Fusion systems of blocks with nontrivial strongly closed
subgroups
Heguo Liu1 , Xingzhong Xu1,2 *, Jiping Zhang3

Abstract. In this paper, we find some exotic fusion systems which have nontrivial strongly closed subgroups, and we prove these fusion systems are also
not realizable by p-blocks of finite groups.

arXiv:1701.01817v1 [] 7 Jan 2017

Key Words: saturated fusion system, finite p-group, block
2000 Mathematics Subject Classification: 20C20

1. Introduction
To get the p-local information of finite group G, one can forget some information
of this group and change the structure of a group to a category. The objects of this
category are all subgroups of a given Sylow p-subgroup S of G, and the morphism
set of Q, P (≤ S) is the set HomG (Q, P ) := {cg |Qg ≤ P }. This category is a
saturated fusion system FS (G) of finite group G over S.
Alperin’s theorem for fusion systems tells us that every morphism in FS (G) can
be generated by some automorphisms of objects in F f cr (see theorem 2.4). So we
can get all the p-local information of G from the automorphisms of objects in F f cr .
The more general concept of a ” saturated fusion system F over a finite p-group S
” was introduced by Lluis Puig (See [6, 11]). Our main result is the following:
Main Theorem. Let p be an odd prime and P = hx, y, z|xp = y p = z p = 1, [x, y] =
z, [z, x] = [z, y] = 1i ∼
= p1+2
+ . Let F1 and F2 be a saturated fusion system over P
and A respectively. Here, A is a finite abelian p-group. Let F = F1 × F2 . Then F
is not a fusion system of a p-block of a finite group if and only if F1 is not a fusion
system of a p-block of a finite group.
This theorem is motivated by Kessar’s problem about fusion systems of blocks:
Does the exotic fusion system is realizable by a fusion system of a block of a finite
group? In [7], she had proven that the Solomon fusion system FSol (3) is not a fusion
system of a 2-block of a finite group. In [8], they used their reduction theorem to
prove that the exotic fusion systems over p1+2
do not occur as fusion systems of
+
p-blocks of finite groups.
These exotic fusion systems which are studied in [7, 8] have no nontrivial strongly
closed subgroups(see [7, Proposition 4.3(iv)],[13, Lemma 4.17]). In this paper, we
fine the exotic fusion systems which have nontrivial strongly closed subgroups, and
1.Department of Mathematics, Hubei University, Wuhan 430062, China
2. Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra,
Spain
3. School of Mathematical Sciences, Peking University, Beijing, 100871, China
Heguo Liu’s E-mail: ghliu@hubu.edu.cn
Xingzhong Xu’s E-mail: xuxingzhong407@mat.uab.cat, xuxingzhong407@126.com
∗ Corresponding author
Jiping Zhang’s E-mail: jzhang@pku.edu.cn
1

2

in Theorem 3.4, we prove these fusion systems are also not realizable by p-blocks
of finite groups. In fact, our result relies on methods in [7, 8] and theory of fusion
systems of blocks built in [4].
The paper is organized as follows. We start in Section 2 with the basic definitions
on fusion systems, Alperin’s theorem for fusion systems and factor fusion systems.
In Section 3, we prove the main theorem (see Theorem 3.4).
2. Preliminaries
In this section we collect some known results that will be needed later. For the
background theory of fusion systems, we refer to [4], [12], [14].
Definition 2.1. A f usion system F over a finite p-group S is a category whose
objects are the subgroups of S, and whose morphism sets HomF (P, Q) satisfy the
following two conditions:
(a) HomS (P, Q) ⊆ HomF (P, Q) ⊆ Inj(P, Q) for all P, Q ≤ S.
(b) Every morphism in F factors as an isomorphism in F followed by an inclusion.
Definition 2.2. Let F be a fusion system over a p-group S.
• Two subgroups P, Q are F -conjugate if they are isomorphic as objects of the
category F . Let P F denote the set of all subgroups of S which are F -conjugate to
P . Since HomF (P, P ) ⊆ Inj(P, P ), we usually write HomF (P, P ) = AutF (P ) and
HomS (P, P ) = AutS (P ).
• A subgroup P ≤ S is f ully automised in F if AutS (P ) ∈ Sylp (AutF (P )).
• A subgroup P ≤ S is receptive in F if it has the following property: for each
Q ≤ S and each ϕ ∈ IsoF (Q, P ), if we set
Nϕ = {g ∈ NS (Q)|ϕ ◦ cg ◦ ϕ−1 ∈ AutS (P )},
then there is ϕ ∈ HomF (Nϕ , S) such that ϕ|Q = ϕ. (where cg : x 7−→ g −1 xg for
g ∈ S)
• A fusion system F over a p-group S is saturated if each subgroup of S is
F -conjugate to a subgroup which is fully automised and receptive. This definition
is due to [12].
Definition 2.3. Let F be a fusion system over a p-group S.
• A subgroup P ≤ S is f ully centralised in F if |CS (P )| ≥ |CS (Q)| for all
Q ∈ PF.
• A subgroup P ≤ S is f ully normalised in F if |NS (P )| ≥ |NS (Q)| for all
Q ∈ PF.
• A subgroup P ≤ S is F -centric if CS (Q) = Z(Q) for Q ∈ P F .
• A subgroup P ≤ S is F -radical if OutF (P ) = AutF (P )/Inn(P ) is p-reduced;
i.e., if Op (OutF (P )) = 1.
• Let F r , F c , F f denote the full subcategories whose objects are F -radical, F centric, and fully normalised in F respectively. Let F f cr = F f ∩ F c ∩ F r and
F cr = F c ∩ F r .

3

• A subgroup P ≤ S is normal in F (denoted P E F ) if for all Q, R ∈ S
and all ϕ ∈ HomF (Q, R), ϕ extends to a morphism ϕ ∈ HomF (QP, RP ) such that
(P )ϕ = P .
• A subgroup P ≤ S is strongly closed in F if no element of P is F -conjugate
to an element of S − P .
Definition 2.4. A saturated fusion system F over a p-group S will be called realizable if F = FS (G) for some finite group G with S ∈ Sylp (G), and will be called
exotic otherwise.
Theorem 2.5. (Alperin’s theorem for fusion systems). Let F be a saturated fusion
system over a p-group S. Then for each morphism ϕ ∈ IsoF (P, R) in F , there exist
sequences of subgroups of S
P = P0 , P1 , . . . , Pk = R, and Q1 , Q2 , . . . , Qk ,
and morphism ψi ∈ AutF (Qi ), such that
(a) Qi is fully normalised in F , F -radical, and F -centric for each i(that is
Qi ∈ F f cr );
(b) Pi−1 , Pi ≤ Qi and ψi (Pi−1 ) = Pi for each i; and
(c) ψ = ψk ◦ ψk−1 ◦ · · · ◦ ψ1 .
The following is devoted to recalling the basic definitions of factor fusion systems,
we refer to [2],[4].
Definition 2.6. Let F be a saturated fusion system over a p-group S, and fix
Q ≤ S and K ≤ Aut(Q). Let NFK (Q) ⊆ F be the fusion system over NSK (Q) where
P, R ≤ NSK (Q),
HomNFK (Q) (P, R) = {ϕ ∈ HomF (P, R)|∃ ϕ ∈ ϕ ∈ HomF (P Q, RQ)
with ϕ|P = ϕ, ϕ(Q) = Q, and ϕ|Q ∈ K}.
Aut(Q)

{1}

(Q) and CF (Q) = NF (Q). If Q is fully
As special cases, set NF (Q) = NF
K-normalized in F , then NFK (Q) is a saturated fusion system.
Definition 2.7. Let F be a saturated fusion system over a p-group S, T a strongly
closed in S with respect to F , and N = NF (T ).
• Set S + = S/T and θ : S −→ S + be the natural map θ : x −→ x+ = xT ∈ S + .
• We define a category F /T : F /T and a morphism θ : N −→ F /T . The
objects of F /T are the subgroups of S + . For P ≤ S and α ∈ HomN (P, S) define
α+ : P + −→ S + by α+ (x+ ) = α(x)+ . This is well defined as T a strongly closed
in S with respect to F . Now define
HomF /T (P + , S + ) = {β + |β ∈ HomN (P T, S)}.
and define θP : HomN (P, S) −→ HomF /T (P + , S + ) by θP (α) = α+ . For α ∈
HomN (P, S), α extends to α̂ ∈ HomN (P T, S) and α̂+ = α+ , so θP is well defined
and surjective.
We next look at the definition of products of fusion systems(see [3, 6]).
Definition 2.8. Let F1 and F2 be a saturated fusion system over finite p-groups
S1 and S2 respectively. F1 × F2 is fusion system over S1 × S2 generated by the set
of all (ϕ1 , ϕ2 ) ∈ Hom(P1 × P2 , Q1 × Q2 ) for ϕi ∈ HomF (Pi , Qi ).
Remark 2.9. Let F1 and F2 be a saturated fusion system over S1 and S2 respectively. Let F = F1 × F2 . Then F /S1 ∼
= F2 .
= CF (S1 )/S1 ∼

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3. Proof of the Main theorem
For sake of completeness, we recall the theory of fusion systems of blocks(see [4,
Part IV]).
Let k be an algebraically closed field of characteristic p, G a finite group and b
a block of kG, that is b is a primitive idempotent of Z(kG). For any p-subgroup
Q of G the canonical projection from kN to kCG (Q) induces an algebra morphism
Q
BrkG
Q from the subalgebra of fixed points of Q, (kG) onto kCG (Q)(see [4, 5]). This
morphism is known in the literature as Brauer morphism.
A (kG, b, G)-Brauer pair is a pair (Q, eQ ) where Q is a p-subgroup of G such
kG
that BrkG
Q (b) 6= 0 and eQ is a block of kCG (Q) such that BrQ (b)eQ 6= 0. Let
(Q, eQ ), (R, eR ) be (kG, b, G)-Brauer pair. We say that (Q, eQ ) is contained in
(R, eR ) if Q ≤ R and for any primitive idempotent i ∈ (kG)R such that BrkG
Q (i)eR 6=
0, we have BrkG
(i)e
=
6
0.
Here,
we
write
(Q,
e
)
≤
(R,
e
).
Recall
[4,
5],
we know
Q
Q
R
Q
that
(1) Given a (kG, b, G)-Brauer pair (R, eR ) and Q ≤ R, there exists a unique
(c, G)-Brauer pair (Q, eQ ) contained in (R, eR ).
(2) All maximal (kG, b, G)-Brauer pair are G-conjugate.
Definition 3.1. Let Q ≤ P be p-subgroups of G.
• Define
kG
P
BrkG
P,Q : BrQ ((kG) ) → kCG (P )
kG
kG
P
by BrkG
P,Q (BrQ (a)) = BrP (a) for a ∈ (kG) . If eQ is a block of kCG (Q) and eQ
kG
kG
kG
is P -stable, then BrP,Q (BrQ (eQ )) = BrP,Q (eQ ) = BrkG
P (eQ ) = eQ .

• Let (Q, eQ ), (P, eP ) be (kG, b, G)-Brauer pair. We write that (Q, eQ ) ✂ (P, eP )
kG
if Q is a normal subgroup of P , eQ is P -stable and BrkG
P,Q (BrQ (eQ ))eP = eP . That
is eQ eP = eP .
• For a (kG, b, G)-Brauer pair (Q, eQ ), let us denote by NG (Q, eQ )the stabilizer
in NG (Q) of eQ .
For further standard notation and terminology, we follow [4], [10].
Definition 3.2. Let (S, eS ) is the maximal (kG, b, G)-Brauer pair. For each Q ≤ S,
let eQ denote the unique block of kCG (Q) such that (Q, eQ ) ≤ (S, eS ). The fusion
category of (kG, b, G) over S, eS is the category F(S,eS ) (kG, b, G) whose objects are
the subgroups of S and which has morphism sets
MorF(S,eS ) (kG,b,G) (Q, R) = HomG ((Q, eQ ), (R, eR ))
for Q, R ≤ S (see [4, Part IV, Defintion 2.21]).
∼ F(S,e ) (kG, b, G), we say that
Definition 3.3. If an abstract fusion system F =
S
fusion system F is block realizable. By Brauer’s Third Main Theorem, any realizable
saturated fusion system is block realizable. Moreover, if F is not block realizable,
then F is exotic. (see [4, Part IV, p. 221]).
Theorem 3.4. Let p be an odd prime and P = hx, y, z|xp = y p = z p = 1, [x, y] =
z, [z, x] = [z, y] = 1i ∼
= p1+2
+ . Let F1 , F2 be a saturated fusion system over P and
A respectively. Here, A is a finite abelian p-group. Let F = F1 × F2 . Then F is
not a fusion system of a p-block of a finite group if and only if F1 is not a fusion
system of a p-block of a finite group.

5

Proof. Let S := P × A. Suppose that F is a fusion system of a p-block of a finite
group, we can set F = F(S,eS ) (kG, b, G) where G is a finite group, b is a block of
G, and (S, eS ) is a maximal (kG, b, G)-Brauer pair.
Since F = F1 ×F2 , we have A is strongly closed in F . Let (A, eA ) be a (kG, b, G)Brauer pair such that (A, eA ) ≤ (S, eS ). By [4, Part IV, Proposition 3.19(b)], we
have
CF (A) = NFK (A) = F(S,eS ) (kCG (A), eA , CG (A))
by [4, Part IV, Proposition 3.8]. Here, K = AutCG (A) (A) = {IdA }.
By Definition 2.7, we have CF (A)/A ∼
= F /A ∼
= F1 . Since A is abelian, we have
A ≤ CG (A). As before, we denote by µA the natural k-algebra homomorphism
µA : kCG (A) −→ k(CG (A)/A)
X
X
λx x 7−→
λx x,
x

x

where x = xA ∈ CG (A)/A. As eA is a block of kCG (A) and |CG (A) : CCG (A) (A)| =
1, then eA := µ(eA ) is a block of k(CG (A)/A) by [10, Ch. 5, Theorem 8.11].
In the following, we will check that
CF (A)/A ⊆ F(S/A,εS ) (kCG (A)/A, eA , CG (A)/A).
Here, εS will be defined in the following text.
First, we will prove that there is an CG (A)-poset equivalence between the subset
of (kCG (A), eA , CG (A))-Brauer pairs consisting of those pairs who contains (A, eA ),
and the subset of (kCG (A)/A, eA , CG (A)/A)-Brauer pairs by the map Φ
(T, eT ) 7−→ Φ(T, eT ).
Here, (T, eT ) is a (kCG (A), eA , CG (A))-Brauer pair such that (T, eT ) ≤ (S, eS ). We
will define the Φ(T, eT ) in the following.
Since A ✂ T ≤ S, set T = T /A. It is easy to see that CG (T )/A ≤ CCG (A)/A (T ),
we can set that CCG (A)/A (T ) = LT /A. Here, LT ≤ CG (A) and CG (T ) ✂ LT . Since
LT acts on T /A and A trivially, we have LT /CG (T ) is a p-group by [4, Appendix
A, Lemma A.2]. Since (T, eT ) is a (kCG (A), eA , CG (A))-Brauer pair, thus (T, eT )
is also a (kG, b, G)-Brauer pair by [4, Part IV, Lemma 3.5]. That is eT is a block
of kCCG (A) (T ) = kCG (T ). Then there exists a unique block εT of kLT such that
εT covers eT by [10, Ch.5, Corollary 5.6].
Set εT := µA (εT ), since LT /CG (T ) is a p-group, thus |LT : CLT (T )| is power of
prime. As CLT (T ) ≤ CLT (A), we can see that |LT : CLT (A)| is power of prime.
Hence εT := µA (εT ) is a block of k(LT /A) = kCCG (A)/A (T ) by [10, Ch. 5, Theorem
8.11].
kC (A)

6
Since (T, eT ) is a (kCG (A), eA , CG (A))-Brauer pair, thus BrT G (eA )eT =
kCG (A)
(eA )· εT 6= 0.
0. As eT is covered by a unique εT , it implies that BrT
Since A ✂ CG (A), we have KerµA ⊆ J(kCG (A)) by [10, Ch.5, Lemma 2.7]. So
kC (A)
µC (BrT G (eA )εT ) 6= 0. Also eA := µA (eA ) is a block of kCG (A) and
kCG (A)

(kCG (A))T

BrT

/ kCG (T )
µA

µA


(kCG (A)/A)T


BrT

/ kCC (A)/A (T )
G

6

commutes, thus
kCG (A)

µA (BrT

kCG (A)/A

(eA )) = BrT

(eA ).

kC (A)/A

(eA )εT 6= 0. Therefore, (T , εT ) is a (kCG (A)/A, eA , CG (A)/A)Hence BrT G
Brauer pair. So we can define Φ(T, eT ) = (T , εT ) and Φ is injective.
We show that the map Φ is inclusion preserving. By [4, Part IV, Proposition
2.14], we only need to prove it when (A, eA ) ≤ (R, eR ) ✂ (T, eT ). So we have R ✂ T ,
kC (A)
BrT,RG (eR )eT = eT and eR is T -stable. That is eR eT = eT . Hence εR εT 6= 0
because εR , εT cover eR , eT respectively. Now, we can see that 0 6= µA (εR εT ) =
εR εT . Also let t ∈ T , εtR is a block of kLR because T acts on LR . (Here, the
definition of LR like LT above.) But there exists a unique block εR of kLR such
that εR covers eR , thus εtR = εR because εtR covers etR = eR . (Here, eR is T -stable.)
Hence εR is T -stable. Therefore, (R, εR ) ✂ (T , εT ), that means Φ(R, eR ) ✂ Φ(T, eT ).
So Φ is a poset bijection.
g

Let g ∈ CG (A), we have Φ(T, eT )g = (T , εT )g = (T , εgT ). Since εT cover eT ,
thus εgT covers egT . Here, µA (εT )g = µA (εgT ). That is Φ(T, eT )g = Φ(T g , egT ). Hence
Φ is compatible with CG (A)-action.
Now, we will prove that (S, εS ) is a maximal (kCG (A)/A, eA , CG (A)/A)-Brauer
pair.
Suppose that there exists a (kCG (A)/A, eA , CG (A)/A)-Brauer pair (R/A, εR )
kCG (A)/A
kC (A)
(S, εS ). Hence, R S and BrR/A
(eA )εR 6= 0. So BrR G (eA )εR 6= 0. There
kC (A)

exists a block eR of kCG (R) such that εR covers eR and BrR G (eA )eR 6= 0. So
(R, eR ) is a (kCG (A), eA , CG (A))-Brauer pair. By [4, Part IV, Lemma 3.5], we
have (R, eR ) is a (kG, b, G)-Brauer pair. That is a contradiction. Hence (S, εS ) is a
maximal (kCG (A)/A, eA , CG (A)/A)-Brauer pair. So we can see that Φ is surjective.
By [4, Part IV, Proposition 3.2], we have F(S/A,εS ) (kCG (A)/A, eA , CG (A)/A) is
a saturated fusion system over S/A.
Second, let A ≤ R ≤ S, we can see that
NCG (A) (R, eR )/A ≤ NCG (A)/A (R, εR ).
That is
CF (A)/A ⊆ F(S/A,εS ) (kCG (A)/A, eA , CG (A)/A).
Since CF (A)/A is a saturated fusion system over S/A ∼
= p1+2
+ , if CF (A)/A is not
a fusion system of a p-block of a finite group, then CF (A)/A is one of the following
saturated fusion systems:
Name

OutF (71+2
+ )

|F cr |

AutF (V )

RV1

62 : 2

6+2

SL2 (7) : 2, GL2 (7)

RV2

D16 × 3

4+4

SL2 (7) : 2, SL2 (7) : 2

RV3

SD32 × 3

8

SL2 (7) : 2

by [8]. And this table also can be fined from [13, Lemma 4.17].
Here, CF (A)/A ⊆ F(S/A,εS ) (kCG (A)/A, eA , CG (A)/A). Hence either
CF (A)/A = F(S/A,εS ) (kCG (A)/A, eA , CG (A)/A),
or CF (A)/A ∼
= RV3 by [8, 13].
= RV2 and F(S/A,εS ) (kCG (A)/A, eA , CG (A)/A) ∼

7

However, RV1 , RV2 , RV3 are not fusion systems of a p-blocks of finite groups.
That is a contradiction. Hence, CF (A)/A ∼
= F1 is a fusion system of a p-block of a
finite group.
Conversely suppose that F1 is a fusion system of a p-block of a finite group. Since
P ∼
= p1+2
+ , we have that F1 is not exotic by [8, Theorem 6.4]. Let F1 = FP (G1 )
for a finite group G1 with P ∈ Sylp (G1 ). Set G = G1 × (A ⋊ AutF (A)), then
P × A ∈ Sylp (G). So F ∼
= FP ×A (G). That is F is a fusion system of a p-block of a
finite group.

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