Morphisms of Fusion Systems
Author:
Karl Amundsson
karlam@kth.se
Supervisor:
Tilman Bauer
SA104X - Degree Project in Engineering Physics, First Level
Department of Mathematics
Royal Institute of Technology (KTH)
May 20, 2014
Abstract
A fusion system on a finite group G with a Sylow p-subgroup P is a cat-egory on P with all subgroups of P as objects and group homomorphisms induced by conjugation in G as morphisms and was first introduced by L. Puig around 1990 in order to aid his research in finite group theory. The idea turned out to be fruitful and today, the theory of fusion systems is an active field in mathematics, with applications to topology, representation theory and finite group theory.
In this paper, we will, among other things, see how fusion systems aid in solving problems in finite group theory. We begin with an introduction to the theory with basic examples and then proceed to prove two famous theorems named after Burnside and Frobenius. However, to finish the proof of Frobenius’ theorem, we will require the focal subgroup theorem, whose proof requires transfer theory and is thus discussed. Afterwards, we introduce abstract- and saturated fusion systems, in which one disregard the underlying group, and later prove that every fusion system on a finite group is saturated. We end with a discussion of morphisms of fusion systems, utilizing the concept previously developed, and generalize the isomorphism theorems to saturated fusion systems.
The presentation is well adapted for undergraduate students with lim-ited knowledge of group- and category theory and no previous knowledge of fusion systems is assumed.
Contents
1 Introduction 1
2 Fusion Systems 2
3 Transfer and the Focal Subgroup Theorem 6
4 Proof of Frobenius 4⇒1 9
5 Abstract Fusion Systems 10
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 the-ory. 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 [10] 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 sys-tems 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. It should be noted that Section 1 to 5 have been written together with Eric Ahlqvist and Oliver Gäfvert.
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 pn
m, 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.
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 writexg = xgx−1 and gx = 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 sub-groups 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
sub-group of P . Then we define the automorphism sub-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.
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)}.
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}.
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
centralizesxQ ≤ 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
CG(xQ) and hence we can find a c ∈ CG(xQ) such that P =cxP . This means
that cx ∈ NG(P ) and since cxu = c(xux−1)c−1 = xux−1= 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.
Proposition 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 p0-subgroup. This maximal normal p0-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.
Lemma 2.12. 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 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 FP(G) is induced by conjugation in P . The last implication
of this theorem will be proved in section 4.
Theorem 2.13 (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 ∈ Sylp(G) and K ∩ P = 1. As P ∼= G/K is a p-group, we see
that K = {x ∈ G | (|x|, p) = 1}. It follows that for any H ≤ G, H ∩ K = {x ∈ H | (|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 subgroups 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
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) = 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 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≤ K2and 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 theo-rem we want to use Alperin’s fusion theotheo-rem. For this we need a property of the so called focal subgroup which was first introduced by Higman [8]. 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
σg(i) = j. Then xigx−1j = 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−1σ g(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 [7].
The next theorem will be necessary in our proof of the focal subgroup theo-rem.
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:
1. xigrix−1i ∈ H for some positive integers ri, 0 ≤ i ≤ t.
2. Pt
i=1ri= n = |G : H|
3. τ (g) = φ(Q
i∈Ixigrix−1i )
In the proof of this theorem we follow the proof in Daniel Gorenstein [7]. 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
i=1
ri = n = |G : H|
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
σg, xigjis a coset representative of H in G corresponding to the (j + 1)th coset
of the ith cycle of σg. Hence
form a complete set of coset representatives for H in G. But then xigri ∈ Hxi
by definition of ri and hence xigrix−1i ∈ H which proves (1).
Now we prove (3). Lets use R as coset representatives and compute τ (g). Let yk = xigj (k depend on i and j) and consider ykg = hk,gyk0
g, hk,g ∈ H. If j < ri− 1, then ykg = xigj+1= yk0 g. Since xig j+1is a coset representative in R and the xigj+1∈ Hyk0 g if and only if yk0g = xig j+1, by definition of h k,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 = xigri−1 for which we
have
ykg = xigri∈ Hxi⇒ ykg = (xigrix−1i )xi, (xigrix−1i ) ∈ H.
Hence xi= yk0
g and we get hk,g = xig
rix−1
i for each yk. Therefore we get that
τ (g) = φ t Y i=1 xigrix−1i !
and since φ is a homomorphism we get (3).
Theorem 3.4 (The Focal Subgroup Theorem, [8]). Let G be a finite group, P ∈ Sylp(G) and let G0 be the commutator subgroup of G. Then
P ∩ G0 = h[x, g] = x−1xg: x ∈ P, g ∈ G, xg∈ P i = hx−1φ(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
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 xixrix−1i ! ≡ t Y i=1 xixrix−1i (mod P ∗).
Since P/P∗ is abelian, we have τ (x) = φ t Y i=1 xrix−rix ixrix−1i ! ≡ t Y i=1 xri ! t Y i=1 xrix ixrix−1i ! (mod P∗). But since x−rix ixrix−1i = [x ri, x i] ∈ P0≤ P∗ we have that τ (x) ≡ t Y 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∗.
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 -normalized which is an important property for 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 P ∈ Sylp(G). Let A, Ag⊆ P , for some g ∈ G.
Then there exists elements x1, x2, ..., xn, subgroups Q1, Q2, ..., Qn∈ Sylp(G) and
a 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,
4. A ⊆ P ∩ Q1 and Ax1x2...xi⊆ 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
conjuga-tion 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 ∩ Qiis 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) CG(P ∩ Qi) ∼ = NP(P ∩ Qi) NP(P ∩ Qi) ∩ CG(P ∩ Qi) ∼ = AutP(P ∩ Qi) (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 ) ∈ Sylp(NG(P )). But NG(P ) was
assumed to be a p-group and hence NG(P ) = NP(P ) which implies that y ∈ P .
Definition 4.5. We will denote by Op(G), the smallest normal subgroup of the
finite group G such that G/Op(G) is a p-group.
Proposition 4.6 (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. 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 = P0 (2) and we have that P0< P . Now consider the group G0Op
(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 Op(G) ≤ ker τ since
P/P0 is a p-group. Thus G0Op(G) ≤ ker τ and G0Op(G) is a proper normal
subgroup of G. Now let H = G0Op(G). If we can show that the hypothesis passes to H, then we are done by induction since ∃NC H such that p - |N | and |H : N | and |G : H| are both powers of p. If Q is a p-group of P ∩ H then NH(Q) = H ∩ NG(Q) and CH(Q) = H ∩ CG(Q) and since NG(Q)/CG(Q) is a
p-group ⇒ NH(Q)/CH(Q) is a p-group.
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 realizable 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 classification of finite simple groups. The fusion systems that are not realizable on any finite group G, are called exotic fusion systems.
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: 1. For each g ∈ P such thatgQ ≤ R, c
g: Q → R defined by cg(x) =gx 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 homo-morphisms.
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 and 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
Nφ= {y ∈ NP(Q)| ∃z ∈ NP(φ(Q)) such that φ(yu) =zφ(u), ∀u ∈ Q} (4)
Remark. Note that QCP(Q) E NφE NP(Q).
Remark. Nφ is the preimage in NP(Q) of AutP(Q) ∩φAutP(R).
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
Proof. Suppose that φ : R → Q is an isomorphism in F . Now
Nφ= {x ∈ NP(R)| ∃y ∈ NP(Q) such that φ(xrx−1) = yφ(r)y−1, ∀r ∈ R},
(5) thus cx−1◦ φ−1◦ cy◦ φ centralizes R and cy◦ φ ◦ cx−1◦ φ−1 centralizes φ(R) =
Q ⇒ φ ◦ cx = φ
0
◦ 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) ∈ 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 H1C 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, H1is 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
any y ∈ Sg, 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(Qg) ∩ P = CP(Qg) and we
conclude that |CP(Q)| ≤ |S| ≤ |CP(Qg)|. 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 instead of centraliz-ers.
We will now define what is meant by a saturated fusion system. The ax-ioms 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.
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) CG(Q) ∼ = NP(Q) NP(Q) ∩ CG(Q) =NP(Q) CP(Q) ∼ = AutP(Q). (7)
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).
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 ≤ Aut
P(R). Now let ψ = α ◦ φ. Now this says exactly that for any
x ∈ NP(Q) there is an y ∈ NP(R) such that ψ ◦ cx◦ψ−1= 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) ≤
AutP(Q) ∩φ
−1
AutP(R).
Proof. Let x ∈ S. For all g ∈ Q we have that xg ∈ Q and thus φ(xφ(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), thereforeφ(cQ(S)) = cR(φ(S)) ≤ AutP(R).
And since cQ(S) ≤ AutP(Q) we have that cQ(S) ≤ AutP(Q) ∩φ
−1
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
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 ψAut
P(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 −1 AutP(Q) ≤ AutP(S) ∩φ −1ψ−1 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
5.13 we have that ψ(NP(Q)) is the full preimage in NP(R) ofψAutP(Q). Also
θ(Nφ) is the preimage in NP(R) of cR(θ(Nφ)) =θ(cS(Nφ)) ≤ψAutP(Q) since φ(c
S(Nφ)) ≤ AutP(Q) and thus θ(Nφ) ≤ ψ(NP(Q)) so that ψ −1
|θ(N
φ)◦ θ is a
map from Nφ to NP(Q) extending φ = ψ−1θ.
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 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)| 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
Morphisms
The way one compare structures in abstract algebra is by studying morphisms. Luckily, a fusion system is just a certain kind of category and so it is possible to define and study morphisms between them, which is the goal of this section. The following notion will be crucial.
Definition 6.1. Let F be a fusion system on a finite p-group P , and let Q ≤ P . 1. If for all ϕ ∈ HomF(Q, P ), we have that ϕ(Q) = Q, we say that Q is
weakly F -closed.
2. If for all R ≤ Q and for all ϕ ∈ HomF(R, P ) we have that ϕ(R) ≤ Q, we
say that Q is strongly F -closed in F .
Remark. The reason we call them weakly- and strongly F -closed is because when you consider a fusion system on a finite group G, the concept coincide with that of groups, i.e., Q ≤ P is weakly (strongly) FP(G) closed if and only
if Q is weakly (strongly) closed in P with respect to G. As a remainder, a subgroup H of a finite group G is said to be weakly closed in K with respect to G if gHg−1 ≤ K =⇒ H = gHg−1 for all g ∈ G and strongly closed in K
with respect to G if gH0g−1≤ K =⇒ gH0g−1 ≤ H for all H0≤ H and for all
g ∈ G.
Example 6.2. Consider the fusion system F = FP(G) in Example 2.7. First
note that if a subgroup is F -isomorphic to a group other than itself, it cannot be weakly F -closed. To see this, suppose Q is F -isomorphic to the subgroup R 6= Q. Also, suppose Q is weakly F -closed, so that it is weakly closed in P with respect to G, i.e., gQg−1 ≤ P implies Q = gQg−1. Now, let c
x: Q → R
be a F -isomorphism, x ∈ G. Then xQx−1 = R ≤ P , so that by definition, Q = xQx−1 = R, which is a contradiction. Conversely, suppose Q is only F isomorphic to itself. Since ϕ : Q → xQx−1 is an isomorphism in F for every x ∈ G, it follows that Q = xQx−1. Hence, the weakly F -closed subgroups are A = {1, (12), (34), (12)(34)}, B = h(1324)i, C = {1, (12)(34), (13)(24), (14)(23)} and D8. They are also strongly F -closed, which follows from the fact the every
subgroup of order 2 of each of the subgroups A, B and C respectively only are F -isomorphic to other subgroups of order 2 of A, B and C respectively.
The above reasoning can be stated as a theorem.
Theorem 6.3. Let G be a finite group and P a Sylow p-subgroup of G. Then Q ≤ P is weakly FP(G)-closed if and only if Q is only FP(G)-isomorphic to
itself.
David Craven gave the following definition in [5], which however is not com-plete.
Definition 6.4. Let P and Q be finite p-groups and let ϕ : P → Q be a group homomorphism. If ψ : A → B is a group homomorphism between A, B ≤ P , we define the function ψϕ: ϕ(A) → ϕ(B) by ψϕ(ϕ(a)) = ϕ(ψ(a)).
Remark. The reason the definition is incomplete is because it is possible for ψϕ to not be well-defined, i.e., it can happen that ϕ(a) = ϕ(b) but
ψϕ(ϕ(a)) 6= ψϕ
(ϕ(b)). For example, take ϕ : Z2× Z2 → Z2 defined by
ϕ(x, y) = x and ψ : Z2× Z2 → Z2× Z2 defined by ψ(x, y) = (x + y, 0). Now,
observe that ϕ(0, 0) = ϕ(0, 1) = 0, but ψϕ(ϕ(0, 1)) = ϕ(ψ(0, 1)) = ϕ(1, 0) = 1
which is not equal to ψϕ(ϕ(0, 0)) = ϕ(ψ(0, 0)) = ϕ(0, 0) = 0.
Remark. Since
ψϕ(ϕ(a)ϕ(b)) = ψϕ(ϕ(ab)) = ϕ(ψ(ab)) = ϕ(ψ(a)ψ(b)) = ϕ(ψ(a))ϕ(ψ(b)) = ψϕ(ϕ(a))ψϕ(ϕ(b)),
Definition 6.5. Let F and E be fusion systems of finite p-groups P and Q respectively. Then we define a morphism between F and E to be a pair (ϕ, Φ) such that ϕ : P → Q is a group homomorphism and for all ψ ∈ HomF(A, B),
we have that Φ(ψ) = ψϕ∈ Hom
E(ϕ(A), ϕ(B)).
Remark. We will sometimes speak only of a morphism Φ : F → E . It is then assumed that we have a group homomorphism ϕ : P → Q such that everything in the above definition is fulfilled. Notice also that ψϕ is well-defined in this
case, since it is Φ(ψ) by definition.
In order to be able to prove the isomorphism theorems for fusion systems, we must first introduce a couple of definitions and theorems, since at the moment, we cannot even formulate them.
Definition 6.6. Let F and E be fusion systems on finite p-groups P and Q respectively. Suppose Φ : F → E is a morphism.
1. The kernel of Φ, denoted ker Φ, is a normal subgroup of P and defined by ker Φ = ker ϕ.
2. Φ is injective if ker Φ = 1.
3. Φ is surjective if ϕ is surjective and for all ψ ∈ HomE(R, S), there exists
ψ ∈ HomF(R, S) such that Φ(ψ) = ψ for some subgroups R and S of P .
4. Φ is an isomorphism if it is injective and surjective. As usual, we write this as F ∼= E .
Remark. From this definition, one sees that two fusion systems F and E on the finite p-groups P and Q respectively are isomorphic if there exists a group isomorphism ϕ : P → Q such that
HomE(ϕ(R), ϕ(S)) = ϕ ◦ HomF(R, S) ◦ ϕ−1|ϕ(R),
for all subgroups R and S of P .
Theorem 6.7 ([5]). Let F be a fusion system on a finite p-group P , and let ϕ : P → Q be a group homomorphism, where Q is a finite p-group, such that ψϕ : ϕ(A) → ϕ(B) is well-defined for every ψ ∈ HomF(A, B). Then ker ϕ is
strongly F -closed if and only if ψϕ is injective for all ψ ∈ HomF(A, B).
Proof. First suppose ker ϕ is strongly F -closed and let ψ ∈ HomF(A, B). Note
that ψ is injective by definition of HomF(A, B). Take any a ∈ A and suppose
ψϕ(ϕ(a)) = ϕ(ψ(a)) = 1. We want to show that ϕ(a) = 1. Let θ : A → ψ(A)
denote the corestriction of ψ to its image. As ψ(a) ∈ ker ϕ and the fact that ker ϕ is strongly F -closed, we get that
θ−1(ker ϕ ∩ ψ(A)) ⊆ ker ϕ. Since ψ(a) ∈ ker ϕ and a ∈ θ−1(ψ(a)), it follows that
a ∈ θ−1(ker ϕ ∩ ψ(A)) ⊆ ker ϕ,
and so ϕ(a) = 1. Conversely, suppose ψϕ is injective for all ψ ∈ Hom
F(A, B).
We want to show that ker ϕ is strongly F -closed, so take any N ⊆ ker ϕ and any ψ ∈ HomF(N, P ). Then ψϕ(ϕ(N )
| {z }
1
) = 1 = ϕ(ψ(N )) so that ψ(N ) ⊆ ker ϕ and we are done.
The above theorem can be weaken to the case when ker ϕ is weakly F -closed and the same proof goes through with strongly changed to weakly and N changed to ker ϕ.
Theorem 6.8. Let F be a fusion system on a finite p-group P , and let ϕ : P → Q be a group homomorphism, where Q is a finite p-group, such that ψϕ : ϕ(A) → ϕ(B) is well-defined for every ψ ∈ Hom(A, B), where ker ϕ
is a subgroup of A and B. Then ker ϕ is weakly F -closed if and only if ψϕ is
injective for all ψ ∈ HomF(A, B), where ker ϕ is a subgroup of A and B.
If F and E two fusion systems on finite p-groups P and Q respectively, and Φ : F → E is a morphism, we have that Φ(ψ) = ψϕ ∈ HomE(ϕ(A), ϕ(B)), so
that whenever ψ is in F , ψϕ is injective. Thus we get the following corollary which will be used when we state the first isomorphism theorem.
Cor 6.9. Let F and E be fusion systems on finite p-groups P and Q respectively. Then if Φ : F → E is a morphism, ker Φ is strongly F -closed.
The isomorphism theorems for groups and rings involve quotient groups and rings but we have yet no notion of quotient category or quotient fusion system. It turns out that there primarily are two different ways one can define quotient of fusion systems and we do so now after introducing a definition that will help formulate our quotient systems.
Definition 6.10. Let P be a finite p-group. By U (P ) we will denote the universal fusion system, i.e., the fusion system on P such that
HomU (P )(Q, R) = {f : Q → R | f injective}.
Definition 6.11. Let F be a fusion system on a finite p-group P and let Q ≤ P be strongly F -closed. Let ϕ : P → P/Q be defined by ϕ(x) = xQ = x. This homomorphism is commonly called the natural homomorphism. Then we define ¯FQ to be the subgraph of U (P/Q) consisting of all objects of U (P/Q)
and morphisms
HomF¯Q(R/Q, S/Q) = {ψ : ϕ(R) → ϕ(S) | ψ ∈ HomF(R, S), QR = R, QS = S},
where ψ(ϕ(r)) = ϕ(ψ(r)) = ψ(r)Q. We define h ¯FQi to be the subgraph of
U (P/Q) with the same objects as ¯FQ and morphisms consisting of all finite
composition of morphisms from ¯FQ.
Remark. ψ is well-defined in this case. To see this, take ϕ(r1) and ϕ(r2) such
that ϕ(r1) = ϕ(r2). Then r1r2−1 ∈ ker ϕ = Q so that, since Q is strongly
F -closed, ψ(r1r−12 ) ∈ Q. It follows that ψ(r1)Q = ψ(r2)Q, i.e., ψ(ϕ(r1)) =
ψ(ϕ(r2)) which means that ψ is well-defined.
Notice also that we cannot weaken the condition on Q to weakly F -closed, since ψ will not in general be defined on the whole of Q.
Remark. If Q is strongly F -closed, then Q is normal since if g ∈ P , then cg: Q → P is in F so that by definition cg(Q) = gQg−1≤ Q. Since conjugation
Remark. The reason we introduced h ¯FQi is because it can happen that ¯FQ is
not a category. For example, suppose we have two morphisms φ : A → B and ψ : C → D such that QB = QC for some strongly F -closed subgroup Q. The images φϕ : QA/Q → QB/Q and ψϕ : QC/Q → QD/Q in ¯F
Q are
then composable but it need not be the case that ψϕ◦ φϕ : QA/Q → QD/Q
arises from a morphism in F . We solve this problem by including all finite compositions of morphisms from ¯FQ, since then we get a category by definition.
Note however that it is possible for ¯FQ to be a category and in this case, we of
course have ¯FQ= h ¯FQi.
Theorem 6.12. Let F be a fusion system on a finite p-group P and suppose Q ≤ P is strongly F -closed. Then h ¯FQi is a fusion system on P/Q.
Proof. In the proof below, if Q is a subgroup contained in R let R = R/Q. Consider Homh ¯F
Qi(R, S) and suppose
gQR = gR ≤ S. Then gR ≤ S by the
lattice theorem, so we know that cg ∈ HomF(R, S). Since also QR = R and
QS = S it follows that cg ∈ Homh ¯FQi(R, S), so that (1) in Definition 5.1 is
satisfied. Next, we prove that the associated isomorphism and its inverse to any ψ ∈ Homh ¯FQi(R, S) is in Homh ¯FQi(R, ψ(R) and Homh ¯FQi(ψ(R), R) respectively. Let ψ ∈ Homh ¯F
Qi(R, S). Then we can find morphisms ψi∈ HomF¯Q(Ri, Si), i =
1, 2, . . . , n, such that ψ = ψn◦ ψn−1◦ · · · ◦ ψ1. To each ψi, there is a morphism
ψi ∈ HomF(Ri, Si) such that ψi = ψiϕ and QRi = R, QSi = S. Let θi ∈
HomF(Ri, ψi(Ri)) be the associated isomorphism to ψi. Since each θi is the
associated isomorphism to ψi,
θ = θn◦ θn−1◦ · · · ◦ θ1,
is the associated isomorphism to ψ. Hence, axiom (2) in 5.1 is true. To see that (3) is true, just replace θi by θi−1 in the equation above.
Theorem 6.13. Let F be a fusion system on a finite p-group P and let Q ≤ P be strongly F -closed. Then
1. The natural map ΦQ: F → h ¯FQi is a morphism between F and h ¯FQi.
2. ΦQ is surjective if and only if ¯FQ= h ¯FQi.
Remark. By the natural map ΦQ : F → h ¯FQi, we mean that the underlying
group homomorphism ϕ : P → P/Q is natural, i.e., ϕ(x) =x.
Proof. 1. Let ϕ : P → P/Q denote the underlying natural group homomor-phism. By definition, if ψ ∈ HomF(R, S), then
ψ ∈ Homh ¯FQi(RQ, SQ) = Homh ¯FQi(ϕ(R), ϕ(S)), so Φ is a functor.
2. First suppose that ΦQ is surjective. Then ϕ is surjective by definition.
Every morphism in h ¯FQi is compositions of morphisms of the form ψϕ,
where ψ is in F . Thus, the only way every morphism in h ¯FQi can be the
Next, we introduce the second kind of quotient system.
Definition 6.14. Let F be a fusion system on a finite p-group P and QE P . Let the underlying group homomorphism ϕ : P → P/Q be the natural one. Then we define the factor system F /Q to be the category whose objects coincide with the objects from ¯FQ and whose morphisms are defined by
HomF /Q(R, S) = {ψ : R → S | ψ ∈ HomF(R, S), ψ(Q) = Q}.
Remark. ¯ψ is well-defined in this case, by exactly the same reason as for why ¯
ψ is well-defined in ¯FQexcept we change strongly F -closed to weakly F -closed.
This is possible because we know that ψ is defined on the whole of Q. Also, by Theorem 6.8, each morphism in F /Q is injective.
Remark. The difference between F /Q and h ¯FQi is thus which morphisms you
choose to transform into the quotient. In h ¯FQi, we transform every morphism
ψ : R → S, even those for which Q is not a subgroup of R and S while in F /Q we only take the morphisms for which Q is a subgroup of R and S. These choices also means that we have to put different restrictions on Q. Q must be strongly F -closed when we construct h ¯FQi, since the kernel of ϕ is Q and by
Theorem 6.7, the kernel is strongly F -closed if and only if ψ is injective for all ψ in F , i.e., a member of U (P/Q). In F /Q, we do not transform all morphisms from F and so we need not require that ψ is injective for all ψ in F . This means that we can relax the condition on Q and still remain well-definedness.
Remark. If Q is weakly F -closed, the condition ψ(Q) = Q is redundant. Theorem 6.15. Let F be a fusion system on a finite p-group P and suppose Q is a normal subgroup of P . Then F /Q is a fusion system on P/Q.
Proof. The proof is very similar to Theorem 6.12. Consider HomF /Q(R, S) and
suppose gQR = gR ≤ S. Then gR ≤ S by the lattice theorem, so we know
that cg ∈ HomF(R, S). Since Q is normal, we know that cg(Q) = Q and so
cg ∈ HomF /Q(R, S). Hence, (1) in Definition 5.1 is fulfilled. Next, take any
ψ ∈ HomF /Q(R, S). Then we can find a ψ ∈ HomF(R, S) such that ψ = ψϕ
and ψ(Q) = Q. Let θ : R → ψ(R) be the associated isomorphism. Obviously, θ(Q) = Q, so θ : R → ψ(R) is the associated isomorphism to ψ so that (2) in 5.1 is satisfied. To see that the associated inverse is included, just replace θ with θ−1: ψ(R) → R. Hence, (3) is true and we are done.
Before we can prove the first isomorphism theorem, we need two lemma, the first due to Frattini.
Lemma 6.16 (Frattini’s argument). Let G be a finite group and let H be a normal subgroup of G. Then G = HNG(P ), for any P ∈ Sylp(H).
Proof. Let g ∈ G. Since H is normal, gP ≤ H, so that gP ∈ Sylp(H). But
all Sylow p-subgroups are conjugate, so there is an h ∈ H such that gP =hP , henceh−1gP = P and so h−1g ∈ NG(P ). Thus g ∈ HNG(P ) and G = HNG(P )
since g was arbitrary.
Lemma 6.17. Let P be a finite p-group and let H be a proper subgroup. Then H is a proper subgroup of NP(H).
Proof. The theorem is clear when H = 1, so assume otherwise. Let H act on G/H, the set of left cosets. Notice that the orbit of H is simply {H} and since the orbits partition G/H, there must exist another orbit of size relatively prime to p. The only possibility is another orbit of size 1, since the order of G/H is a power of p. Consequently, there exists g ∈ G \ H such that hgH = gH for all h ∈ H. Hence, g−1hg ∈ H for all h ∈ H and so g−1Hg ≤ H. But conjugation is injective, so that gHg−1 = H and we are done, since g 6∈ H.
Remark. Interestingly, in Lemma 6.17, we actually proved that if G is any finite group, H a proper p-subgroup and |G : H| divisible by p, then H is a proper subgroup of NG(H).
The next proof uses another formulation of Alperin’s fusion theorem, other than the one given in Theorem 4.3, which we state here for easy reference. A proof can be found in [5].
Theorem 6.18 (Alperin’s Fusion Theorem). Let F be a saturated fusion system on a finite p-group P , and let ϕ : R → S be an F -isomorphism. Then there exists
(a) a sequence of F -isomorphic subgroups R = R0, R1, . . . , Rn = S,
(b) a sequence U1, U2, . . . , Un of fully normalized, F -radical, F -centric
sub-groups, with Ri−1, Ri≤ Ui,
(c) and a sequence of F -automorphisms ϕi of Ui with ϕi(Ri−1) = Ri, such that
(ϕm◦ ϕm−1◦ · · · ◦ ϕ1)|R= ϕ.
Theorem 6.19 ([5]). Let F be a saturated fusion system on a finite p-group P , and let Q ≤ P . If Q is strongly F -closed, then F /Q = ¯FQ. Thus, the natural
map F → F /Q is a morphism by Theorem 6.13.
Proof. For brevity, we will write φ : QR/Q → QS/Q for the induced map in ¯
FQ from φ : R → S in F . If ψ : R/Q → S/Q is a map in F /Q, then ψ = ψ ϕ 0
for some ψ0: R → S in F such that ψ0(Q) = Q. Then, obviously, QR = R and
QS = S so that ψϕ0 is in ¯FQ. Thus, we need to show that every morphism in ¯FQ
is in F /Q. Note that if to every morphism φ : R → S in F , we can find another morphism ψ : QR → P such that φ = ψ, we would be done. To see this, simply observe that ψ, and therefore φ, is in F /Q since ψ(Q) = Q as Q is strongly F -closed. The proof is by induction on n = |P : R|. Note that we can suppose that Q 6≤ R, since otherwise φ(Q) = Q so that φ ∈ HomF /Q(R/Q, S/Q). In
particular, n > 1 as n = 1 implies P = R and P contains Q. Also note that we can assume that φ is a bijection since it is an injection. Since F is saturated, we can therefore apply Alperin’s fusion theorem and write φ = (φm◦ . . . φ2◦ φ1)|R,
where φi : Ui → Ui is an F -automorphism such that Ui is fully F -normalized.
There are also a sequence of F -isomorphic subgroups R0= R, R1, . . . , Rm= S
such that Ri−1, Ri≤ Ui and φi(Ri−1) = Ri.
Suppose first that φi is in F /Q for i = 1, 2, . . . , m. Then, since φ is the composition of φi restricted to QR/Q, φ is in F /Q. Thus without loss of
generality, we can assume that φi does not lie in F /Q for some i = 1, 2, . . . , m.
Since |P : Ui| ≤ |P : Ri| = |P : R|, we may also assume that |Ui| = |R| by the
is fully normalized and φ is an automorphism of R. Let K be the kernel of the map AutF(R) → Aut(QR/Q) given by φ 7→ φ. Let
T = NPK(R) = {g ∈ NP(R) | cg|P ∈ K}.
Since R is fully normalized, AutP(R) ∈ Sylp(AutF(R)) by Theorem 5.15 and
Definition 5.3. Thus, K ∩ AutP(R) = AutT(R) ∈ Sylp(K). By the Frattini
argument, it follows that AutF(R) = KNAutF(R)(AutT(R)).
We will need two results before we can continue. 1. Q ∩ T = NQK = NQ(R) and RNQ(R) > R.
The fact that Q ∩ T = NK
Q is evident from T = NPK(R). Obviously,
NK
Q(R) ⊆ NQ(R). Let g ∈ NQ(R) and x ∈ R. Note that gxQ = xQ
since gxg−1x−1 ∈ Q. It follows that cg is the identity automorphism of
Aut(QR/Q) since cg(xQ) = cg(x)Q = xQ. Thus, cg ∈ K and NQK(R) =
NQ(R). The second assertion follows from NQ(R) = NQ(Q ∩ R) > Q ∩ R
by Lemma 6.17.
2. ψ ∈ NAutF(R)(AutT(R)) =⇒ T ⊆ Nψ.
Since Nψ = c−1R (AutP(R) ∩ (ψ−1◦ AutP(R) ◦ ψ)), where cR : NP(R) →
AutP(R) is defined by cR(g) = cg, we see that T ⊆ Nψ if and only
if cR(T ) = AutT(R) ⊆ AutP(R) ∩ (ψ−1 ◦ AutP(R) ◦ ψ). Obviously,
AutT(R) ⊆ AutP(R). As ψ ∈ NAutF(R)(AutT(R)),
ψ(Aut
T(R)) = AutT(R).
Finally, since AutF(R) = KNAutF(R)(AutT(R)), we can write φ = χψ,
where χ ∈ K and ψ ∈ NAutF(R)(AutT(R)). As χ is the identity automorphism
in QR/Q, it follows that φ = ψ. If Nψ > R, then ψ can be extended to
ψ0 : Nψ → P , where |P : Nψ| < |P : R| = n, to that by induction, ψ0 lies in
F /Q. Hence, ψ lies in F /Q, which we assumed it did not. It remains to check the case when T ≤ Nψ ≤ R. Then Q ∩ T = NQ(R) ≤ R. This contradicts
RNQ(R) > R and so we are done.
Cor 6.20 (First Isomorphism Theorem, [5]). Let F and E be saturated fusion systems on the finite p-groups P and Q respectively. If Φ : F → E is a morphism, then F / ker Φ ∼= Im Φ.
Proof. Let ϕ1 be the underlying group homomorphism to Φ : F → E and
let ϕ2 : P → P/ ker ϕ1 be the underlying natural homomorphism to ΦK :
F → F /K, where K = ker Φ. Notice that by the previous theorem, it is enough to find an isomorphism between Φ(F ) and ΦK(F ) = F /K since ΦK is
surjective by Theorem 6.13. We will construct the isomorphism from the group homomorphism ϕ : ϕ1(P ) → ϕ2(P ) defined by ϕ(ϕ1(x)) = ϕ2(x) = x ker ϕ1.
First we need to check that ϕ is well-defined, so take ϕ1(x) and ϕ1(y) such
that ϕ1(x) = ϕ1(y). Then ϕ(ϕ1(x)) = x ker ϕ1 and ϕ(ϕ1(y)) = y ker ϕ1. Since
ϕ1(xy−1) = ϕ1(x)ϕ1(y)−1 = 1, it follows that x ker ϕ1 = y ker ϕ1 so that ϕ
is well-defined. Next, we define F : Φ(F ) → ΦK(F ) by F (Φ(ψ)) = Φ(ψ)ϕ.
To see that Φ(ψ)ϕ actually is in Φ
K(F ), suppose ψ : A → B. Then Φ(ψ)ϕ :
ϕ(ϕ1(A)) → ϕ(ϕ1(B)) = Φ(ψ)ϕ: ϕ2(A) → ϕ2(B). In fact, if a ∈ A, then
so we have that F (Φ(ψ)) = ΦK(ψ). Also, F is seen to be well-defined by exactly
the same method as above. Thus F is a functor. From F (Φ(ψ)) = ΦK(ψ), it
is evident that F is surjective since ϕ is surjective. It remains to show that F is injective so take any ϕ1(a) ∈ ker F . Then ϕ(ϕ1(a)) = ϕ2(a) = 1 so that
a ∈ ker ϕ2. But ϕ2: P → P/ ker ϕ1is the natural homomorphism, so a ∈ ker ϕ1.
Thus, ϕ1(a) = 1 and the proof is complete.
Definition 6.21. Let F be a fusion system on a finite p-group P . Then we say that E is a subsystem of F if E is a fusion system according to Definition 5.1. E will be said to be on Q ≤ P if all the objects of E are all the subgroups of Q. Theorem 6.22 (Second Isomorphism Theorem, [5]). Let F be a saturated fu-sion system on a finite p-group P and let Q ≤ P be strongly F -closed. Suppose E is a saturated subsystem of F on the group R ≤ P . Let EQ/Q = ΦQ(E ),
where ΦQ: F → F /Q is the natural map. Then E Q/Q ∼= E /(Q ∩ R).
Proof. The natural isomorphism QR/Q → R/(Q ∩ R) induces an isomorphism Φ : U (QR/Q) → U (R/(Q ∩ R)). Then Φ|EQ/Q : E Q/Q → Φ(E Q/Q) is also an
isomorphism, so we need to show that Φ(E Q/Q) = E /(Q ∩ R). First take any φ : S/Q → T /Q in E Q/Q. Since ΦQ(E ) = E Q/Q, we know that φ = ΦQ(ψ) =
ψϕfor some ψ ∈ Hom
E(S, T ) such that S = QS, T = QT and ϕ : P → P/Q the
natural homomorphism. But Φ(φ) = φϕ1 : ϕ
1(S/Q) → ϕ1(T /Q), where ϕ1 :
QR/Q → R/(Q∩R) is the natural isomorphism. Since ϕ1(S/Q) = S/(Q∩R), we
get Φ(φ) ∈ HomE/(Q∩R)(S/(Q∩R), T /(Q∩R)) and hence Φ(E Q/Q) ⊆ E /(Q∩R).
Next, take a morphism θ : S/(Q ∩ R) → T /(Q ∩ R) in E /(Q ∩ R). Then there exists a χ ∈ HomE(S, T ) such that θ = χϕ2, where ϕ2 : R → R/(Q ∩ R) is
the natural homomorphism. The image χ = ΦQ(χ) : QS/Q → QT /Q satisfies
Φ(χ) = θ and so E /(Q ∩ R) ⊆ Φ(E Q/Q). Hence, E /(Q ∩ R) = Φ(E Q/Q) and the proof is complete.
Theorem 6.23 (Third Isomorphism Theorem, [5]). Let F be a saturated fusion system on a finite p-group P , and let Q and R be strongly F -closed subgroups of P . Then (F /Q)/(R/Q) ∼= F /R.
Proof. Since P/R ∼= (P/Q)/(R/Q) by the third isomorphism theorem, we can without loss of generality assume that E = (F /Q)/(R/Q) and F /R are on the same group. Thus, we want to show that every morphism in E are in F /R and conversely. First take any φ ∈ HomF /R(S/R, T /R). Then we can
find a φ ∈ Hom(S, T ) with image φ = φϕ0, where ϕ
0 : P → P/R is the natural
homomorphism. Let φ0: S/Q → T /Q be the image of φ in F /Q, φ0 = φϕ1, where
ϕ1: P → P/Q is the natural homomorphism. Then the image φ00: S/R → T /R
in E of φ0 is φ00= (φ0)ϕ2, where ϕ
2: P/Q → P/R is the natural homomorphism.
Since φ00(sR) = ϕ2(φ0(sQ)) = ϕ2(φ(s)Q)) = φ(s)R = φ(sR), it follows that
F /R ⊆ (F /Q)/(R/Q). Conversely, take any φ00 : S/R → T /R in E . Then φ00
is the image of a morphism φ0 : S/Q → T /Q in F /Q and φ0 is the image of a morphism φ : S → T in F . Let φ be the image of φ in F /R. As above, φ = φ00 and so (F /Q)/(R/Q) ⊆ (F /R).
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