A map.

A Some properties of ﬁnite groups

Lemma A.1. 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. 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 since conjugation preserves the order we get

[h, k] = h kh−1k−1 = hkh−1 k−1 ∈ P ∩ K = 1, | {z } | {z } ∈H ∈K so that hk = kh. Lemma A.2. If p is a prime and P is a p-group, then Z(P ) 6= 1. Proof. This is an easy application of the class equation.

Lemma A.3. Let p be a prime and P a p-group. If H < P then H < NP (H).

30 Proof. We proceed by induction on |P |. If |P | = p, then Z(P ) = P so suppose |P | > p. Let H be a proper subgroup of P . First suppose Z(P ) 6≤ H, then H < hZ(P ),Hi. But Z(P ) normalizes every subgroup of P and hence hZ(P ),Hi ≤ NP (H). Hence we may assume that Z(P ) ≤ H. Since Z(P ) is non-trivial, |P/Z(P )| < |P | and hence, by induction, H/Z(P ) < NP (H)/Z(P ). By the lattice isomorphism theorem we now have that H < NP (G) and the induction is complete. Lemma A.4. 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 −1 have that gHg = cg(H) = H. Lemma A.5. Let P be a p-group and φ ∈ Aut(P ) of order prime to p. If 1 = P0 P1 ,..., Pn = P is a sequence of subgroups all normal in P , and E E E such that φ ≡ IdPi (mod Pi−1). Then φ = IdP . Pi Proof. We use induction on n and hence it suffice to prove the statement for n = 2. Suppose that 1 P1 P and φ ∈ Aut(P ) such that p - |φ|. By assumption, E E φ = IdP1 and φ(g)P1 = gP1 for all g ∈ P . Hence we have that φ(g) = gy for P1 k k some y ∈ P1 and since y is fixed under φ we must have that φ (g) = gy for all k ∈ Z. Let m = |φ|. We have that g = φm(g) = gym which implies that ym = 1. But since gcd(m, p) = 1 and y have p order, this implies that y = 1 and hence φ must be the identity map on P . ∼ ∼ Proposition A.6. Aut(D8) = D8 and Inn(D8) = Z/2Z ⊕ Z/2Z. 4 2 3 Proof. D8 = hr, s : r = s = 1, sr = r si. The only elements of order 4 are r 3 3 2 2 and r and hence for any φ ∈ Aut(D8), φ(r) ∈ {r, r }. Hence φ(r ) = φ(r) = 2 3 2 2 2 r , since (r ) = r . This means that hr i = Z(D8) is characteristic in D8. There are 5 elements of order 2, namely {r2, s, rs, sr, r2s} and r2 is fixed under any automorphism. Hence there are 2 possibilities for φ(r) and 4 possibilities for φ(s) and since every automorphism is determined by how the generators r and s are mapped, |Aut(D8)| ≤ 8. 8 2 7 Now consider D16 = ha, b : a = b = 1, ba = a bi, which has a subgroup 2 D8 = ha , bi. Since |D16 : D8| = 2, D8 E D16. The only elements commuting 2 4 4 with both a and b are 1 and a and hence CD16 (D8) = ha i. Hence we have that ∼ 4 ∼ AutD16 (D8) = ND16 (D8)/CD16 (D8) = D16/ha i = D8. Thus since |Aut(D8)| ≤ ∼ 8 we have Aut(D8) = AutD16 (D8) = D8. ∼ 2 2 For the second part, Inn(D8) = D8/Z(D8) = D8/hr i. And since D8/hr i 2 ∼ is a group of order 4 which is not cyclic, D8/hr i = Z/2Z ⊕ Z/2Z. ∼ Proposition A.7. Aut(Q8) = S4.

4 4 2 2 −1 Proof. Q8 = {1, -1, i, -i, j, -j, k, -k} = hi, j : i = j = 1, i = j , ji = i ji. Hence Q8 has one element of order 1, one of order 2 and six of order 4. We ﬁrst want to ﬁnd an upper limit for |Aut(Q8)|. Every automorphism of φ ∈ Aut(Q8) is determined by how the generators i and j are mapped,

31 since φ(i) and φ(j) will generate φ(Q8). Also, the conjugacy classes of Q8 are {1}, {−1}, {i, -i}, {j, -j}, {k, -k} and any automorphism of Q8 will permute these. So ﬁrst we choose where to map {i, -i} (3 choises), then how to map {j, -j} (2 choises), then how to map i inside {i, -i} (2 choises) and then how to map j in {j, -j} (2 choises). This gives us that |Aut(Q8)| ≤ 3 · 2 · 2 · 2 = 24. If we can ﬁnd a set M of 4 elements such that Aut(Q8) acts on M as S4 acts on {1, 2, 3, 4} then we are done. Let M = S1 = {{i, j, k}, {-i, -j, -k}},S2 = {{i, -j, -k}, {-i, j, k}},

S3 = {{-i, -j, k}, {i, j, -k}},S4 = {{-i, j, -k}, {i, -j, k}} and let Aut(Q8) act on M, by φ(S1) = {{φ(i), φ(j), φ(k)}, {φ(-i), φ(-j), φ(-k)}} and equally for S2 and S3, where φ ∈ Aut(Q8). We know that S4 is generated by (12) and (1234) so we want to ﬁnd the corresponding permutation of M arising from the action of Aut(Q8). Note that any automorphism on Q8 is a permutation of the elements of Q8 and hence we can write it in cycle notation. Consider the map ψ : S4 → Aut(Q8) deﬁned by

(12) 7→ σ1 = (i -i)(j k)( -j -k) = (S1 S2)

(1234) 7→ σ2 = (i -j -i j)(k -k) = (S1 S2 S3 S4) Now (12)(1234) = (234) and ψ((12))ψ((1234)) = (i -i)(j k)(-j -k)(i -j -i j)(k -k) = (i -k j -i k -j). It is not hard to se that the action of (i -k j -i k -j) on M is exactly (S2 S3 S4) which implies that ψ((12)(1234)) = ψ((12))ψ((1234)). From here it is not hard conclude that ψ is a homomorphism and hence ψ(S4) = hσ1, σ2i. ∼ This shows that hσ1, σ2i = S4 and since hσ1, σ2i ≤ Aut(Q8) which has order ∼ ≤ 24, we conclude that Aut(Q8) = S4.

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