When l(B) = 1, what can we say about counting height zero
characters?
Theorem 0.1. Let B be a block of a solvable group and suppose l(B) = 1.
Let D be the defect group for B. Then k0 (B) ≤ |D : D0 |.
Of course this follows immediately from Olsson’s conjecture, which is
known to be true for solvable groups, though we give an independent proof.
The following result is perhaps more interesting:
Theorem 0.2. Let B be a block of a solvable group and suppose l(B) = 1.
Let D be the defect group for B, and let b ∈ Bl(NG (D)|D) be the Brauer
correspondent for B. Then k0 (B) = |D : D0 | if and only if b is nilpotent.
One might ask (and I conjecture this at the end of this set of notes) if the
assumption that l(B) = 1 is really necessary to get this result? Mark has
shown me an example that demonstrates that this result is not true without
the assumption that l(B) = 1. One might also ask if we can strengthen the
result to k0 (B) = |D : D0 | if and only if B is nilpotent. I do not know the
answer to that yet.
First we set some notation. For a non-negative integer h and a block B of
G, we let kh (B) denote the number of ordinary characters of B of height h.
Moreover, if D is a p-group, we let kh (D) denote the number of characters
of D of degree ph .
The following lemma will prove our results in the case that G has p-length
one.
Lemma 0.3. Let G be a solvable group of p-length one. Suppose B is a
block of G with defect group D and l(B) = 1. Then:
(1) k(B) ≤ k(D).
(2) k0 (B) ≤ |D : D0 |.
(3) For any non-negative integer h, we have kh (B) ≤ kh (D).
(4) k0 (B) = |D : D0 | if and only if B is nilpotent.
Proof. Write N = Op0 (G) and let β ∈ Irr(N ) be covered by the block B, so
that β is a constituent of ϕN , where ϕ is the unique Brauer character of the
block B. Let T be the stabilizer of β in G, and assume first that T is proper
in G. Then the block b of T corresponding to B via Clifford induction also
has a unique Brauer character, and T certainly has p-length one. Thus the
result holds in T . Recall that the induction map is a bijection that preserves
decomposition numbers, defect groups, block nilpotence, and heights in this
situation, so we are done in this case.
Thus we may assume that β is invariant in G, and thus D is a Sylow
p-subgroup of G and ϕ has p0 -degree. Moreover, B consists of all of the
characters of G above β. We see that ϕH = α, where H is a p-complement
in G and α ∈ Irr(H). Since ϕ is the unique Brauer character of B and
1
2
ϕH = α, we see that α is the unique character of H lying over β, and thus
β is fully ramified in H. Note then that α(1)/β(1) = |H : N |1/2 .
By replacing (G, N, β) with an isomorphic character triple, we may assume that N is a central p0 -subgroup of G (note that the hypotheses hold
after replacement, and that Op0 (G/N ) is trivial). Write M/N = Op (G/N ),
and thus we may write M = N × D, where D is a normal Sylow p-subgroup
of G. Moreover, β is linear and α(1) = |H : N |1/2 .
Suppose δ ∈ Irr(D) and write X = Gδ . Observe that |G : X| is a
p0 -number, and thus HX = G and |H : H ∩ X| = |G : X|. Note that
M = DN ⊆ X since N is central in G. By Theorem 2.1 of Navarro’s
“Actions and characters in blocks” we see that there is a bijection from
rdz(X|δ) to dz(X/D), where in this case the defect zero characters of X/D
are all of the characters of X/D, and thus dz(X/D) = Irr(X/D). Moreover,
rdz(X|δ) by definition (because |X : D| is a p0 -number) consists of the
characters ν of X above δ such that ν(1)/δ(1) is a p0 -number, so that in fact
rdz(X|δ) = Irr(X|δ). Thus Navarro’s map (which we denote by N here) is
really a bijection from Irr(X/D) to Irr(X|δ) in this setting.
Let βb ∈ Irr(M ) denote the unique p0 -special extension of β to M , so that
we may consider βb as β × 1. We claim (proof later) that µ ∈ Irr(X|δ) lies
above β×δ if and only if N (µ) ∈ Irr(X/D) lies above β. Thus |Irr(X|β×δ)| =
b
|Irr(X|β)|.
Write S = X ∩ H. Now we have that
b = |Irr(S|β)|,
|Irr(G|β × δ)| = |Irr(X|β × δ)| = |Irr(X|β)|
where the last equality follows from Lemma 4.1 of “Characters of π-separable
groups”. Recall that α is fully ramified with respect to N , so that each
character of Irr(S|β) is a constituent of αS , and thus
|Irr(G|β × δ)| = |Irr(S|β)| = |{ξ ∈ Irr(S)|[αS , ξ] 6= 0}| ≤ |H : S| = |G : X|.
Notice that χ ∈ Irr(B) has height h if and only if χM lies over β × γ
where γ ∈ Irr(D) is such that γ(1) = ph . Let Irrh (G|β) denote the set of
characters of G lying above β for which χ(1)p = ph . Now for each G-orbit
of characters of D of degree ph , choose one orbit representative γ and let [γ]
denote the orbit containing δ. Therefore
X
X
kh (B) = |Irrh (G|β)| =
|Irr(G|β × δ)| ≤
|G : Gγ | = kh (D)
[γ]
[γ]
and we have proven the third statement. This trivially implies the second
statement, and summing over all values of h yields the first statement.
We now work towards proving the fourth statement. Suppose that k0 (B) =
|D : D0 |. Note that the only inequality in our proof so far was
|{ξ ∈ Irr(S)|[αS , ξ] 6= 0}| ≤ |H : S| = |G : X|,
3
where we recall that S = Hγ Thus if k0 (B) = |D : D0 | , we have that for
every linear character λ of D,
|{ξ ∈ Irr(S)|[αS , γ] 6= 0}| = |H : S|,
where S = Hλ . By comparing degrees, we see that every character of S
lying under α must necessarily induce α. If |H : S| = |H : N |1/2 = α(1),
then every character of S lying under α must be linear. But this means β
extends to S in this case, and every character of S lying over β (all of which
lie under α, as β is fully ramified) must be linear. Therefore S/N is abelian.
Note that H/N is acting coprimely and faithfully on the group D, and thus
H/N acts coprimely and faithfully on the group D/Φ(D). Moreover, the
above paragraph shows that if λ is a character of D/Φ(D) with |Hλ : N | =
|H : N |1/2 , then Hλ /N is abelian. By applying Lemma 4.1 of (Cossey-IsaacsLewis) we see that this is a contradiction, unless H/N is trivial. Therefore
N = H and the group G is nilpotent. Therefore the block B is nilpotent.
Now assume the block B is nilpotent. By Lemma 2 of Navarro’s “Nilpotent Characters” we see that G has a normal p-complement, and thus
G = H × D, and thus it is clear that k0 (B) = |D : D0 |.
We need the following lemma, which is essentially Theorem 6.2 of “The
Alperin Argument Revisited”, though it is not explicitly stated this way in
that paper.
Lemma 0.4. Let B be a block of a solvable group with l(B) = 1 and let D
be the defect group of D. Let b be the block of NG (D) that corresponds to B
under the Brauer correspondence. Then l(B) = 1.
Proof. Let N = Op0 (G) and let β ∈ Irr(N ) be covered by B. If Gβ < G,
let B0 be the block of T = Gβ corresponding to B under Clifford induction.
Note D is a defect group for B0 . Let b0 be the block of NT (D) corresponding
to B0 via the Brauer correspondence. It is a basic fact about block induction
in solvable groups that b0 induces to B0 via Clifford induction.
The fact that l(B) = 1 implies that l(B0 ) = 1. By the inductive hypothesis, we see that l(b0 ) = 1. By Clifford induction, we see that l(b) = 1.
Thus we may assume that β is invariant in G and thus D is a Sylow psubgroup of G. By basic group theory, we see that CN (D) = Op0 (NG (D))
and that there is a p-complement H of G such that K = H ∩ NG (D) is a
p-complement of NG (D). Let ϕ be the unique Brauer character in B. Then
ϕH = α ∈ Irr(H) and α is fully ramified with respect to β. Thus, in the
terminology of “The Alperin Argument Revisited”’ we see that (G, ϕ, β) is
a uniqueness triple. If we let β ∗ be the Glauberman correspondent of β in
CN (D), then β ∗ is invariant in NG (D). Moreover, by Theorem 6.2 of that
paper, we have that β ∗ is fully ramified with respect to the p-complement
K of NG (D). Thus the unique block b of NG (D) lying over β ∗ also has a
unique Brauer character, and we are done.
We can now prove our main result.
4
Theorem 0.5. Let G be a block of a solvable group G with defect group D
and suppose l(B) = 1. Then k0 (B) ≤ |D : D0 | and equality occurs if and
only if b is nilpotent, where b is the block of NG (D) corresponding to to B
under the Brauer correspondence.
Proof. Note that by the previous lemma, l(b) = 1. The hypothesis that
l(B) = 1, the hypothesis on the number of height zero characters, and
the hypothesis that the block is nilpotent are each preserved by Clifford
induction of blocks, so we might as well assume that the block B covers a
G-invariant character of N = Op0 (G) and that the defect group is a Sylow psubgroup of G (see for instance Lemma 1 of “Nilpotent characters”). Recall
(see Theorem 15.12 of Manz and Wolf for instance) under these conditions,
k0 (B) = k0 (b). Thus k0 (B) ≤ |D : D0 | by the first statement of our main
lemma. If we have k0 (B) = |D : D0 |, then k0 (b) = |D : D0 | and thus by our
first main lemma (applied to NG (D)) we see that b is nilpotent. Similarly,
if b is nilpotent, then we see that k0 (B) = k0 (b) = |D : D0 |.
It is quite possible that there are stronger statements that can be proven
here. For instance, it would be nice if the following were true:
Conjecture 1. Suppose B is a block of a solvable group G such that l(B) =
1. Then k0 (B) = |D : D0 | if and only if B is nilpotent.
I don’t know how to prove that, or if it is true. One would need to show
probably that b is nilpotent for every subpair of B, and then show that this
is enough to get B is nilpotent. Or possibly some other approach. I suspect
it would require a deeper understanding of B-subgroups than I possess. (I
know it is not the case that if B = bG and B is nilpotent, then b is nilpotent.
In the example in the “Nilpotent characters” paper the nonnilpotent block
b that induces B is not a “subpair” block, so this is not a counterexample
to our conjecture.)
Also, it would be nice to be able to remove the hypothesis that l(B) = 1.
However, the following conjecture is not true:
Conjecture 2. Let B be a block of a solvable group G with defect group
D. Then k0 (B) = |D : D0 | if and only if b is nilpotent, where b is block of
NG (D) corresponding to B under the Brauer correspondence.
As a counterexample, consider the faithful action of the semidihedral
group H of order 16 on the elementary abelian group D of order 9, and
let p = 3. Let G be the semidirect product formed by this action. Since
Op0 (G) = 1, there is only one block in this group, and its defect group is
the abelian normal Sylow 3-subgroup D. One can show that |Irr(G)| =
|Irr(D)| = 9, so k0 (B) = |D : D0 |. However, this group has more than one
3-Brauer character (in fact it has seven), so the block cannot be nilpotent.