Fusion systems of blocks with nontrivial strongly closed subgroups

0.14 MB·

by  Heguo Liu

Fusion systems of blocks with nontrivial strongly closed
subgroups
Heguo Liu1, Xingzhong Xu1,2*, Jiping Zhang3
Abstract. Inthispaper,wefindsomeexoticfusionsystemswhichhavenon-
trivial strongly closed subgroups, and we prove these fusion systems are also
notrealizablebyp-blocksoffinitegroups.
Key Words: saturatedfusionsystem,finitep-group,block
2000 MathematicsSubject Classification: 20C20
7
1
1. Introduction
0
2
Togetthep-localinformationoffinitegroupG,onecanforgetsomeinformation
n
ofthis groupandchangethe structureofagroupto acategory. The objectsofthis
a
J category are all subgroups of a given Sylow p-subgroup S of G, and the morphism
7 set of Q,P(≤ S) is the set HomG(Q,P) := {cg|Qg ≤ P}. This category is a
saturated fusion system F (G) of finite group G over S.
S
]
R Alperin’s theorem for fusionsystems tells us that everymorphismin F (G) can
S
G be generated by some automorphisms of objects in Ffcr(see theorem 2.4). So we
. cangetallthep-localinformationofGfromtheautomorphismsofobjectsinFfcr.
h
The more generalconcept of a ” saturated fusion system F over a finite p-groupS
t
a ” was introduced by Lluis Puig (See [6, 11]). Our main result is the following:
m
Main Theorem. Letp bean oddprime andP =hx,y,z|xp =yp =zp =1,[x,y]=
[
z,[z,x] = [z,y] = 1i ∼= p1+2. Let F and F be a saturated fusion system over P
+ 1 2
1 and A respectively. Here, A is a finite abelian p-group. Let F =F ×F . Then F
1 2
v
is not a fusion system of a p-block of a finite group if and only if F is not a fusion
7 1
system of a p-block of a finite group.
1
8
1 This theorem is motivated by Kessar’s problem about fusion systems of blocks:
0 Does the exotic fusion system is realizable by a fusion system of a block of a finite
.
1 group? In[7],shehadproventhattheSolomonfusionsystemFSol(3)isnotafusion
0 system of a 2-block of a finite group. In [8], they used their reduction theorem to
7 prove that the exotic fusion systems over p1+2 do not occur as fusion systems of
1 +
p-blocks of finite groups.
:
v
i Theseexoticfusionsystemswhicharestudiedin[7,8]havenonontrivialstrongly
X
closed subgroups(see [7, Proposition 4.3(iv)],[13, Lemma 4.17]). In this paper, we
r fine the exoticfusionsystems whichhavenontrivialstronglyclosedsubgroups,and
a
1.DepartmentofMathematics, HubeiUniversity,Wuhan430062, China
2. Departament de Matema`tiques, Universitat Aut`onoma de Barcelona, E-08193 Bellaterra,
Spain
3. SchoolofMathematicalSciences, PekingUniversity,Beijing,100871, China
HeguoLiu’sE-mail: [email protected]
XingzhongXu’sE-mail: [email protected], [email protected]
∗Correspondingauthor
JipingZhang’sE-mail: [email protected]
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].
Thepaperisorganizedasfollows. WestartinSection2withthebasicdefinitions
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 fusion system F over a finite p-group S is a category whose
objects are the subgroups of S, and whose morphism sets Hom (P,Q) satisfy the
F
following two conditions:
(a) Hom (P,Q)⊆Hom (P,Q)⊆Inj(P,Q) for all P,Q≤S.
S F
(b) Every morphism in F factors as an isomorphism in F followed by an inclu-
sion.
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 PF denote the set of all subgroups of S which are F-conjugate to
P. Since Hom (P,P) ⊆ Inj(P,P), we usually write Hom (P,P) = Aut (P) and
F F F
Hom (P,P)=Aut (P).
S S
• A subgroup P ≤S is fully automised in F if Aut (P)∈Syl (Aut (P)).
S p F
• A subgroup P ≤S is receptive in F if it has the following property: for each
Q≤S and each ϕ∈Iso (Q,P), if we set
F
N ={g ∈N (Q)|ϕ◦c ◦ϕ−1 ∈Aut (P)},
ϕ S g S
then there is ϕ ∈ Hom (N ,S) such that ϕ| = ϕ. (where c : x 7−→ g−1xg for
F ϕ Q g
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 fully centralised in F if |C (P)| ≥ |C (Q)| for all
S S
Q∈PF.
• A subgroup P ≤ S is fully normalised in F if |N (P)| ≥ |N (Q)| for all
S S
Q∈PF.
• A subgroup P ≤S is F-centric if C (Q)=Z(Q) for Q∈PF.
S
• A subgroup P ≤ S is F-radical if Out (P) = Aut (P)/Inn(P) is p-reduced;
F F
i.e., if O (Out (P))=1.
p F
• Let Fr,Fc,Ff denote the full subcategories whose objects are F-radical, F-
centric, and fully normalised in F respectively. Let Ffcr = Ff ∩Fc ∩ Fr and
Fcr =Fc∩Fr.
3
• A subgroup P ≤ S is normal in F (denoted P E F) if for all Q,R ∈ S
and all ϕ∈Hom (Q,R), ϕ extends to a morphism ϕ∈Hom (QP,RP) such that
F F
(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 real-
izable if F = F (G) for some finite group G with S ∈ Syl (G), and will be called
S p
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∈Iso (P,R) in F, there exist
F
sequences of subgroups of S
P =P ,P ,...,P =R, and Q ,Q ,...,Q ,
0 1 k 1 2 k
and morphism ψ ∈Aut (Q ), such that
i F i
(a) Q is fully normalised in F, F-radical, and F-centric for each i(that is
i
Q ∈Ffcr);
i
(b) P ,P ≤Q and ψ (P )=P for each i; and
i−1 i i i i−1 i
(c) ψ =ψ ◦ψ ◦···◦ψ .
k k−1 1
Thefollowingisdevotedtorecallingthebasicdefinitionsoffactorfusionsystems,
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 NK(Q)⊆F be the fusion system over NK(Q) where
F S
P,R≤NK(Q),
S
Hom (P,R)={ϕ∈Hom (P,R)|∃ ϕ∈ϕ∈Hom (PQ,RQ)
NFK(Q) F F
with ϕ| =ϕ,ϕ(Q)=Q, and ϕ| ∈K}.
P Q
As special cases, set N (Q) = NAut(Q)(Q) and C (Q) = N{1}(Q). If Q is fully
F F F F
K-normalized in F, then NK(Q) is a saturated fusion system.
F
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 =N (T).
F
• 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 ∈ Hom (P,S) define
N
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
Hom (P+,S+)={β+|β ∈Hom (PT,S)}.
F/T N
and define θ : Hom (P,S) −→ Hom (P+,S+) by θ (α) = α+. For α ∈
P N F/T P
Hom (P,S), α extends to αˆ ∈Hom (PT,S) and αˆ+ =α+, so θ is well defined
N N P
and surjective.
We next look at the definition of products of fusion systems(see [3, 6]).
Definition 2.8. Let F and F be a saturated fusion system over finite p-groups
1 2
S and S respectively. F ×F is fusion system over S ×S generated by the set
1 2 1 2 1 2
of all (ϕ ,ϕ )∈Hom(P ×P ,Q ×Q ) for ϕ ∈Hom (P ,Q ).
1 2 1 2 1 2 i F i i
Remark 2.9. Let F and F be a saturated fusion system over S and S respec-
1 2 1 2
tively. Let F =F ×F . Then F/S ∼=C (S )/S ∼=F .
1 2 1 F 1 1 2
4
3. Proof of the Main theorem
For sakeof completeness, we recallthe 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 ofG the canonicalprojectionfrom kN to kC (Q) induces analgebramorphism
G
BrkG fromthesubalgebraoffixedpointsofQ,(kG)Q ontokC (Q)(see[4,5]). This
Q G
morphism is known in the literature as Brauer morphism.
A (kG,b,G)-Brauer pair is a pair (Q,e ) where Q is a p-subgroup of G such
Q
that BrkG(b) 6= 0 and e is a block of kC (Q) such that BrkG(b)e 6= 0. Let
Q Q G Q Q
(Q,e ),(R,e ) be (kG,b,G)-Brauer pair. We say that (Q,e ) is contained in
Q R Q
(R,e )ifQ≤Randforanyprimitiveidempotenti∈(kG)R suchthatBrkG(i)e 6=
R Q R
0,wehaveBrkG(i)e 6=0. Here,wewrite(Q,e )≤(R,e ). Recall[4,5],weknow
Q Q Q R
that
(1) Given a (kG,b,G)-Brauer pair (R,e ) and Q ≤ R, there exists a unique
R
(c,G)-Brauer pair (Q,e ) contained in (R,e ).
Q R
(2) All maximal (kG,b,G)-Brauer pair are G-conjugate.
Definition 3.1. Let Q≤P be p-subgroups of G.
• Define
BrkG :BrkG((kG)P)→kC (P)
P,Q Q G
by BrkG (BrkG(a)) = BrkG(a) for a ∈ (kG)P. If e is a block of kC (Q) and e
P,Q Q P Q G Q
is P-stable, then BrkG (BrkG(e ))=BrkG (e )=BrkG(e )=e .
P,Q Q Q P,Q Q P Q Q
• Let (Q,e ),(P,e ) be (kG,b,G)-Brauer pair. We write that (Q,e )✂(P,e )
Q P Q P
if Q is a normal subgroup of P, e is P-stable and BrkG (BrkG(e ))e =e . That
Q P,Q Q Q P P
is e e =e .
Q P P
• For a (kG,b,G)-Brauer pair (Q,e ), let us denote by N (Q,e )the stabilizer
Q G Q
in N (Q) of e .
G Q
For further standard notation and terminology, we follow [4], [10].
Definition3.2. Let(S,e )isthemaximal(kG,b,G)-Brauerpair. ForeachQ≤S,
S
let e denote the unique block of kC (Q) such that (Q,e ) ≤ (S,e ). The fusion
Q G Q S
category of (kG,b,G) over S,e is the category F (kG,b,G) whose objects are
S (S,eS)
the subgroups of S and which has morphism sets
Mor (Q,R)=Hom ((Q,e ),(R,e ))
F(S,eS)(kG,b,G) G Q R
for Q,R≤S (see [4, Part IV, Defintion 2.21]).
Definition 3.3. If an abstract fusion system F ∼= F (kG,b,G), we say that
(S,eS)
fusionsystemF isblockrealizable. ByBrauer’sThirdMainTheorem, anyrealizable
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 = yp = zp = 1,[x,y] =
z,[z,x] = [z,y] = 1i ∼= p1+2. Let F ,F be a saturated fusion system over P and
+ 1 2
A respectively. Here, A is a finite abelian p-group. Let F = F ×F . Then F is
1 2
not a fusion system of a p-block of a finite group if and only if F is not a fusion
1
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 (kG,b,G) where G is a finite group, b is a block of
(S,eS)
G, and (S,e ) is a maximal (kG,b,G)-Brauer pair.
S
SinceF =F ×F ,wehaveAisstronglyclosedinF. Let(A,e )bea(kG,b,G)-
1 2 A
Brauer pair such that (A,e ) ≤ (S,e ). By [4, Part IV, Proposition 3.19(b)], we
A S
have
C (A)=NK(A)=F (kC (A),e ,C (A))
F F (S,eS) G A G
by [4, Part IV, Proposition 3.8]. Here, K =Aut (A)={Id }.
CG(A) A
By Definition 2.7, we have C (A)/A ∼=F/A ∼= F . Since A is abelian, we have
F 1
A≤C (A). As before, we denote by µ the natural k-algebra homomorphism
G A
µ :kC (A)−→k(C (A)/A)
A G G
Xλxx7−→Xλxx,
x x
wherex=xA∈C (A)/A. Ase isablockofkC (A)and|C (A):C (A)|=
G A G G CG(A)
1, then e :=µ(e ) is a block of k(C (A)/A) by [10, Ch. 5, Theorem 8.11].
A A G
In the following, we will check that
C (A)/A⊆F (kC (A)/A,e ,C (A)/A).
F (S/A,εS) G A G
Here, ε will be defined in the following text.
S
First,wewillprovethatthereisanC (A)-posetequivalencebetweenthesubset
G
of(kC (A),e ,C (A))-Brauerpairsconsistingofthosepairswhocontains(A,e ),
G A G A
and the subset of (kC (A)/A,e ,C (A)/A)-Brauer pairs by the map Φ
G A G
(T,e )7−→Φ(T,e ).
T T
Here,(T,e )isa(kC (A),e ,C (A))-Brauerpairsuchthat(T,e )≤(S,e ). We
T G A G T S
will define the Φ(T,e ) in the following.
T
Since A✂T ≤S, set T =T/A. It is easy to see that C (T)/A≤C (T),
G CG(A)/A
we can set that C (T)=L /A. Here, L ≤C (A) and C (T)✂L . Since
CG(A)/A T T G G T
L acts on T/A and A trivially, we have L /C (T) is a p-group by [4, Appendix
T T G
A, Lemma A.2]. Since (T,e ) is a (kC (A),e ,C (A))-Brauer pair, thus (T,e )
T G A G T
is also a (kG,b,G)-Brauer pair by [4, Part IV, Lemma 3.5]. That is e is a block
T
of kC (T) = kC (T). Then there exists a unique block ε of kL such that
CG(A) G T T
ε covers e by [10, Ch.5, Corollary 5.6].
T T
Set ε :=µ (ε ), since L /C (T) is a p-group, thus |L :C (T)| is power of
T A T T G T LT
prime. As C (T) ≤ C (A), we can see that |L : C (A)| is power of prime.
LT LT T LT
Henceε :=µ (ε )isablockofk(L /A)=kC (T)by[10,Ch. 5,Theorem
T A T T CG(A)/A
8.11].
Since (T,e ) is a (kC (A),e ,C (A))-Brauer pair, thus BrkCG(A)(e )e 6=
T G A G T A T
0. As e is covered by a unique ε , it implies that BrkCG(A)(e )· ε 6= 0.
T T T A T
Since A✂C (A), we have Kerµ ⊆ J(kC (A)) by [10, Ch.5, Lemma 2.7]. So
G A G
µ (BrkCG(A)(e )ε )6=0. Also e :=µ (e ) is a block of kC (A) and
C T A T A A A G
(kC (A))T BrkTCG(A) //kC (T)
G G
µA µA
(cid:15)(cid:15) (cid:15)(cid:15)
(kC (A)/A)T //kC (T)
G Br CG(A)/A
T
6
commutes, thus
µ (BrkCG(A)(e ))=BrkCG(A)/A(e ).
A T A T A
Hence BrkCG(A)/A(e )ε 6= 0. Therefore, (T,ε ) is a (kC (A)/A,e ,C (A)/A)-
A T T G A G
T
Brauer pair. So we can define Φ(T,e )=(T,ε ) and Φ is injective.
T T
We show that the map Φ is inclusion preserving. By [4, Part IV, Proposition
2.14],weonlyneedtoproveitwhen(A,e )≤(R,e )✂(T,e ). SowehaveR✂T,
A R T
BrkCG(A)(e )e = e and e is T-stable. That is e e = e . Hence ε ε 6= 0
T,R R T T R R T T R T
because ε ,ε cover e ,e respectively. Now, we can see that 0 6= µ (ε ε ) =
R T R T A R T
ε ε . Also let t ∈ T, εt is a block of kL because T acts on L . (Here, the
R T R R R
definition of L like L above.) But there exists a unique block ε of kL such
R T R R
thatε coverse ,thusεt =ε becauseεt coverset =e . (Here,e isT-stable.)
R R R R R R R R
Henceε isT-stable. Therefore,(R,ε )✂(T,ε ),thatmeansΦ(R,e )✂Φ(T,e ).
R R T R T
So Φ is a poset bijection.
Let g ∈ C (A), we have Φ(T,e )g = (T,ε )g = (Tg,εg). Since ε cover e ,
G T T T T T
thusεg coverseg. Here,µ (ε )g =µ (εg). ThatisΦ(T,e )g =Φ(Tg,eg). Hence
T T A T A T T T
Φ is compatible with C (A)-action.
G
Now, we will prove that (S,ε ) is a maximal (kC (A)/A,e ,C (A)/A)-Brauer
S G A G
pair.
Suppose thatthere exists a(kC (A)/A,e , C (A)/A)-Brauerpair (R/A,ε )(cid:13)
G A G R
(S,ε ). Hence, R(cid:13)S and BrkCG(A)/A(e )ε 6=0. So BrkCG(A)(e )ε 6=0. There
S R/A A R R A R
exists a block e of kC (R) such that ε covers e and BrkCG(A)(e )e 6= 0. So
R G R R R A R
(R,e ) is a (kC (A),e ,C (A))-Brauer pair. By [4, Part IV, Lemma 3.5], we
R G A G
have(R,e )is a(kG,b,G)-Brauerpair. Thatisacontradiction. Hence(S,ε )isa
R S
maximal(kC (A)/A,e ,C (A)/A)-Brauerpair. SowecanseethatΦissurjective.
G A G
By [4, Part IV, Proposition 3.2], we have F (kC (A)/A,e ,C (A)/A) is
(S/A,εS) G A G
a saturated fusion system over S/A.
Second, let A≤R≤S, we can see that
N (R,e )/A≤N (R,ε ).
CG(A) R CG(A)/A R
That is
C (A)/A⊆F (kC (A)/A,e ,C (A)/A).
F (S/A,εS) G A G
SinceC (A)/AisasaturatedfusionsystemoverS/A∼=p1+2,ifC (A)/Aisnot
F + F
a fusionsystemofa p-blockofafinite group,then C (A)/A is oneofthe following
F
saturated fusion systems:
Name Out (71+2) |Fcr| Aut (V)
F + F
RV 62 :2 6+2 SL (7):2,GL (7)
1 2 2
RV D ×3 4+4 SL (7):2,SL (7):2
2 16 2 2
RV SD ×3 8 SL (7):2
3 32 2
by [8]. And this table also can be fined from [13, Lemma 4.17].
Here, C (A)/A⊆F (kC (A)/A,e ,C (A)/A). Hence either
F (S/A,εS) G A G
C (A)/A=F (kC (A)/A,e ,C (A)/A),
F (S/A,εS) G A G
or C (A)/A∼=RV and F (kC (A)/A,e ,C (A)/A)∼=RV by [8, 13].
F 2 (S/A,εS) G A G 3
7
However, RV ,RV ,RV are not fusion systems of a p-blocks of finite groups.
1 2 3
That is a contradiction. Hence, C (A)/A∼=F is a fusion system of a p-block of a
F 1
finite group.
ConverselysupposethatF isafusionsystemofap-blockofafinitegroup. Since
1
P ∼= p1+2, we have that F is not exotic by [8, Theorem 6.4]. Let F = F (G )
+ 1 1 P 1
for a finite group G with P ∈ Syl (G ). Set G = G ×(A⋊Aut (A)), then
1 p 1 1 F
P ×A∈Syl (G). So F ∼=F (G). That is F is a fusion system of a p-block of a
p P×A
finite group. (cid:3)
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