1 Introduction
Two of the most important counting conjectures
in the representation theory of finite groups are
the Alperin–McKay and the Alperin Weight Conjectures, both
concerned with invariants of blocks.
The Alperin–McKay conjecture states that
the numbers of ordinary irreducible characters
with height zero of b and b′ coincide
for any p-block b of a finite group G, where b′
denotes the Brauer correspondent of b.
In this situation the Blockwise Alperin Weight conjecture
claims that the number of irreducible p-Brauer characters
of b coincides with the number of weights of b.
Recall that a weight of b is a G-conjugacy class of pairs
(Q,ϕ) with a p-subgroup Q≤G and an irreducible character ϕ
of NG(Q)/Q such that ϕ lies in a block of
NG(Q)/Q with defect zero and the lift of ϕ to
NG(Q) belongs to a block c with cG=b, where cG is
defined using Brauer induction of blocks.
While the conjectures are known only for specific classes of
blocks of finite groups, there is now a new approach that might
lead to general proofs of the conjectures.
In [38, 37]
it is shown that these conjectures hold if every finite non-abelian simple group satisfies the inductive Blockwise Alperin Weight (BAW) condition
and the inductive Alperin–McKay (AM) condition.
In the case of the Alperin Weight conjecture
the proof also leads to an approach for blocks with a given defect group.
Recall that for any finite group G and a block
B of G with defect group D
the Blockwise Alperin Weight conjecture holds,
if the inductive Blockwise Alperin Weight (BAW) condition holds
with respect to ℛD for all simple groups involved in G,
where ℛD is the set of all finite groups involved in D,
see [38, Theorem B].
(A group D1 is involved in D if there exist groups
H1⊲H2≤D such that H2/H1 is isomorphic to D1.)
Analogously one can establish blockwise versions of
the inductive Alperin–McKay (AM) condition, see
Definitions 3.2 and 7.2 below.
We verify those conditions for blocks with cyclic defect groups,
and also verify the inductive BAW condition for nilpotent blocks.
In a few cases the inductive conditions have been checked
for specific finite simple groups and with respect to certain p-groups,
see [36], [37], [38], and [11].
The inductive BAW condition for a prime p could be verified for most
sporadic finite simple groups,
the alternating groups and finite simple groups of Lie type
defined over a field of characteristic p, see
[6], [27], [36] and [38].
In the present paper we prove that the inductive Blockwise Alperin
Weight (BAW) condition and the inductive Alperin–McKay (AM) condition hold
for all p-blocks of finite quasisimple groups with cyclic defect groups
under the assumption that p is odd.
While in most of the anterior proofs
the knowledge on the representation theory of the specific
quasisimple group was key,
we apply here (well-)known results on blocks with cyclic defect
groups for the proof without relying on the classification
of finite simple groups.
Theorem 1.1Let p be an odd prime, q a prime, S a finite non-abelian simple group, G its universal
covering group and G¯ its universal p′-covering group.
Then the inductive Alperin–McKay (AM) condition
from
Definition 7.2
holds for all p-blocks of G with cyclic defect groups and the
inductive Blockwise Alperin Weight (BAW) condition
from Definition 3.2
holds for every q-block of G¯ with cyclic defect groups.
This extends the earlier results
from
[37, Corollary 8.3 (b)]
and [38, Proposition 6.2] on blocks
with cyclic defect groups,
where in addition the outer automorphism groups had to be cyclic.
Analogously Theorem 7.1 of [34] establishes there an
equivariant bijection for the non-blockwise version of
the inductive Alperin Weight condition
for quasisimple groups with cyclic Sylow p-subgroups,
and thereby proves this inductive condition in the case where the
outer automorphism group of the quasisimple group is cyclic.
The main part of the paper is devoted to developing methods to
control the Clifford theory of the considered characters.
The assumption that the prime is odd allows the application of rationality arguments.
In [24] the case p=2 is considered with different methods.
Theorem 1.1 implies the
inductive condition of Dade’s conjecture recently proposed by
the second author, see [39] for the inductive condition and the proof of the implication.
The result of [28] on Dade’s final conjecture
for blocks with cyclic defect groups is related but does not directly
provide the more involved requirements of the inductive AM condition.
The main part of the proof is devoted to proving that the
inductive Alperin–McKay condition holds, especially the condition
on the existence of
certain extensions. Applying the following criterion we show
that also the inductive Blockwise Alperin Weight condition
holds for the considered blocks.
Theorem 1.2Let S be a finite non-abelian simple group, G its
universal covering group and B a p-block of G with
abelian defect group D.
Let B′∈Bl(NG(D)) with (B′)G=B.
Assume that B and D satisfy the following:
The inductive AM condition from Definition
7.2 holds for B with
the group M:=NG(D) and a bijection
ΛBG:Irr0(B)→Irr0(B′),
where
Irr0(B) is the set of irreducible ordinary characters of
G in B with height zero,
The decomposition matrix C′ associated with
𝒮:=(ΛBG)-1({χ′∈Irr(B′):D≤ker(χ′)})
is a unitriangular matrix after suitable ordering of
the characters.
Then the inductive BAW condition holds for B¯,
where B¯ is the unique p-block of the universal p′-covering
group of S dominated by B.
Exploiting the fact that the defect groups of nilpotent
blocks of quasisimple groups are abelian according to An
and Eaton ([4, 5]),
we prove in addition that the inductive BAW condition holds
for nilpotent blocks. (Although the proof given here uses
mainly the theory of nilpotent blocks, it relies on the results
of [4, 5],
that uses the
classification of finite simple groups.)
Theorem 1.3The inductive BAW condition
holds for nilpotent blocks of finite quasisimple groups.
A main ingredient for the proof of Theorem 1.1
is the study of blocks with cyclic defect groups
due to various authors. Nevertheless the existing results do not allow
an immediate approach to
condition (iii) of the inductive AM condition
from Definition 7.2, hence
more considerations are required.
Furthermore, we use
[23] to simplify the checking of the
extensibility part of the conditions.
This provides an example of how the conditions, especially
the last technical part on the existence of certain extensions
with additional properties, can be verified.
This article is structured in the following way: We introduce
the main notation
in Section 2. In Section 3 we prove
the inductive BAW condition for nilpotent blocks and thereby
give the proof of
Theorem 1.3.
After recalling various results about blocks with cyclic
defect groups in Sections 4 and 5
we extend them
in Section 6 in order to determine the Clifford theory of
characters in blocks with cyclic defect groups
by means of local data. We use that in Section 7
to verify the inductive AM condition for those blocks.
In Section 8 we show Theorem 1.2
and thereby give a criterion saying when the inductive
AM condition implies the inductive BAW condition.
Namely, we prove Theorem 1.1 in Sections 6 and 7.
We conclude with the proof of the inductive BAW
condition for blocks with cyclic defect groups
as an application of Theorem 1.2.
2 Notation
In this section we explain most of the notation used later.
For characters and blocks we use mainly the notation of
[30] and [31].
Let p be a prime. Let (𝒦,𝒪,k) be a
p-modular system that is “big enough” with respect to
all finite groups occurring here. That is to say,
𝒪 is a complete discrete valuation ring of
rank one such that its quotient field 𝒦 is
of characteristic zero,
and its residue field k=𝒪/rad(𝒪) is of
characteristic p, and that 𝒦 and k are
splitting fields for all finite groups occurring in this paper.
All considered groups are finite.
For a finite group G
we denote by Bl(G) the set of p-blocks of G.
For N⊲G and b∈Bl(N) we denote by Bl(G∣b)
the set of p-blocks of G covering b. We write
Bl(G∣D) for the set of p-blocks of G with defect group D.
We write Irr(G) and IBr(G), respectively, for the sets of
irreducible ordinary and Brauer characters of G.
For a character ϕ∈IBr(G)∪Irr(G) we denote by bl(ϕ)
the p-block of G to which ϕ belongs. For N⊲G we say that
B¯∈Bl(G/N) is dominated
by B∈Bl(G) and write
B¯⊆B, if all irreducible characters of B¯ lift to
characters of B, see [31, p. 198] or
[30, p. 360].
We denote by dz(G) the set of
irreducible ordinary characters of G that belong to a block with defect zero.
When Q is a p-subgroup of G and B∈Bl(G),
we denote by dz(NG(Q)/Q,B) the set of defect zero
characters χ¯∈Irr(NG(Q)/Q) such that
bl(χ¯) is dominated by a p-block B′∈Bl(NG(Q))
with (B′)G=B.
By ccG(x) we denote the conjugacy class of G containing
x∈G, and for any subset X⊆G we denote by X+
the sum ∑x∈Xx in 𝒪G or kG.
We write G0 for the set of all
p-regular elements of G. The restriction of a character
χ of G to G0 is denoted by χ0.
Assume B∈Bl(G). Then we denote by
λB:Z(kG)→k the associated central function, see
[31, p. 48].
We denote by Irr(B) and IBr(B), respectively, the sets of
characters of Irr(G) and IBr(G) which belong to B.
For ϕ∈IBr(B) and χ∈Irr(B)
we write also λϕ and λχ instead of λB. If H is
a subgroup of a finite group G and if χ and ν are
characters of G and H respectively, then we denote by χH and
νG the restriction of χ to H and
the induction of ν to G, respectively.
By Irr(G∣ν) we denote the set of irreducible
constituents of νG and if H⊲G, we write Irr(H∣χ)
for the constituents of χH.
For H⊲G, ϕ∈IBr(H) and ψ∈IBr(G)
we denote by IBr(G∣ϕ) the set of irreducible
constituents of ϕG, and by IBr(H∣ψ) the set
of irreducible constituents of ψH.
Let N⊲G and b∈Bl(N). Then let
G[b]:={g∈Gb:λb(g)(cc〈N,g〉(y)+)≠0 for some y∈gN},
where for every g∈G we denote
by b(g) an arbitrary block in Bl(〈N,g〉∣b).
Note that by this definition, G[b] is well-defined,
since it does not depend on the choice of the blocks b(g), and
G[b]⊲Gb, see [29].
For a group A acting on a set X we denote by AX′ the
stabilizer of X′ in A, where X′⊆X.
For a ring R we denote by R× the set of all
units in R, so it becomes a group by multiplication.
Later we use also vertices and other methods seeing blocks as algebras.
The relevant notation will be introduced later.
3 The inductive blockwise Alperin weight condition for nilpotent blocks
In this section we introduce a condition on blocks of quasisimple groups
that partitions the inductive Blockwise Alperin Weight condition (BAW condition for short), see Definition 3.2
and prove that the inductive BAW condition holds
for all nilpotent blocks of finite quasisimple groups.
Recall that the universal p′-covering groupG of
a finite perfect
group S is the quotient Y/Z(Y)p where Y is the universal
covering group of S and Z(Y)p the Sylow p-subgroup of
Z(Y). (Recall that the universal covering group of a perfect group S
is a perfect group Y such that Y/Z(Y)≅S and |Z(Y)| is maximal.)
In [38] two versions of the inductive BAW condition
are presented, one general version and a version relative to a
set of finite p-groups, see [38, Definitions 4.1 and 5.17]. Additionally we present here a blockwise
version of the conditions and give its relation to the established
conditions.
Notation 3.1A p-subgroup Q of a finite group G is radical
if Op(NG(Q))=Q, and
we denote by Rad(G) the set of radical p-subgroups of G,
and Rad(G)/∼G denotes an arbitrary G-transversal
of Rad(G), namely it is a set of representatives
of G-orbits of Rad(G) by the G-conjugation.
Recall, for a group A acting on G we denote by
AB the stabilizer of B∈Bl(G).
Definition 3.2Definition 3.2 (Inductive BAW condition for a p-block,
see also [38, Definition 5.17])
Let p be a prime, S a finite non-abelian simple group,
G a p′-covering group of S, and B∈Bl(G) such that
Z(G)∩kerϕ is trivial for any ϕ∈IBr(B).
We say that the inductive BAW condition holds for B
if the following statements are satisfied:
There exist subsets
IBr(B∣Q)⊆IBr(B) for Q∈Rad(G) with
IBr(B∣Q)a=IBr(B∣Qa) for every a∈Aut(G)B, and
IBr(B)=⋃˙Q∈Rad(G)/∼GIBr(B∣Q)
(disjoint union).
For every Q∈Rad(G) there exists a bijection
ΩQG:IBr(B∣Q)→dz(NG(Q)/Q,B)
such that ΩQG(ϕ)a=ΩQaG(ϕa)
for every a∈Aut(G)B and ϕ∈IBr(B∣Q).
For every Q∈Rad(G) and every character
ϕ∈IBr(B∣Q),
there exist a finite group A:=A(ϕ,Q)
and characters ϕ~∈IBr(A)
and ϕ~′∈IBr(NA(Q))
with the following properties:
The group A
satisfies G⊲A, A/CA(G)≅Aut(G)ϕ,
CA(G)=Z(A) and p∤|Z(A)|.
ϕ~∈IBr(A) is an extension of ϕ.
ϕ~′∈IBr(NA(Q¯)) is an extension
of ΩQG(ϕ)0.
For every J with
G≤J≤A
the characters ϕ~ and ϕ~′ satisfy
bl(ϕ~J)=bl(ϕ~NJ(Q¯)′)J.
If G^ is another p′-covering group of S
and B^ a p-block of G^ such that
G^/ker(ϕZ(G^))=G
for some ϕ∈IBr(B^),
then we say that the inductive BAW condition holds for B^,
if the inductive BAW condition from above hold for the
block of G dominated by B^.
Note that if the defect group of B is abelian, the set
dz(NG(Q)/Q,B) is non-empty
for any Q∈Rad(G) if and only if Q is a defect group of B,
see [2, proof of Consequence 2].
Hence in condition (i) of Definition 3.2 the sets
IBr(B∣Q) are chosen to be empty unless Q is a defect group
of B. In the case that Q is a defect group of B the characters
dz(NG(Q)/Q,B) correspond to the Brauer characters of
the Brauer correspondent of NG(Q) of B.
This definition of the inductive BAW condition for a block
is a partitioning of the one established in
[38, Definition 4.1].
For a set of p-groups ℛ
we denote by Blℛ(G) the set of all p-blocks of G
having a defect group in ℛ.
Lemma 3.3Let S be a finite non-abelian simple group,
S^ its universal p′-covering group
and R a set of p-groups. Assume that for some
Aut(S^)-transversal
in BlR(S^) the inductive BAW
condition holds. Then the inductive BAW condition from
[38, Definition 5.17]
holds for S with respect to R.
Proof.
In [38, Definition 5.17] the Brauer characters
of the universal p′-covering group of S are considered, while in Definition
3.2 the Brauer characters of p′-covering groups G of S are
considered that are faithful on Z(G).
Via lifting those two sets of characters correspond to each other.
Also the blocks are in correspondence.
Note that for every Q∈ℛ,
every character in
ΩQG(IBr(B∣Q)∩IBr(G∣ν0))
lifts to a character in Irr(NG(Q)∣ν)
for every ν∈Irr(Z(G)) since B and
bl(ϕ′) both cover the same block of the p′-group
Z(G), whenever ϕ′ is a lift of a character in
dz(NG(Q)/Q,B). Apart from those requirements
both conditions coincide.
∎
When verifying the above condition for any block,
Definition 3.2 (iii) is crucial.
The group A(ϕ,Q) as required in the first half
of the condition can be
constructed for all characters ϕ∈IBr(G).
Lemma 3.4Let p be a prime, S a finite non-abelian simple group, G
a p′-covering group of S, and ϕ∈IBr(G) such that
kerϕ∩Z(G)=1.
Then there exists a finite group A which satisfies the following:
G⊲A,
A/CA(G)≅Aut(G)ϕ,
CA(G)=Z(A), and p∤|Z(A)|.
Proof.
In a first step we construct a projective k-representation
𝒫 of Aut(G)ϕ and then determine
a central extension of Aut(G)ϕ using the factor set of
𝒫. Finally, we prove that the finite
group obtained thereby has the properties claimed in the statement.
Let 𝒟 be a k-representation of G associated with
ϕ. Let 𝕋 be a full representative system of
Inn(G)-cosets in Aut(G)ϕ. For t∈𝕋
we define 𝒫(t) by the following:
Let Y:=G⋊〈t〉. Then there exists
an extension 𝒟~ of 𝒟 to Y
as (linear) representation.
We set 𝒫(t):=𝒟~(t).
Further, we choose an Z(G)-section
rep:Inn(G)→G,
i.e., a map rep:Inn(G)→G
such that for x∈Inn(G)=G/Z(G) the element rep(x) induces the automorphism x on G via conjugation and rep(1Inn(G))=1G.
We obtain a projective k-representation of Inn(G) by
𝒫(i)=𝒟(rep(i)) for every i∈Inn(G).
Let 𝒫:Aut(G)ϕ→GLϕ¯(1)(k) be given by
𝒫(it)=𝒟(rep(i))𝒫(t) for every i∈Inn(G) and t∈𝕋.
Straightforward calculations show that 𝒫 is
a projective representation. Let α be its factor set and let
C be the subgroup of k× that is generated by
α(a,a′) for a,a′∈Aut(G)ϕ. By the construction
of 𝒫 the values of α are roots of unity of finite order and hence
C is a finite group.
Like in [31, proof of Theorem (8.28)]
the factor set α defines a central extension
A of Aut(G)ϕ:
The elements of A are the pairs (a,c)
with a∈Aut(G)ϕ and c∈C, and multiplied by
(a1,c1)(a2,c2)=(a1a2,α(a1,a2)c1c2) for every ai∈Aut(G)ϕ and ci∈C.
Let ν:Z(G)→C be the morphism
such that for z∈Z(G) the matrix 𝒟(z) is
a scalar matrix with diagonal entries ν(z).
Then G is isomorphic to a normal subgroup of A,
via the isomorphism rep(g)z↦(g,ν(z)). This proves (i).
We have CA(G)=C. Accordingly we have that
A/CA(G)≅Aut(G)ϕ and CA(G)=C=Z(A).
As C is a finite subgroup of the multiplicative group k×,
p∤|C|.
We observe that 𝒫 lifts to a representation 𝒬 of A, defined by
𝒬(a,c)=c𝒫(a) for every a∈Aut(G)ϕ and c∈C.
By straightforward calculations we see that the Brauer
character of A afforded by 𝒬 is an extension of ϕ.
∎
Lemma 3.5Let N⊲G and L⊲G with N≤L and let ϕ~∈IBr(L)
such that ϕ:=ϕ~N∈IBr(N). Assume that ϕ~ is
G-invariant and that for every prime q≠p there exists
an extension ψq of ϕ to some Kq≤G with (ψq)Kq∩L=ϕ~Kq∩L, where Kq satisfies
N≤Kq and Kq/N∈Sylq(G/N). Then ϕ~ extends to G.
This is an extension of [31, Theorem (8.11)], that proves
the above for L=N.
Proof.
By [31, Theorem (8.11)] the character ϕ~ extends to
LKp, where Kp≤G satisfies N≤Kp and
Kp/N∈Sylp(G/N). According to [31, Theorem (8.29)]
it suffices to prove that ϕ~ extends to LKq for
every prime q≠p. Let 𝒟 be a k-representation of
L whose associated Brauer character is ϕ~.
We construct a k-representation of LKq extending
𝒟. According to [31, Theorem (8.16)], for every prime
q≠p there exists a k-representation 𝒬q of Kq
associated with ψq such that
𝒬q(x)=𝒟(x) for every x∈Kq∩L.
Let x∈Kq. Then there exists a k-representation 𝒟~ of 〈L,x〉 that extends 𝒟. Both
(𝒬q)〈L∩Kq,x〉 and 𝒟~〈L∩Kq,x〉
are extensions of (𝒬q)L∩Kq. By [31, Theorem (8.16)] the matrices 𝒬q(x) and 𝒟~(x) satisfy
𝒟~(x)=ζx𝒬q(x)
for some ζx∈k×. Since 𝒟(x) satisfies
𝒟(l)𝒟~(x)=𝒟(lx) for every l∈L,
the matrix 𝒬q(x)=ζx-1𝒟~(x)
has the analogous property
𝒟(l)𝒬q(x)=𝒟(lx) for every l∈L.
This implies that there exists a k-representation 𝒟q of
LKq with
𝒟q(lk)=𝒟(l)𝒬q(k) for every l∈L and k∈Kq.
Accordingly ϕ~ can be extended to LKq, and hence to G.
∎
We recall some properties of nilpotent blocks of quasisimple
groups, before we verify the inductive BAW condition for those blocks.
Note that the proof of the following result relies on the
classification of finite simple groups.
Theorem 3.6Theorem 3.6 (An–Eaton [4, Theorem 1.1]
and [5, Theorem 1.1])
Let S be a finite non-abelian simple group,
G the universal p′-covering group of S,
and B∈Bl(G) a nilpotent block.
Then B has abelian defect groups.
Furthermore, from the work of Külshammer–Puig [25]
we can deduce the following about the extensibility of characters
in nilpotent blocks.
Proposition 3.7Let N⊲G and b∈Bl(N) a G-invariant nilpotent block
with defect group D.
Let b′∈Bl(NN(D)) be the Brauer correspondent of b.
Assume that p∤|G/N|.
If the unique character ϕ∈IBr(b) extends to G,
i.e., there exists a character ϕ~∈IBr(G) with ϕ~N=ϕ,
then for the unique character ϕ′∈IBr(b′)
there exists a character ϕ~′∈IBr(NG(D)) with
(ϕ~′)NN(D)=ϕ′ such that
bl((ϕ~′)NJ(D))J=bl(ϕ~J) for every J with N≤J≤G.
Proof.
Set H:=NG(D), L:=NG[b](D) and M:=NN(D).
Further, set B:=bl(ϕ~)∈Bl(G∣b),
𝔅:=bl(ϕ~G[b])∈Bl(G[b]∣b), and
b′:=bl(ϕ′)∈Bl(M).
Let B′∈Bl(H) and 𝔅′∈Bl(L)
be the Brauer correspondents of B and 𝔅, respectively.
In the following we view blocks as bimodules.
Now, since p∤|G[b]/N|, it follows from
[23, Lemma 3.4]
that (𝔅)↓≅N×Nb as
k[N×N]-bimodules
and hence 𝔅 is nilpotent.
Then, by the definition of nilpotent blocks,
𝔅′ and b′ are both nilpotent blocks
since 𝔅 and b are nilpotent.
Clearly, we have B∈Bl(G∣b)∩Bl(G∣𝔅)
and 𝔅∈Bl(G[b]∣b).
Hence, by [13, Theorem 3.5] or [29, Theorem 3.5 (i)], B is the unique block
of G covering 𝔅.
In the following we denote by 1C the block idempotent of a block C over k.
Moreover, [13, Theorem 3.5] (or [26, Corollary 4])
implies that 1B=1𝔅.
Then, by the theorem of Harris–Knörr [17] (or [31, Theorem (9.28)]),
there exists a unique block
B′∈Bl(H∣b′) with (B′)G=B.
Analogously we see 𝔅′∈Bl(L∣b′). Note that the blocks b, 𝔅, B, b′, 𝔅′
and B′ have D as defect group since p∤|G/N|,
see [31, Theorem (9.26)].
Note that according to [13, Corollary 12.6] (or [29, Theorem 3.13])
we see that L=H[b′]. According to
[13, Theorem 3.5] (or [29, Theorem 3.5]), B′
is the unique block of H covering 𝔅′ and
hence the associated idempotents satisfy 1B′=1𝔅′, just as above.
Since 𝔅′ is nilpotent, we have
IBr(𝔅′):={ψ} for some ψ∈IBr(L).
Since 𝔅↓≅N×Nb,
it follows from [26, Theorem 8] and
[18, Theorem 4.1] that
𝔅′↓≅M×Mb′.
Hence, by [18, Theorem 4.1], the map
IBr(𝔅′)→IBr(b′) given by
θ↦θM is a bijection. This yields that
ψM=ϕ′.
Since both 𝔅 and 𝔅′ are nilpotent,
it holds by [25, 1.20.3] that
there exist an 𝒪-algebra 𝒜 of finite rank
and positive integers n and n′ such that
B≅Matn(𝒜) and B′≅Matn′(𝒜) as 𝒪-algebras.
Thus, the Morita equivalence between B and B′ induces a bijection
Π:IBr(B)→IBr(B′)
such that
Π(θ)(1)=n′nθ(1) for each θ∈IBr(B).
Now, since IBr(𝔅)={ϕ~G[b]},
ϕ~(1)=ϕ(1) and D is a defect group of 𝔅,
it holds by a result of Puig [35, (1.4.1)] that
𝔅≅Matϕ(1)(𝒪D)
as 𝒪-algebras, and hence
rank𝒪𝔅=|D|ϕ(1)2.
On the other hand, since 1B=1𝔅, we have
rank𝒪B=|G:G[b]|rank𝒪𝔅.
Similarly, since ψ(1)=ϕ′(1), it follows that
rank𝒪𝔅′=|D|ϕ′(1)2 and rank𝒪B′=|H:H[b′]|rank𝒪𝔅.
Note that H[b′]=L=G[b]∩H, and
G=Gb=NH=G[b]H by the Frattini argument, hence
|G:G[b]|=|H:H[b′]|. Thus,
rank𝒪Bn2=rank𝒪𝒜=rank𝒪B′(n′)2.
Accordingly
(n′n)2=rank𝒪B′rank𝒪B=rank𝒪𝔅′⋅|H:H[b′]|rank𝒪𝔅⋅|G:G[b]|=rank𝒪𝔅′rank𝒪𝔅=|D|ϕ′(1)2|D|ϕ(1)2=(ϕ′(1)ϕ(1))2.
This yields that
ψ(1)=ϕ′(1)=n′nϕ(1).
Let ϕ~′:=Π(ϕ~)∈IBr(B′). Since ϕ~′(1)=ψ(1),
B′ covers 𝔅′ and
IBr(𝔅′)={ψ}, we know that
ϕ~L′=ψ and hence ϕ~M′=ϕ′.
Thus ϕ′ has an extension belonging to B′.
Recall that ϕ~′ was constructed such that
bl(ϕ~G[b])=bl(ψ)G[b]=bl(ϕ~L′)G[b].
Since p∤|G:N|, we can apply [23, Lemma 2.4]
and obtain that ϕ~ and ϕ~′ satisfy
bl(ϕ~〈N,x〉)=bl(ϕ~〈M,x〉′)〈N,x〉 for every x∈L0.
Note that this implies for every x∈H0 using [29, Theorem 3.5] the following equalities:
bl(ϕ~〈N,x〉)=bl(ϕ~〈N,x〉∩G[b])〈N,x〉=bl(ϕ~〈M,x〉∩L′)〈N,x〉
=(bl(ϕ~〈M,x〉∩L′)〈M,x〉)〈N,x〉=bl(ϕ~〈M,x〉′)〈N,x〉.
According to [23, Lemma 2.5 (a)] this proves
bl(ϕ~J)=bl(ϕ~NJ(D)′)J for every J with N≤J≤G.∎
We apply this to verify that the inductive BAW condition holds for nilpotent
blocks of quasisimple groups.
Proof of Theorem 1.3.
Without loss of generality we may assume that the block B of
the quasisimple group G has a non-central abelian defect group.
As mentioned after Definition 3.2
this implies that for any Q∈Rad(G) the set dz(NG(Q)/Q,B)
is non-empty if and only if Q is a defect group of B,
see [2, proof of Consequence 2].
Hence the sets from condition (i) of Definition 3.2 can be chosen such that the requirement trivially holds.
We can assume that any ϕ∈IBr(B) is faithful on Z(G).
(Otherwise we can descend to a quotient of G. Note that
by [30, Theorem 5.8.8],
B corresponds to a unique block of the quotient with an
abelian defect group. By the definition of nilpotent blocks
we see that this block is again nilpotent.)
Let B′∈Bl(NG(D)) be the Brauer correspondent of B.
Then, by the definition of nilpotent blocks,
B′ is also nilpotent.
The characters in dz(NG(D)/D,B) lift to characters in Irr(B′).
By the theory of nilpotent blocks
due to Broué and Puig [7],
see [40, Theorem (52.8)],
there is exactly one character in Irr(B′) that contains
D in its kernel. We identify dz(NG(D)/D,B) with
IBr(B′), since the characters in dz(NG(D)/D,B)
correspond to characters in IBr(B′).
There exists an Aut(G)D,B-equivariant bijection
ΩDG:IBr(B)→dz(NG(D)/D,B)
since both sets contain exactly one character.
Hence it suffices to check condition (iii) of Definition 3.2. Let ϕ∈IBr(B).
Lemma 3.4 yields that there is a finite group A
such that G⊲A,
Z(A)=CA(G), A/Z(A)≅Aut(G)ϕ, p∤|Z(A)|,
and that ϕ extends to A, namely there is
a character ϕ~∈IBr(A) with ϕ~G=ϕ.
Thus, conditions (1)–(2) of Definition 3.2 (iii) are satisfied.
Let ϕ′:=ΩDG(ϕ)0∈IBr(NG(D)).
According to [23, Theorem C (c) (2)], we have that
ϕ′ extends to NA[B](D)
and some extension ϕ′~∈IBr(NA[B](D)) satisfies
bl(ϕ′~NJ(D))J=bl(ϕ~J) for every G≤J≤A[B].
We check that we can apply Lemma 3.5.
For every prime q we denote by Aq a group with
G≤Aq≤A and Aq/G∈Sylq(A/G).
For p≠q there exists an extension ψq of ϕ′ to NAq(D) according to Proposition 3.7 such that
bl((ψq)NJ(D))J=bl(ϕ~J)J for every G≤J≤Aq.
Hence the character
ψq satisfies
(ψq)NAq[B](D)=(ϕ′~)NAq[B](D).
Hence ϕ′~ extends to NA(D) by Lemma 3.5.
Accordingly Definition 3.2 (iii) is satisfied for ϕ.
∎
4 Brauer characters in blocks with cyclic defect groups – Recall
In this section we recall some known results about
the Brauer characters of blocks with cyclic defect groups that are relevant
for our later considerations.
Based on the work of Dade [12, 14],
blocks having cyclic defect groups seem well-understood.
For Brauer characters the Green correspondence gives a natural
bijection with many additional properties.
For the Green correspondence we use the notation as introduced in
[30, Section 4.4].
Although Dade gave the following statement already in [14], we recall for the sake of completeness its proof since its details are used later.
Lemma 4.1Let B∈Bl(G) with a cyclic defect
group D. Let B′∈Bl(NG(D)) be the Brauer correspondent of B.
Then there exists an Aut(G)B,D-equivariant bijection
Π:IBr(B)→IBr(B′)
such that for ϕ∈IBr(B) and a simple kG-module
V affording ϕ, the character Π(ϕ) is the
irreducible Brauer character of NG(D) afforded by
the head of fV, where f=f(G,D,NG(D)) is
the Green correspondence with respect to (G,D,NG(D)).
Proof.
The existence of a bijection can be deduced
from [14, Lemma 4.7] together with
[31, Theorem (9.9)].
Since any simple kG-module in B has D as its vertex,
see [12] or [22, Corollary 3.7],
fV is defined. Then it is known that fV belongs
to B′, where B′∈Bl(NG(D)) is the Brauer correspondent
of B, see [30, Corollary 5.3.11]. Since any
indecomposable kNG(D)-module in B′ is uniserial according
to [1, Section 19], the head of fV is a
simple kNG(D)-module in B′. Thus Π is a well-defined
map and bijective, see [1, Chapter V].
The bijection Π is Aut(G)B,D-equivariant
since the
Green correspondence has the analogous equivariance property.
∎
The theory of Dade from [14] and the Green
correspondence provide several tools in this situation.
In order to explore the bijection in Lemma 4.1
we recall some facts about Green correspondence.
Notation 4.2For H≤G and a kH-module V we denote by VG the
induced kG-module, and for a kG-module W we denote by WH
the restriction of W to a kH-module.
We denote by vx(V) the vertex of V, as defined in
[30, p. 270].
For kG-modules W and W′ we write W′∣W when W′ is
(isomorphic to) a direct
summand of W.
Further, for subgroups H1,H2 of G we write H1⊆GH2
if H1 is G-conjugate to a subgroup of H2.
Lemma 4.3Let N⊲G with p∤|G:N|. Further, let V be an
indecomposable kN-module with vertex D and suppose that
there is a kG-module V~ such that V~N≅V as
kN-modules. For H:=NG(D) and M:=NN(D) the
following holds:
Let f~ and f be the Green correspondences
with respect to (G,D,H) and (N,D,M), respectively.
Then,
(f~V~)M is the direct sum of fV and some indecomposable
modules that do not have D as their vertices.
f~V~ is a direct summand of (fV)H.
Proof.
Clearly V~ is indecomposable as a kG-module. Secondly,
D⊆Gvx(V~)
by [30, Lemma 4.3.4 (ii)].
Since V~ is relatively N-projective by
[30, Theorem 4.2.5], we know that
V~∣(V~N)G=VG.
Thus, [30, Lemma 4.3.4 (i)] implies that
D is a vertex of V~.
Because of (a) we can apply Green correspondence and have
V~M=(V~H)M=(f~V~⊕(⊕iY~i))M=(f~V~)M⊕(⊕i((Y~i)M)),
where each Y~i has a vertex which is in the set
𝔜~:=𝔜(G,D,H),
that is defined as in [30, Section 4.4.1].
We easily get by [30, Lemma 4.3.4 (ii)]
and the definition of 𝔜 that any indecomposable
direct summand of each (Y~i)M can not have D as a vertex.
Thus it follows that
V~M=(V~H)M=(f~V~)M⊕(⊕ indec. kM-module
with vertex ≠D).
On the other hand,
V~M=(V~N)M=VM=fV⊕(⊕ indec. kM-module
with vertex ≠D).
Therefore, by the Krull–Schmidt theorem, we know the assertion of (b).
By the proof of (a) it holds that V~∣VG.
Hence, Burry’s theorem in [30, Theorem 4.4.8 (ii)]
implies that f~V~ is a direct summand of (fV)H. This proves (c).
∎
The above statement on Green correspondence is applied
in the situation of extending characters.
Lemma 4.4Let N⊲G with p∤|G:N|. Further, suppose
that V is an indecomposable kN-module such that
there is a kG-module V~
with V~N≅V as kN-modules. Let D be a vertex
of V (and hence V~ is an indecomposable kG-module
with vertex D, see Lemma 4.3
(a)).
For H:=NG(D) and M:=NN(D) it holds that
(f~V~)M≅fV,
where f~ and f are the Green correspondences with
respect to (G,D,H) and (N,D,M), respectively.
Proof.
First, recall that fV is H-invariant by
definition. Now, since M⊲H, the Mackey formula in
[30, Theorem 3.1.9] implies that
((fV)H)M=⊕h∈[M\H/M](((fV)h)Mh∩M)M
=⊕h∈[H/M](((fV)h)Mh∩M)M=⊕h∈[H/M](fV)h,
where [M\H/M] is the set of representatives of the
double cosets of a pair (M,M) in H, and [H/M] is the set of
representatives of the left cosets of M in H,
and also
for a kM-module X we denote by Xh the kM-module
obtained via conjugation with h∈H.
We see that the last term is congruent to
|H:M|-many copies of fV as kM-module.
Now, by Burry’s theorem quoted in Lemma 4.3 (c)
it follows that f~V~|(fV)H.
Hence, (f~V~)M|((fV)H)M,
that is, (f~V~)M|(fV⊕⋯⊕fV). On the other hand it follows from Lemma 4.3 (b)
that (f~V~)M is
a direct sum of fV and indecomposable kM-modules with
vertices different from D. Thus, since fV has D as a vertex,
the Krull–Schmidt theorem implies that (f~V~)M≅fV.
∎
We start with the following easy consequence of Lemma 4.1.
Lemma 4.5Let N⊲G and b∈Bl(N) with a cyclic defect
group D, and let b′∈Bl(NN(D)) be the Brauer correspondent
of b.
There exists a natural NG(D)b-equivariant bijection
Πb,D:IBr(b)→IBr(b′).
Assume p∤|G:N|, and suppose that B∈Bl(G∣b)
and ϕ~∈IBr(B) with ϕ:=ϕ~N∈IBr(b). Then
(ΠB,D(ϕ~))NN(D)=Πb,D(ϕ).
Proof.
Part (a) follows from Lemma 4.1.
Part (b) is a consequence of Lemma 4.4
and the definition of Πb,D using the Green correspondence.
∎
5 Ordinary characters in blocks with cyclic defect groups
In this section we describe the ordinary characters of blocks
with cyclic defect group. We use the Broué–Puig *-construction
and explicitly decompose some generalized characters obtained
that way. We start by recalling the notation for ordinary
characters in those blocks.
Notation 5.1Notation 5.1 (Characters in blocks with cyclic defect groups)
Let G be a finite group and B∈Bl(G) a block with
cyclic defect group D. Let e be the inertial index of B
and Λ a representative set of the
NG(D,bD)-orbits on Irr(D)∖{1D},
where bD∈Bl(CG(D)) with (bD)G=B.
We denote by
χ1,…,χe,{χλ:λ∈Λ}
the ordinary characters of B as in [15, Section 68].
(Note that this notion depends on the choice of bD.)
We write Irrnex(B) for the set {χ1,…,χe} of non-exceptional
characters of B and Irrex(B) for the set
{χλ:λ∈Λ}
of exceptional characters.
We denote by
ϕ1,…,ϕe the irreducible Brauer characters
of B, such that ϕi is a constituent of χi0.
The exceptional characters can be described by using the
*-construction from [8].
5.2 The Broué–Puig *-construction of class functions
In the given situation (D,bD) is
a maximal B-Brauer pair.
Recall that a Brauer element of G is a pair (u,f), where u is a
p-element of G and f is a block of CG(u).
Let χ be any 𝒪-valued
class function defined on G and let ν be any
𝒪-valued class function defined on D that is (G,bD)-stable,
i.e., ν(u)=ν(ux) for any x∈G and (u,f)∈(D,bD)
with (u,f)x∈(D,bD). Then,
the *-construction χ*ν is well-defined,
and actually χ*ν is a generalized character
of G in B, for more details see
[8, Definitions 2.4 and 2.5 and also
Theorem 2.6].
Note that this definition depends on the choice of
(D,bD), see [10, p. 446, paragraph above Remark 1].
Let B be a block with cyclic defect group D and fix
a maximal b-Brauer pair (D,bD).
Let e:=|NG(D,bD):CG(D)| as above and
Π a set of representatives for the NG(D,bD)-conjugacy classes
of D.
For each v∈D with v≠1, by [8, Theorem 1.8],
there is a unique block bv with (v,bv)∈(D,bD).
It is known that each bv is a nilpotent block and hence
has a unique irreducible Brauer character.
Let ϕ(v) be defined by IBr(bv)={ϕ(v)}.
Lemma 5.3Let u∈D with u≠1 and
ℬ:={β∈Bl(CG(u)):βG=B}:={β1,…,βℓ},
where |B|=ℓ.
Then, there is a bijection
ℬ→Π∩ccG(u)
with β↦v whenever
(u,β)=G(v,bv) for some v∈Π in the sense of [30, p. 369].
The inverse map is given by
v↦(bv)g, where g∈G with vg=u and
bv is defined above.
Proof.
This follows from direct computations using
properties of the
(maximal) B-Brauer pairs described in
[3, Theorem 3.10 and Proposition 4.21]. Recall that D is abelian.
∎
In the following let Λ be a representative set
of the NG(D,bD)-conjugacy classes of Irr(D)∖{1D}.
For each λ∈Λ let
ηλ:=∑g∈[NG(D,bD)/CG(D)]λg=(λD⋊E)D
for E:=NG(D,bD)/CG(D).
Then ηλ is (G,bD)-stable in the sense of
[8, Definition 2.5]. We sketch its proof here.
Now, assume that (u,bu),(v,bv)∈(D,bD) such that
(u,bu) and (v,bv) are conjugate in G.
Since D is abelian,
it follows from [3, Proposition 4.21] that
NG(D,bD) controls fusions in D, namely,
u and v are E-conjugate. Therefore,
ηλ(u)=ηλ(v) since
ηλ=(λD⋊E)D by definition.
The generalized decomposition numbers of B for u∈D
with u≠1 satisfy
dχi,ϕ(u)u=ϵi∈{±1}
and
dχλ,ϕ(u)u=ϵηλ(u)
with some ϵ∈{±1} (independent of λ),
see [12] or [15, Theorem 68.1].
The following statement is an unpublished result by
Atumi Watanabe, whose proof we reproduce for completeness.
Proposition 5.4Proposition 5.4 (Watanabe)
For any λ∈Λ,
ϵ1(χ1*ηλ)=ϵ(e-1)χ1-∑i=2eϵiχi+ϵχλ.
Proof.
We verify the above equation first for p-singular elements and later for p-regular elements.
Let g∈G be a p-singular element. According to [31, Corollary (5.9)] we can assume that there exist some u∈D with u≠1
and s∈CG(u)p′ such that g is G-conjugate to us.
For v∈Π∩ccG(u) we fix some gv∈G with v:=ugv.
Before determining ϵ1(χ1*ηλ)(us) note that
ηλ(u)(∑ϕ∈IBr(CG(u))bl(ϕ)G=Bϕ(s))=∑v∈Π∩ccG(u)ηλ(v)ϕ(v)(sgv)
follows from Lemma 5.3
since ηλ is (G,bD)-stable.
The definition of the *-construction and the one of χ1(v,bv)
(in [8, Definition 2.4]) give
ϵ1(χ1*ηλ)(us)=ϵ1∑v∈Πηλ(v)χ1(v,bv)(us)
(5.1)=ϵ1∑v∈Π∩ccG(u)ηλ(v)χ1(v,bv)(us).
Now since χ1(v,bv) is a generalized character of
G and ηλ is (G,bD)-stable, we have
ϵ1(χ1*ηλ)(us)=ϵ1∑v∈Π∩ccG(u)ηλ(v)χ1(v,bv)((us)gv)
=ϵ1∑v∈Π∩ccG(u)ηλ(v)χ1(v,bv)(vsgv)
=ϵ1ηλ(u)∑v∈Π∩ccG(u)χ1(v,bv)(vsgv).
By the definition of χ1(v,bv) together with
Brauer’s Second Main Theorem, see [31, Theorem (5.2)]
for example, we obtain
ϵ1(χ1*ηλ)(us)=ϵ1ηλ(u)∑v∈Π∩ccG(u)χ1(vsgvbv)
=ϵ1ηλ(u)∑v∈Π∩ccG(u)∑ϕ∈IBr(CG(v))dχ1,ϕvϕ(sgvbv)
=ϵ1ηλ(u)∑v∈Π∩ccG(u)∑ϕ∈IBr(bv)ϵ1ϕ(sgv).
In the last step we use the before mentioned
generalized decomposition numbers.
As mentioned above
IBr(bv)={ϕ(v)}
and hence
(5.2)ϵ1(χ1*ηλ)(us)=ηλ(u)∑v∈Π∩ccG(u)ϕ(v)(sgv).
Now we consider the value of ϵ1(e-1)χ1-∑i=2eϵiχi+ϵχλ
on the element us using again Brauer’s Second Main Theorem and
generalized decomposition numbers
(ϵ1(e-1)χ1-∑i=2eϵiχi+ϵχλ)(us)
=ϵ1(e-1)χ1(us)-∑i=2eϵiχi(us)+ϵχλ(us)
=∑ϕ∈IBr(CG(u))bl(ϕ)G=B(ϵ1(e-1)dχ1,ϕuϕ(s)-∑i=2eϵidχi,ϕuϕ(s)+ϵdχλ,ϕuϕ(s))
=∑ϕ∈IBr(CG(u))bl(ϕ)G=B((e-1)ϵ12-∑i=2eϵi2+ϵ2ηλ(u))ϕ(s)
=ηλ(u)∑ϕ∈IBr(CG(u))bl(ϕ)G=Bϕ(s)
=ηλ(u)∑v∈Π∩ccG(u)ϕ(v)(sgv),
where in the last step we use equation (5.1). Together with (5.2) we see that
(5.3)ϵ1(χ1*ηλ)(g)=(ϵ1(e-1)χ1-∑i=2eϵiχi+ϵχλ)(g)
for any p-singular element g∈G.
Next we consider the values of the two generalized characters
for t∈G0. Let u∈D be a non-trivial element. Block
orthogonality from [31, Corollary (5.11)] implies
0=∑χ∈Irr(B)χ(u)¯χ(t)=∑i=1eχi(u)¯χi(t)+∑λ∈Λχλ(u)¯χλ(t)
=∑i=1e(∑ϕ∈IBr(CG(u))bl(ϕ)G=Bdχi,ϕu¯ϕ(1))χi(t)+∑λ∈Λ(∑ϕ∈IBr(CG(u))bl(ϕ)G=Bdχλ,ϕu¯ϕ(1))χλ(t).
Using the generalized decomposition numbers we see
0=∑i=1eϵi(∑ϕϕ(1))χi(t)+∑λ∈Λϵηλ(u)¯(∑ϕϕ(1))χλ(t)
=(∑i=1eϵiχi(t)+∑λ∈Λϵηλ(u)χλ(t))∑ϕϕ(1).
Dividing by the positive integer ∑ϕ∈IBr(CG(u))bl(ϕ)G=Bϕ(1) we obtain
(5.4)∑i=1eϵiχi(t)+∑λ∈Λϵηλ(u)χλ(t)=0.
The value χλ(t) is independent of λ∈Λ, since the value of the (generalized) decomposition numbers is independent of λ. Hence for any λ0∈Λ equation (5.4) implies
(5.5)0=∑i=1eϵiχi(t)+ϵχλ0(t)∑λ∈Ληλ(u)=∑i=1eϵiχi(t)-ϵχλ0(t),
where in the last step we use
∑λ∈Ληλ(u)=∑λ∈ΛλD⋊E(u)=∑λ′∈Irr(D)-{1D}λ′(u)=-1,
for E:=NG(D,bD)/CG(D).
Recall t∈G0, hence we obtain for λ∈Λ from the definitions
(χ1*ηλ)(t)=∑v∈Πηλ(v)χ1(v,bv)(t)=ηλ(1)χ1(t)=eχ1(t),
since ηλ(1)=|Λ|λ(1)=|Λ|=e.
Using this together with equation (5.5) we see
ϵ1(χ1*ηλ)(t)=ϵ1eχ1(t)=eϵ1χ1(t)-∑i=1eϵiχi(t)+ϵχλ(t)
=(ϵ1(e-1)χ1-∑i=2eϵiχi+ϵχλ)(t).
This proves the statement.
∎
In our later considerations we use the following
well-known facts about the rationality of characters in
blocks with cyclic defect groups.
Lemma 5.5Let B be a p-block with cyclic non-trivial defect group D. Further, let χ1∈Irr(B) be defined as in Notation 5.1.
For any odd prime p the non-exceptional characters of
B are the p-rational characters in Irr(B).
Proof.
This follows from [15, Theorem 68.1 (8)].
∎
Using this statement Proposition 5.4 implies the following.
Lemma 5.6The character χλ is the unique
constituent of χ1*ηλ that is
not p-rational. Further, it has multiplicity ±1.
Proof.
This follows from Proposition 5.4 using Lemma 5.5.
∎
In order to deduce later from the inductive
AM condition for those blocks, the inductive BAW condition
we use the following property of the
decomposition matrix that is well known.
Theorem 5.7Let B∈Bl(G) be a block with cyclic defect group D. Further, let the characters in Irr(B) and IBr(B) be denoted as above.
Then we can label the non-exceptional characters
χ1,…,χe and the irreducible Brauer
characters ϕ1,…,ϕe of B such that
the associated decomposition matrix is unitriangular, i.e.
(χi)0=∑j=1idi,jϕj
for some non-negative integers di,j with di,i=1 and di,j=0 whenever j<i. In particular, for p≠2 there exists some Aut(G)B,D-equivariant bijection
Irrnex(B)→IBr(B).
Proof.
We have that the bijection Irrnex(B)→IBr(B) is Aut(G)B,D-equivariant according to the considerations in [11, proof of Theorem 7.4] since Irrnex(B) is an Aut(G)B,D-stable set.
∎
6 Beyond blocks with cyclic defect groups
In this section we apply the results recalled from the previous section to
describe the Clifford theory of characters belonging to blocks
with cyclic defect groups.
As a result of this section we see that various properties of
characters belonging to a
block with cyclic defect groups are already determined locally.
Using Notation 5.1 we establish equivariant bijections
between the characters of Brauer corresponding
blocks with cyclic defect groups. Further, we study the existence
of certain extensions.
Since we are using various rationality arguments, we highly make use of the assumption that p is odd.
Using the assumption that p is odd, we see that there is
a natural bijection between
characters of a block with cyclic defect groups and
those of its Brauer correspondent, once a maximal Brauer pair of the considered block is fixed.
Proposition 6.1Let p be an odd prime, N a finite group, and
b∈Bl(N) with cyclic defect group D.
Then there exists an Aut(N)D-equivariant bijection
Λb,D:Irr(b)→Irr(b′),
where b′∈Bl(NN(D)) is the Brauer correspondent of b.
Furthermore, for every ν∈Irr(Z(N)) the bijection satisfies
the inclusion
Λb,D(Irr(b)∩Irr(N∣ν))⊆Irr(NN(D)∣ν).
The bijection is a consequence of Dade’s work in
[12, 14], see also [15, Section 68].
For later applications we present a detailed construction of the bijection.
Proof.
As mentioned above, we have Irr(b)=Irrnex(b)∪Irrex(b).
Recall that Irrnex(b) coincides with the set of
p-rational characters in Irr(b). Hence Irrnex(b)
and Irrex(b) are Aut(N)b-stable.
Note that any p-rational character of N is trivial on
the Sylow p-subgroup of Z(N).
Let a be the integer with |D|=pa, and let Di be the subgroup
of D with |D:Di|=pi for each integer i. We assume that Da-1 is non-central.
Otherwise the non-exceptional
characters are considered as characters of G/Da-1. According to [14, Lemma 4.10] there exists an
Aut(N)b-equivariant unique bijection Irrnex(b) to
Irrnex(ca-1)
constructed using Green correspondence, where ci∈Bl(NN(Di)) is the block with (ci)N=b.
The characters in Irrnex(ca-1) can be identified
with the characters in the block
c¯a-1∈Bl(NN(Da-1)/Da-1)
that is dominated by ca-1. By [32, Lemma (3.3)] the block c¯a-1 has defect
group D/Da-1. Successively applying this procedure,
we obtain an Aut(N)b-equivariant bijection
Λb,D,nex:Irrnex(b)→Irrnex(b′).
Let Z(N)p′ be the Hall p′-subgroup of Z(N).
Since b and b′ cover the same p-block of
Z(N)p′, the bijection satisfies
Λb,D,nex(Irr(b)∩Irr(N∣ν))⊆Irr(NN(D)∣ν) for every ν∈Irr(Z(N)).
The exceptional characters of b′ are denoted
by χλ′ for 1≠λ∈Irr(D) using
the block bD∈Bl(CG(D)). Furthermore,
we define Λb,D:Irr(b)→Irr(b′)
to be the map that coincides with Λb,D,nex on
Irrnex(b) and satisfies Λb,D(χλ)=χλ′ for every λ∈Λ.
Let Z(N)p be the Sylow p-subgroup of Z(N).
The characters χλ, χλ′ satisfy the formulas from
[32, p. 1135]. Accordingly for
ν∈Irr(Z(N)p) with λ∈Irr(D∣ν)
the characters χλ, χλ′ are contained
in Irr(N∣ν) and Irr(NN(D)∣ν), respectively.
In order to finish the proof it remains to prove that Λb,D
is Aut(N)D,b-equivariant on Irrex(b).
Let ϕ∈IBr(CN(D)) such that bl(ϕ)=bD.
Then Aut(N)D,b is generated by automorphisms induced
by NN(D) and Aut(N)D,ϕ. After choosing bD, and hence ϕ the character χ1*ηλ is uniquely defined
where ηλ is the sum of characters that are
NN(D)ϕ-conjugate to λ. For λ≠1 the
character χλ is the unique non-p-rational
constituent of χ1*ηλ according to Lemma 5.6. Hence the stabilizer
of χλ in Aut(N)D,ϕ coincides with
Aut(N)D,ϕ,ηλ. Furthermore, for
σ∈Aut(N)D,ϕ and 1≠λ∈Irr(D) the character (χλ)σ
coincides with χλσ since χλ is the unique non-p-rational constituent of
χ*λ and hence (χλ)σ and χλσ are non-p-rational constituents of
(χ*ηλ)σ=χ*(ηλ)σ=χ*ηλσ (see [10, Remark 1]). An analogous statement holds for χλ′ and hence the map Λb,D is an
Aut(N)b,D-equivariant bijection. This proves the statement.
∎
This bijection is compatible with “going-to-quotients”.
Corollary 6.2Let p be an odd prime, N a finite group, and Z≤Z(N) a non-trivial p-group. Let
b∈Bl(N) be a p-block with cyclic defect group D.
Let b¯ be the unique block of N/Z dominated by b,
χ¯∈Irr(b¯) and χ∈Irr(b) the lift
of χ¯. Let bD be a block of CN(D) with (bD)N=b and b¯D/Z
the block of CN/Z(D/Z) with (b¯D/Z)N/Z=b¯ that covers a block
of CN(D)/Z that is dominated by bD.
Let Λb¯,D/Z and Λb,D(χ) be the bijections from Proposition 6.1 defined using the blocks bD and b¯D/Z.
Then Λb¯,D/Z(χ¯) lifts to Λb,D(χ).
Proof.
Note that once the block bD is chosen, b¯D/Z is uniquely defined since
CG/Z(D/Z)/(CG(D)/Z) is a p-group.
Moreover, by the assumption we are in the situation considered in
[12, p. 27]: we have e=1 and the characters χ¯λ¯ with
λ¯∈Irr(D/Z) lift to the characters χλ of G.
Further, the character in Irrnex(b¯) lifts to the one of
Irrnex(b), since they are the only p-rational characters in those blocks.
The characters of b¯ can be labelled using the block b¯D/Z.
By the definition of Λb,D and
Λb¯,D/Z the statement also holds for exceptional characters,
since the character χλ¯∈Irrex(b¯) lifts to χλ, where λ is the lift of λ¯.
∎
The bijection Λb,D also preserves the
“extensibility of characters”, more
precisely it maps characters in b that extend to G,
to characters in b′ with a similar property.
Proposition 6.3Let p be an odd prime, N⊲G with p∤|G:N|, and
b∈Bl(N) a p-block with a cyclic non-central
defect group D. Further, let B∈Bl(G∣b)
and χ~∈Irr(B) with
χ:=χ~N∈Irr(b). Then there is an extension
χ~′ of Λb,D(χ) to NG(D) such that
bl(χ~′)G=bl(χ~).
Proof.
Let the characters in Irr(b)∪IBr(b) be labelled
as in Theorem 5.7.
First assume that χ is non-exceptional
and hence χ=χi for some i.
According to Theorem 5.7
the character ϕi is invariant in G and is
a constituent of χ0 with multiplicity one.
For any extension χ~ of χ to G the class function χ~0
has a constituent that is an extension of ϕi to G.
Let b′∈IBr(NN(D)) be the Brauer correspondent of b.
By Lemma 4.5 (b) we know that some character in
IBr(b′) extends to NG(D).
Let B′ be a block of NG(D) to which
this extension belongs.
Note that B′ has D as a defect group according to
[31, Theorem (9.26)].
In the next step we show that all characters in
IBr(B′) have the same degree: One can see that for a character ϕ∈IBr(CG(D)) with bl(ϕ)NG(D)=B′
the group NG(D)ϕ/CG(D) is a p′-group by
[1, Chapter 15, Theorem 4] and as
subgroup of Aut(D) it is cyclic.
Accordingly ϕ extends to NG(D)ϕ and all characters in
IBr(NG(D)ϕ∣ϕ) are extensions of ϕ. This
implies that all characters in IBr(B′)=IBr(NG(D)∣ϕ)
have the same degree.
Hence every character in IBr(b′) extends to NG(D). Since all ordinary
non-exceptional characters are lifts of those characters,
Λb,D(χ) extends to NG(D), as well.
Next we consider the case where χ∈Irrex(b).
Since Λb,D is NG(D)-equivariant,
χ′:=Λb,D(χ) is exceptional and
NG(D)-invariant.
Let λ∈Irr(D) be such
that χ′=χλ′. (Note that then λ is non-trivial.)
By the definition of χλ′ this implies
NG(D)=NN(D)NG(D)λ.
Considering the structure of Aut(D), we see that λ
is stabilized only by automorphisms whose order is a power of p.
Since p∤|G/N|, this proves
NG(D)λ=NN(D)λCG(D).
Let ϕ∈IBr(CN(D)) such that bl(ϕ) is
covered by b′, and let ϕ~ be some extension of
ϕ to NN(D)ϕ. Then we denote by
Cϕ and Cϕ~ their stabilizers in
C:=CG(D). As in [29, Lemma 3.12]
we see that Cϕ acts on the set of extensions of
ϕ to NN(D)ϕ by multiplication with a
linear character of NN(D)ϕ/CN(D). Note
that NN(D)ϕ/CN(D) is a cyclic p′-group,
and hence Cϕ/Cϕ~ is cyclic as well. From
[29, Lemma 3.12 and Theorem 3.13] we know that
NG(D)[b′]=NN(D)Cϕ~,
because Cϕ~ is the stabilizer of a bilinear
form defined in [29, subsection before Lemma 3.12],
see also [13, Corollary 12.6]
for the original proof. Since χ′ is an exceptional
character, we can set χ′:=ψN for some
ψ∈Irr(CG(D)∣λ), where
1≠λ∈Irr(D) and bl(ψ)=bl(ϕ). Further, note that ψ is the unique character in
Irr(CG(D)∣λ) with ψ0=ϕ.
Since χ′ is NG(D)-invariant,
NG(D)=NG(D)χ′=NN(D)NG(D)ψ
=NN(D)(NG(D)λ∩NG(D)ϕ)
=NN(D)CG(D)ϕ.
According to
[23, Theorem C (a) (2)]
combined with [29, Theorem 4.1],
χ′ has a unique extension χ~′ to
NG(D)[b′] such that
bl(χ~′)G[b]=bl(χ~G[b]).
According to [26, Proposition 9] we
have NNG(D)[b′]=G[b], and hence there is a
unique block b~′∈Bl(NG(D)[b′]∣b′) with
(b~′)G[b]=bl(χ~G[b]),
see [31, Theorem (9.28)]
(or [17]). The block
bl(χ~G[b]) is G-invariant by definition.
Hence its Harris–Knörr correspondent b~′ is
NG(D)-invariant, as well. By the definition
of χ~′ this implies that χ~′ is NG(D)-invariant.
Since NG(D)/NG(D)[b′] is isomorphic to
Cϕ/Cϕ~ and hence cyclic, χ~′ has
an extension χ^′ to NG(D),
see [19, Corollary (11.22)].
According to [13, Lemma 3.3 and Proposition 1.9]
(see also [29, Theorem 3.5 (i)]),
this extension satisfies
bl(χ^′)G=bl(χ~).
∎
A similar statement also holds when G/N is a p-group but its proof uses different methods.
Proposition 6.4Let p be an odd prime, N⊲G such that G/N is a p-group,
and b∈Bl(N) a p-block with cyclic defect group D.
If χ∈Irr(b) extends to G,
then Λb,D(χ) extends to NG(D).
Furthermore, the extensions χ~∈Irr(G) of χ
and χ~′∈Irr(NG(D)) of
Λb,D(χ) can be chosen such that
Irr(Z(G)∣χ~)=Irr(Z(G)∣χ~′).
For the proof of the above statement a significant factor is whether
χ is exceptional or not.
Lemma 6.5If in the situation of Proposition 6.4
a non-exceptional character χ∈Irr(b) extends to G,
then Λb,D(χ) extends to NG(D).
Furthermore, the extensions χ~∈Irr(G) of χ
and χ~′∈Irr(NG(D)) of
Λb,D(χ) can be chosen to be p-rational and to satisfy
Irr(Z(G)∣χ~)=Irr(Z(G)∣χ~′).
Proof.
We prove here the stronger statement that if χ or Λb,D(χ)
is NG(D)-invariant extensions χ~ and χ~′ with the above mentioned properties exist.
Since χ is a non-exceptional
character, χ is p-rational. Analogously by the definition
of Λb,D the character χ′:=Λb,D(χ) is
non-exceptional and hence p-rational, as well. Since
χ′ is NG(D)-invariant, χ′ extends to
some p-rational character of NG(D) according to
[19, Theorem (6.30)].
Analogously we see that χ extends to
some p-rational character of G.
In this situation let χ~ and
χ~′ be p-rational extensions of χ and χ′,
respectively. Accordingly
Irr(Z(G)∣χ~) and Irr(Z(G)∣χ~′)
contain only p-rational characters. This implies
that both ker(χ~) and ker(χ~′) contain the Sylow
p-subgroup of Z(G). Further, every p′-subgroup
Z≤Z(G) is contained in N and
the blocks bl(χ) and bl(χ′)
cover the same block of Z.
This proves
Irr(Z(G)∣χ~)=Irr(Z(G)∣χ~′).∎
This result is used in the considerations on exceptional characters.
Lemma 6.6If in the situation of Proposition 6.4 an exceptional character χ∈Irr(b) has an extension χ~∈Irr(G),
then Λb,D(χ) extends to NG(D).
Furthermore, there exists an extension χ~′∈Irr(NG(D)) of
Λb,D(χ) with
Irr(Z(G)∣χ~)=Irr(Z(G)∣χ~′).
Proof.
We first assume that b is not nilpotent.
We construct some generalized character
that proves to have an extension of χ as constituent.
Let χ~1 be the p-rational extension of
the non-exceptional character χ1 that exists by Lemma 6.5. Let χ~1′ be the p-rational extension of the non-exceptional character χ1′. Let λ∈Irr(D) with χ=χλ, b~ the p-block of G covering N and D~ the defect group of b~ with D≤D~.
Since Λb,D is NG(D)-equivariant, χλ is G-invariant if and only if Λb,D(χλ) is NG(D)-invariant. We see that Λb,D(χλ) is NG(D)-invariant if and only if λ is D~-invariant.
We fix for the further considerations a maximal
b~-Brauer pair (D~,bD~) as in [9, Section 3].
Further, let E be the
p-complement in NG(D~,bD~)/CG(D).
Then G=N⋊CD~(E) according to [42, Lemma 4 (i)].
The group Z:=CD~(E)∩Z(G) is normal in G. In the following we assume that Z=1 otherwise we can replace G by G/Z.
Hence the character λ∈Irr(D) has a canonical extension λ~∈Irr(D~) with CD~(E)≤ker(λ~).
Let ηλ~ be the sum of NG(D~,bD~)-conjugates of λ~.
We see that χ~1*ηλ~ is a generalized character and satisfies
(χ~1*ηλ~)N=χ1*ηλ
according to [9, Theorem 1]. Furthermore, the character χλ has
multiplicity ±1 in χ1*ηλ by Lemma 5.6
and is G-invariant.
Let aα (α∈Irr(G)) be the integers such that
χ~1*ηλ~=∑α∈Irr(G)aαα.
Then we have
∑α∈Irr(G|χλ)aαα(1)χλ(1)=±1.
Since G/N is a p-group, the quotient α(1)/χλ(1)
is a p-power unless α is an extension of χλ. Hence
there is at least one constituent of χ~1*ηλ~
that is an extension χ~ of χλ. Since by assumption Z(G)≤N we have
Irr(Z(G)∣χ~)=Irr(Z(G)∣χ)=Irr(Z(G)∣χ′)=Irr(Z(G)∣χ~′).
If b is nilpotent, then b′ is also nilpotent. Let
χ1∈Irrex(b′).
The character χ1 is NG(D)-invariant
since b′ is NG(D)-invariant and χ1 is the unique
p-rational character. Like in the proof of Lemma 6.5,
χ1 extends to some p-rational χ~1∈Irr(G), see [19, Theorem (6.30)].
Since b′ is nilpotent, we have χ~=χ~1*ν~ for some
character ν~∈Irr(D~), where D~ is a defect group of
bl(χ~) containing D.
Without loss of generality we can assume that χ~1*ν~
is constructed with a Brauer pair as described in [9, Section 3], and hence we have
χ=χ~N=(χ~1)N*ν~D=χ1*ν with
ν=ν~D, see [9, Theorem 1 (ii)].
∎
7 The inductive Alperin–McKay condition
for blocks with cyclic defect groups
The inductive Alperin–McKay condition (or AM condition, for short) from [37, Definition 7.2] can be seen as a set of properties
that should be satisfied by all height zero characters. A relative version with
respect to p-groups has been introduced in [11, Section 7.1].
We refine this further to a condition on p-blocks.
As before the inductive AM condition holds for a
finite non-abelian simple group S if it holds for
any prime p and any p-block of S,
and the inductive
AM condition holds for S with respect to a defect
group if it holds for all p-blocks with this specific
defect group. It brings forth a successive approach to the
inductive AM condition. We start by giving a blockwise version
of the inductive AM condition.
Notation 7.1For a p-block B we denote by Irr0(B) the set of
height zero characters in Irr(B).
Note that for a finite non-abelian simple group S, its
universal covering group G and its universal p′-covering group,
the associated automorphism groups can be identified,
see [16, Corollary 5.1.4 (c)].
Further, for Z≤Z(G) there is a natural embedding
of Aut(G/Z) into Aut(S)=Aut(G).
So it makes sense to denote by Aut(S)χ the stabilizer of
χ in Aut(S) for any character χ∈Irr(G).
In the following we state a blockwise version of
the inductive AM condition from [37, Definition 7.2].
Definition 7.2Let S be a finite non-abelian simple group, G its universal
covering group and B∈Bl(G) with defect group D.
We say that the inductive AM condition holds for B,
if the following statements hold:
There exists an
Aut(G)B,D-stable group M
with NG(D)≤M⪇G.
Let B′∈Bl(M) with (B′)G=B.
There exists an Aut(G)B,D-equivariant bijection
ΛB,DG:Irr0(B)→Irr0(B′),
such that
ΛB,DG(Irr0(B)∩Irr(G∣ν))⊆Irr(M∣ν) for every ν∈Irr(Z(G)).
For every χ∈Irr0(B) there exist a finite group
A:=A(χ) and characters χ~ and χ~′
such that
for Z:=ker(χ)∩Z(G) and
G¯:=G/Z the group A satisfies
G¯⊲A,A/CA(G¯)=Aut(G)χ,CA(G¯)=Z(A).
(More precisely, A/CA(G¯)≅Aut(G)χ makes sense since A/CA(G¯) can be identified with a subgroup of Aut(G¯) which is a subgroup of Aut(S)=Aut(G).)
χ~∈Irr(A) is an extension of
the character χ¯∈Irr(G¯), that lifts to χ.
χ~′∈Irr(M¯NA(D¯)) is
an extension of χ¯′, where
for D¯:=DZ/Z and M¯:=M/Z
the character χ¯′∈Irr(M¯)
is the one that lifts to
χ′:=ΛB,DG(χ)∈Irr0(B′).
The characters satisfy
Irr(CA(G¯)∣χ~)=Irr(CA(G¯)∣χ~′)
and
bl(χ~J)=bl((χ~′)M¯NJ(D¯))J for every J with G¯≤J≤A.
If in the above situation the bijection ΛB,DG satisfies
ΛB,DG(χ)(1)p′≡±|G:NG(D)|p′χ(1)p′modp for every χ∈Irr(B),
then we say that the Isaacs–Navarro-refinement
(or IN-refinement, for short)
of the inductive AM condition holds forB,
see
[37, Definitions 7.2 and 7.6]. (This forms
an inductive condition for the Isaacs–Navarro conjecture from [21].)
As in Lemma 3.4, for every χ∈Irr0(B) we construct, in the following,
a finite group A that satisfies (1) and (2) of
Definition 7.2 (iii).
Lemma 7.3Let p be a prime, S a finite non-abelian simple group
and G the universal covering group of S.
Let χ∈Irr(G) and G¯:=G/(ker(χ)∩Z(G)).
Then there exists a finite group A with
G¯⊲A,
A/CA(G¯)=Aut(G)χ
and CA(G¯)=Z(A),
the character χ¯∈Irr(G¯) associated
to
χ
extends to A.
Proof.
The construction given in Lemma 3.4
can easily be transferred to this situation,
and we obtain A using the same method.
∎
In the remaining section we prove the first part of Theorem 1.1.
In order to be able to apply some considerations
in future work we separate the statements that can be applied in general
from those that are specific to blocks with cyclic defect groups.
The following statement is an analogue of Lemma 3.5
for ordinary characters.
Lemma 7.4Let N≤L⊲G with N⊲G and
χ~∈Irr(L) with χ:=χ~N∈Irr(N).
Assume that χ~ is G-invariant and that
for every prime q there exists an extension
ψq of χ to some Kq with
(ψq)Kq∩L=(χ~)Kq∩L,
where Kq satisfies N≤Kq≤G and Kq/N∈Sylq(G/N).
Then χ~ extends to G.
Proof.
This is Lemma 3.5 in the case where p∤|G|.
∎
We apply this statement to construct
an extension by using extensions to certain groups
related to Sylow q-subgroups for primes q≠p.
Proposition 7.5Let N⊲G, let χ~∈Irr(G) be
a character with χ:=χ~N∈Irr(N) and b:=bl(χ). Let M≤N be an
NG(D)-invariant subgroup with NN(D)≤M for some
defect group D of bl(χ).
Suppose there exists some character χ′∈Irr(M)
with bl(χ′)N=bl(χ). For every
prime q let Gq be a group such that N≤Gq≤G and
Gq/N∈Sylq(G/N). Furthermore, let H:=MNG(D) and
Hq:=Gq∩H. Assume that for L:=G[b] the character χ′ has
the following properties:
For every prime q≠p there exists
some extension κq∈Irr(Hq) of χ′ such that
bl(κq,L∩Hq)L∩Gq=bl(χ~L∩Gq).
For ν∈Irr(Z(G)∣χ~) there exists some extension
κp∈Irr(Hp∣νZ(G)∩Hp) of χ′.
Then there exists some extension χ~′∈Irr(H∣ν)
of χ′ such that
bl(χ~J∩H′)J=bl(χ~J) for every J with N≤J≤G.
Proof.
Let χ′⋅ν be the extension of χ′
contained in Irr(MZ(G)∣ν).
Assumption (i) proves that κq is contained in
Irr(Hq∣νZ(G)∩Hq). Hence κq has an extension
to HqZ(G) that is an extension of χ′⋅ν.
Assumption (ii) implies that χ′⋅ν has an extension
to HpZ(G). Hence by [19, Corollary (11.31)]
there exists some extension ψ∈Irr(H) of χ′⋅ν.
Using a construction already applied in the
proof of [33, Proposition 5.12],
we define first successively a character
ϵ:L∩H→ℂ with
bl(ψJ∩HϵJ∩H)J=bl(χ~J) for every J with N≤J≤L.
For every element x∈(L∩H)0 we define
ϵ(x) to be a linear character of
〈M,x〉 with
M≤ker(ϵ(x)) such that
(7.1)bl(ψ〈M,x〉ϵ(x))〈N,x〉=bl(χ~〈N,x〉).
Note that according to [29, Theorem 4.1 (iii)]
the character ϵ(x) exists and is unique.
In the following we denote for an arbitrary element x∈G
by xp∈G the p-element and by xp′∈G the p′-element
such that x=xpxp′=xp′xp. For x∈L∩H we define ϵ(x) with
ϵ〈M,xp′〉(x)=ϵ(xp′) and 〈M,xp〉≤ker(ϵ(x)).
The class function ϵ is defined by
ϵ(x)=ϵ(x)(x) for every x∈L∩H.
Note that by this definition the function ϵ is constant on M-cosets.
Let N≤J≤L be a group with p∤|J:N|.
According to [23, Theorem C (b)],
there exists a character δ∈Irr(J∩H) with
M≤ker(δ) such that
bl(ψJ∩Hδ)J=bl(χ~J).
According to [23, Lemma 2.5],
the character then also satisfies
bl(ψ〈M,y〉δ〈M,y〉)〈N,y〉=bl(χ~〈N,y〉) for every y∈L∩J.
Since ϵ(y) is uniquely defined by equation
(7.1), we see that
ϵ(y)=δ〈M,y〉. By the definition of ϵ this implies
ϵJ∩H=δJ∩H.
Accordingly ϵE is a character for every group
E≤L∩H with p∤|〈M,E〉:M|.
In order to apply Brauer’s characterization of characters,
see e.g. [19, Corollary (8.12)], we have
to consider ϵE for every elementary group
E≤L∩H that is a direct product of some p-group
Ep and a p′-group Ep′. By the definition,
the class function ϵ defined on L∩H
satisfies
ϵ(x)=ϵ(xp) for every x∈L∩H.
By the above ϵEp′ is a character,
and hence ϵE is a character.
The other remaining conditions in Brauer’s
characterization of characters are satisfied as well,
since ϵ(1)=1 and (ϵ,ϵ)=1 where
(⋅,⋅) denotes the inner product on characters.
Hence ϵ∈Irr(L∩H).
Accordingly ψL∩Hϵ is a well-defined
character and the character satisfies by the definition of
ϵ the equation
bl(ψJ∩HϵJ∩H)J=bl(χ~J) for every J with N≤J≤L.
By [29, Theorem 3.5 (i)] (see
[13, Lemma 3.3 and Proposition 1.9]),
any extension ψ~1∈Irr(H)
of ψ1:=ψL∩Hϵ satisfies
bl((ψ~1)J∩H)J=bl(χ~J) for every J with N≤J≤G.
Hence it is sufficient to prove that ψ1 extends to H.
By Lemma 7.4 we only have to check that for
any prime q the character (ψ1)L∩Hq extends
to Hq, where Hq satisfies M≤Hq≤H and
Hq/M∈Sylq(H/M).
For q≠p we note that κq coincides with
(ψ1)Hq∩L by the given construction
and hence assumption (i) implies that (ψ1)Hq∩L
extends to Hq. On the other hand
(ψ1)Hp∩L=ψHp∩L
is Hp-invariant and ψHp is an extension of
ψHp∩L. Accordingly there exists an extension
ψ~1 of ψ1. By definition,
ψ~1 satisfies
ψ~1∈Irr(H∣ν) and
bl((ψ~1)J∩H)J=bl(χ~J) for every J with N≤J≤G.∎
Theorem 7.6Let p be an odd prime, S a finite non-abelian simple group, and G
its universal covering group. Let B∈Bl(G) be a p-block with
cyclic non-central defect group
D.
Then the inductive AM condition holds for B.
Proof.
We verify that the conditions in Definition 7.2
are satisfied with the group M:=NG(D).
Without loss of generality we assume that the characters of Irr(B)
are faithful on the p′-Hall group of Z(G).
Otherwise we replace G by a quotient over a p′-group.
We take
ΛB,D:Irr0(B)→Irr0(B′) to
be the bijection in Proposition 6.1.
Then Definition 7.2 (i)–(ii) are satisfied,
since ΛB,DG has the required
properties according to Proposition 6.1.
It remains to check Definition 7.2 (iii).
Let χ∈Irr0(B), Z:=ker(χ)∩Z(G), G¯:=G/Z and
χ¯∈Irr(G¯) the character associated with χ, i.e., the character
of G¯ that lifts to G.
Note that by assumption Z is a p-group.
Then by Lemma 7.3 we can associate a group A to χ
such that χ¯ extends to some χ~∈Irr(A)
and A/CA(G¯)=Aut(G)χ and CA(G¯)=Z(A).
Further, let B¯∈Bl(G¯) be the unique block of G¯
dominated by B. According to [31, Theorem (9.9)] such a block exists, has DZ/Z as defect group
and is unique since Z≤Z(G), see also
[30, Theorems 5.8.8 and 5.8.11].
For B¯ there exists a further bijection
ΛB¯,D¯G¯, see Proposition 6.1.
According to Corollary 6.2, the character
χ′:=ΛB,D(χ) is a lift of
χ¯′:=ΛB¯,D¯(χ¯)∈Irr(M¯)
with M¯:=M/Z.
Let χ~∈Irr(A) be the above mentioned extension of χ¯.
For every prime q let Aq be a subgroup with
G¯≤Aq≤A and Aq/G¯∈Sylq(A/G¯).
According to Proposition 6.3,
for every prime q≠p and D¯:=DZ/Z there exists an extension
κq∈Irr(NAq(D¯)) of χ¯′ to Aq
such that
bl(κq,L∩Hq)L∩Aq=bl(χ~L∩Aq)
where L:=A[bl(χ¯)] and Hq:=NA(D¯)∩Aq. According to
Proposition 6.4 there exists an extension
κp∈Irr(NAp(D¯)) of χ¯′
such that κp∈Irr(NAp(D¯)∣νAp∩Z(A)) for any ν∈Irr(Z(A)∣χ~).
Now Proposition 7.5 can be applied and proves that the character
χ¯′ has an extension χ~′∈Irr(NA(D¯)) such that
bl(χ~J∩NA(D¯)′)J=bl(χ~J) for every J with G¯≤J≤A.
This proves that for χ, and hence for all characters
in B,
Definition 7.2 (iii) is satisfied.
Accordingly the inductive
AM condition holds for B.
∎
For completeness we mention that the IN-refinement of
the inductive AM condition
for blocks with cyclic defect groups holds.
Theorem 7.7Let p be an odd prime.
The inductive AM condition holds together with
the IN-refinement
for p-blocks of universal covering groups of non-abelian
simple groups with cyclic defect groups.
Proof.
If p is odd, the bijection constructed
in Proposition 6.1
coincides with the bijection used in [21, proof of
Theorem 2.10].
Hence for a block B∈Bl(G) with cyclic defect group D
there exists a bijection ΛB,DG satisfying
the conditions in Definition 7.2 and
ΛB,DG(χ)(1)p′≡±|G:NG(D)|p′χ(1)p′modp for every χ∈Irr(B).∎
As an immediate consequence we obtain the following result that
is a generalization of [37, Corollary 8.3 (b)].
Corollary 7.8Let p be an odd prime and S a finite simple non-abelian group whose universal
covering group has a
cyclic Sylow p-subgroup. Then the inductive AM condition
together with the IN-refinement holds for S with respect
to p, in particular S satisfies the
inductive McKay conditions in [20, Section 10].
Note that according to [16, Table 2.2] this proves the inductive AM
condition for groups of type
D43(q) for primes p with p2∤(q6-1)
and p≥5, see [11, Corollary 7.3].
8 The inductive Blockwise Alperin Weight condition
for blocks with cyclic defect groups
In this section we prove that the inductive blockwise Alperin
weight condition (BAW condition, for short) holds
for blocks with cyclic defect groups and odd primes.
We give the proof in two steps. First, we show that
under certain additional assumptions the inductive AM condition
for a block with abelian defect group
implies the inductive BAW condition for the corresponding block,
see Theorem 1.2.
Secondly, we verify the second part of Theorem 1.1
by applying this statement.
Note that 2-blocks with cyclic defect groups are nilpotent and hence
the inductive BAW condition holds for them according to Theorem 1.3.
It is clear that the last part of the inductive
AM condition and the inductive BAW condition have similarities.
Further, if the considered block has an abelian defect group,
then the involved characters for the local situation belong
to the same blocks.
In order to pass from ordinary characters to
Brauer characters we consider the decomposition matrix and
its submatrices. Theorem 1.2 assumes that there exists a
unitriangular submatrix
and states that the inductive AM condition implies the inductive BAW
condition. It is a generalization of
[27, Theorem 3.8] and [11, Theorem 7.4],
where weaker statements for groups of Lie type are given.
Proof of Theorem 1.2.
Set e:=|IBr(B)| and let {ϕ1,…,ϕe}:=IBr(B)
and 𝒮:={χ1,…,χe}⊆Irr(B) such that
the associated decomposition matrix is unitriangular, see
Theorem 5.7.
Via χi↦ϕi this gives a bijection
between 𝒮 and IBr(B). Furthermore, there exists
a natural correspondence between dz(NG(D)/D,B)
and {χ′∈Irr(B′):D≤ker(χ′)} via
lifting. Together with these bijections,
ΛBG induces a bijection ΩDG:IBr(B)→dz(NG(D)/D,B).
Since the decomposition matrix C′ is unitriangular,
ΩQG is Aut(G)B,D-equivariant
according to the considerations made in [11, proof
of Theorem 7.4].
Since the inductive AM condition holds for B and for
every character χi∈𝒮, there exist a finite group
Ai and characters χ~i and χ~i′ such that:
For Z:=ker(χi)∩Z(G) and G¯:=G/Z
the group Ai satisfies G¯⊲Ai, Ai/CAi(G¯)=Aut(G)χi and
CAi(G¯)=Z(Ai).
χ~i∈Irr(Ai) is an extension of
the character χ¯i∈Irr(G¯) determined by χi.
For D¯:=DZ/Z and M¯:=M/Z let
χ¯i′∈Irr(M¯) be the character defined by
ΛDG(χi)∈Irr0(M∣D).
Then χ~i′∈Irr(M¯NAi(D¯))
is an extension of χ¯i′.
The characters satisfy
Irr(Z(Ai)∣χ~i)=Irr(Z(Ai)∣χ~i′)
and
bl((χ~i)J)=bl((χ~i′)M¯NJ(D))J for every J with G¯≤J≤Ai.
We consider the situation for a fixed χi.
The character ϕi occurs with multiplicity one in
(χi)0 and ϕ¯i∈IBr(G¯) determined by
ϕi is a constituent with multiplicity one
in (χ¯i)0. Note that ϕ¯i is Ai-invariant.
Hence (χ~i)0 has a constituent ψ∈Irr(Ai∣ϕ¯i).
The character
ψG¯ is a constituent of χi0.
Since ϕ¯i occurs with multiplicity one,
this proves ψG¯=ϕ¯i. Hence ψ is
an extension of ϕ¯i. By definition the character ψ satisfies
bl(ψJ)=bl(χ~J) for every J with G¯≤J≤Ai.
Note that since Aut(S)χi=Aut(S)ϕi,
the group Ai associated with χi satisfies
the properties mentioned in Definition 3.2 (iii) (1),
at least after taking the quotient over the p-part of the center.
By the definition of ΩDG the character
ΛDG(χi) is a lift of ΩDG(ϕi).
Let θ′ be the character of M¯
determined by ΛDG(χi). By the inductive AM condition,
χ′ has an extension χ~′∈Irr(NA(D¯)M¯).
Let ψ′:=χ~0. This character is irreducible
since ψ′ is an extension of (χ¯i)0 and
(χ¯i)0 is irreducible.
By the definition of ψ we may conclude
bl(ψNJ(D)′)J=bl((χ~i′)NJ(D))J=bl((χ~i)J)=bl(ψJ)
for every J
with G¯≤J≤Ai.
This proves that ΩQG defines a bijection
with all properties required in Definition 3.2,
and hence B¯ satisfies the inductive BAW condition,
where B¯ is the block of the universal p′-covering of S
group dominated by B.
∎
Under the assumption that the characteristic is odd
we can apply the above criterion for blocks with cyclic defect groups and
prove thereby that those blocks satisfy the inductive BAW condition.
Proof of Theorem 1.1.
Let p be an odd prime and B a p-block of the universal covering group G of a finite
non-abelian
simple group S with cyclic defect group.
According to Theorem 7.7 the inductive AM condition holds for B.
For the verifications of the conditions in Definition 7.2
one uses the bijection ΛB,DG from Propositions
6.1. Hence the group M chosen in the
verification coincides with NG(D).
The decomposition matrix of B has been described in Theorem
5.7 and is unitriangular.
Accordingly the assumptions of Theorem 1.2
are satisfied and hence the inductive BAW condition holds for B.
Let B¯ be a block of G¯:=G/Zp with cyclic defect group where Zp
is the Sylow p-subgroup of Z(G).
Note that B¯ might be dominated by a block B of G with non-cyclic
defect group.
Although we have not proven that B satisfies the inductive AM condition,
the previous section
gives a bijection with the necessary properties. Hence an adapted version of
Theorem 1.2 can be applied in that case and
proves the statement.
∎
and
Britta Späth
