INEQUALITIES FOR SOME BLOCKS OF FINITE GROUPS

GUNTER MALLE AND GABRIEL NAVARRO

Abstract. We analyze some inequalities for numbers of characters in p-blocks of finite groups and prove them in the case of blocks with a normal defect group, as well as in the case of certain blocks in non-abelian simple groups.

1. Introduction

Suppose that B is a p-block of a finite group G with defect group D. Let k(B) = |Irr(B)|, k_{0}(B) be the number of height zero characters in B, l(B) = |IBr(B)| and k(D) the number of irreducible characters of D. This note started with the following observation:

Theorem A. Suppose that B is a p-block of a finite group G with normal defect group D. Then

(1)  k(B)/k_{0}(B) ≤ k(D′) and k(B)/l(B) ≤ k(D) .

Although blocks with normal defect group are of a very special type, it does not seem unreasonable to ask if similar formulas could hold true for general blocks. We do notice that the weaker inequality

k(B)/k_{0}(B) ≤ |D′|

already implies Brauer’s k(B) conjecture (if we assume the Alperin-McKay Conjecture). Or that if D is abelian, both inequalities in Theorem A hold (if we assume Brauer’s height zero conjecture and the Alperin weight Conjecture).

It seems therefore interesting to test these inequalities for blocks which are far from having a normal defect group, and for primes p such that D is far from being abelian.

Theorem B. Suppose that B is a p-block of a symmetric group or of an exceptional group of Lie type in its defining characteristic. Then the inequalities (1) hold for B.

For p-solvable groups, however, it seems possible that these inequalities might be too optimistic.

Date: December 1, 2005.

2. Blocks with a normal defect group

If N ◁ G and τ ∈ Irr(N), we write Irr(G|τ) for the set of complex irreducible characters χ of G which have τ as an irreducible constituent of χ_{N}. If τ is Ginvariant, it is a result of P.X. Gallagher that |Irr(G|τ)| is the number of the socalled τ-good classes of G/N. Recall that xN is τ-good if τ extends to N⟨x, c⟩ for every c ∈ G with [c, x] ∈ N. (See [3].)

We need two lemmas.

Lemma 2.1. Suppose that N, Z ◁ G where (|N|, |Z|) = 1. Let τ ∈ Irr(N) and θ ∈ Irr(Z) be G-invariant. Then

|Irr(G | τ × θ)| ≤ |Irr(G/N | 1_{N} × θ)| .

Proof. Write M = N × Z and η = τ × θ = αβ, where α = τ × 1_{Z} and β = 1_{N} × θ. Suppose that M ⊆ H ⊆ G with H/M abelian and that η extends to γ ∈ Irr(H). We claim that θ extends to H. Let π be the set of primes dividing |N| and let V/M be a π-complement of H/M. Now, since V/N is a π′-group, it follows that τ has a unique extension τ̂ ∈ Irr(V) such that the determinantal order of τ̂ is a π-number. (See [5, Corollary (8.16)].) In particular, τ̂ is H-invariant. Since Z is a π′-group, it easily follows that Z ⊆ ker τ̂. Therefore, τ̂_{M} = α. By [5, Theorem (6.16)], we have that the irreducible constituents of η^{V} have the form τ̂δ for a uniquely determined δ ∈ Irr(V |β). In particular,

γ_{V} = τ̂δ

for some H-invariant δ ∈ Irr(V |β). Also, δ_{M} = β by evaluating degrees. Now, δ ∈ Irr(V/N), (|H/V |, |V/N|) = 1 and again by [5, Corollary (8.16)], we deduce that δ extends to H. This proves the claim. The claim easily implies that whenever xM is η-good, then xM is 1_{N} × θ-good, proving the lemma. □

Lemma 2.2. Suppose that N ◁ G with N ⊆ H ⊆ G and let ψ ∈ Irr(N) be Ginvariant. Then

|Irr(H | ψ)| ≤ |G : H||Irr(G | ψ)| .

Proof. We have that

Irr(H|ψ) = ⋃_{χ∈Irr(G|ψ)} Irr(χ_{H}) ,

where here Irr(ρ) denotes the set of irreducible constituents of a character ρ. If χ ∈ Irr(G), then |Irr(χ_{H})| ≤ |G : H| by [5, Lemma (2.29)], and the lemma follows. □

Now, we are ready to prove Theorem A, which we restate here for convenience.

Theorem 2.3. Suppose that B is a block of G with defect group D ◁ G. Then

k(B)/k_{0}(B) ≤ k(D′) and k(B)/l(B) ≤ k(D) .

Proof. By Theorem 6 of [11], we may assume that D ∈ Syl_{p}(G) is normal in G and that Z = O_{p′}(G) is central in G. By Fong’s Theorem [8, Theorem (10.20)], we have that there exists θ ∈ Irr(Z) such that Irr(B) = Irr(G|θ) and IBr(B) = IBr(G|θ).

Suppose now that N is a normal p-subgroup of G. Let θ̂ ∈ Irr(NZ/N) be such that (θ̂)_{Z} = θ. If τ ∈ Irr(N), let T_{τ} be the stabilizer of τ in G. Then notice that T_{τ} is also the stabilizer of τ × θ in G. Now, by using Clifford’s Theorem, Clifford’s Correspondence ([5, Theorems (6.2) and (6.11)]), and Lemma 2.1 we have that

|Irr(G | θ)| = ∑_{τ∈Irr(N)} \frac{|Irr(G | τ × θ)|}{|G : T_{τ}|} = ∑_{τ∈Irr(N)} \frac{|Irr(T_{τ} | τ × θ)|}{|G : T_{τ}|} ≤

≤ ∑_{τ∈Irr(N)} \frac{|Irr(T_{τ}/N | θ̂)|}{|G : T_{τ}|} = ∑_{τ∈Irr(N)} \frac{|Irr(T_{τ}/N | θ̂)|}{|G/N : T_{τ}/N|} ≤ k(N)|Irr(G/N | θ̂)| ,

where the last inequality follows from Lemma 2.2.

Now, notice that χ ∈ IrrB has height zero if and only if χ(1) is not divisible by p, which happens if and only if D′ ⊆ Ker(χ). If we write N = D′, we have that Irr_{0}(B) = Irr(G | θ̂), and the first part of the theorem follows. Now, D is contained in the kernel of every irreducible Brauer character of G, and if we put N = D, then IBr(B) = IBr(G/D | θ̂) = Irr(G/D | θ̂) and the second part is also proved. □

3. Groups of Lie-type

It now seems interesting to test these inequalities for blocks which are far from having a normal defect group, and for primes p such that D is far from being abelian. A typical example occurs for finite groups of Lie type in their defining characteristic.

We thank the referee for encouraging us to include the following asymptotic result:

Theorem 3.1. Fix a Weyl group W and consider the family {G(q)} of finite groups of Lie type obtained as fixed points under a Frobenius map of a semisimple algebraic group of adjoint type with Weyl W, defined over 𝔽_{q}. Then there is a bound q_{0} = q_{0}(W) such that the inequalities (1) hold for all blocks of G(q) in the defining characteristic p, for all q ≥ q_{0}.

Proof. Let G be a semisimple group of adjoint type with Weyl group W over the algebraic closure of a finite field of characteristic p, defined over 𝔽_{q}. Let G = G^{F} be the group of fixed points under the corresponding Frobenius morphism F. By a result of Humphreys [4] in this case Irr(G) consists of two p-blocks: the principal block B_{0}, and a single block of defect zero containing the Steinberg character. Thus, the inequalities have to be verified for the block B_{0} only.

By Lusztig’s Jordan-decomposition of characters (see [1, 12.9], for example) the irreducible characters of G of p-height zero are parametrized by semisimple conjugacy classes in the dual group G^{∗} = G^{∗}^{F^{∗}}, where G^{∗} is a group in duality with G and F^{∗} the corresponding Frobenius endomorphism. More precisely, to each such semisimple class c belongs at least one so-called semisimple character of G (and precisely one such character if the centralizers of elements in c are connected). Since G is of adjoint type, the dual group G^{∗} is simply connected. Thus by a result of Steinberg (see [1, Th. 3.7.6] and note that under our assumption that G is semisimple, the center of G^{∗} is finite), the number of semisimple conjugacy classes of G^{∗} is given by q^{r}, where r is the semisimple rank of G. So, writing k(G) := |Irr(G)|, we have k(B_{0})/k_{0}(B_{0}) ≤ (k(G) − 1)/q^{r}. Furthermore, note that l(B_{0}) = l(G) − 1 = s(G) − 1 if we write s(G) for the number of semisimple classes of G. Thus our inequalities are proved if we can show that

(k(G) − 1)/q^{r} ≤ k(D′) and (k(G) − 1)/(s(G) − 1) ≤ k(D).

But both k(G) (the number of conjugacy classes of G) and s(G) (the number of semisimple conjugacy classes of G) are asymptotic to q^{r} as q goes to infinity, so for sufficiently large q, the left hand sides are less than 2. Thus, both inequalities hold unless D′ = 1. In the latter case, which only occurs for direct products of groups of type A_{1}, the height zero conjecture is known to be correct. □

In the case of exceptional groups, we can say more:

Proposition 3.2. Let G be an exceptional group of Lie type, of adjoint type. Then the inequalities (1) hold for all blocks of G in the defining characteristic p.

Proof. The numbers of irreducible characters and their degrees are well-known for the Suzuki- and Ree-groups, and the assertion is verified readily. We may hence assume that G is obtained as in the previous theorem as the group of fixed points G^{F} of a simple exceptional adjoint group of Lie type G under a Frobenius endomorphism F : G → G defining an 𝔽_{q}-rational structure on G.

As remarked above, we only have to check the principal block B_{0}, which contains all but one of the irreducible characters of G. We saw in the previous proof that it suffices to check that

(k(G) − 1)/q^{r} ≤ k(D′) and (k(G) − 1)/(s(G) − 1) ≤ k(D).

Now the numbers k(G) and s(G) are explicitly known for all exceptional types, they are available on the web-site [7].

For example, for G = E_{8}(q) we have

\begin{matrix}k(G) = q^{8} + q^{7} + 2q^{6} + 3q^{5} + 9q^{4} + 14q^{3} + 32q^{2} + 47q + 70, & p = 2, \\ k(G) = q^{8} + q^{7} + 2q^{6} + 3q^{5} + 10q^{4} + 16q^{3} + 39q^{2} + 65q + 102, & p = 3, \\ k(G) = q^{8} + q^{7} + 2q^{6} + 3q^{5} + 10q^{4} + 16q^{3} + 40q^{2} + 67q + 111, & p = 5, \\ k(G) = q^{8} + q^{7} + 2q^{6} + 3q^{5} + 10q^{4} + 16q^{3} + 40q^{2} + 67q + 112, & p ≥ 7, \end{matrix}

and s(G) = q^{8} (see also the article [2] for this case). It follows that (k(G) − 1)/q^{r} < 5 for q = 2 and (k(G) − 1)/q^{r} < 2 for q ≥ 3, and the same bounds result for (k(G)−1)/(s(G)−1). Since the defect group D of B_{0}, that is, the Sylow p-subgroup of G, is non-abelian of order q^{120}, and |D′/D′′| ≥ q^{7}, this proves the claim for E_{8}(q).

Similarly, all other types can be handled. We list all cases where one of the quotients in question is at least 2:

(^{2}G_{2}(3), 2), (G_{2}(2), 5), (G_{2}(3), 2.75), (G_{2}(4), 2.07), (^{2}F_{4}(2), 3.2),

(^{3}D_{4}(2), 2.26), (F_{4}(2), 6.3), (F_{4}(3), 3.4), (F_{4}(4), 2.1), (E_{6}(2), 2.8),
(^{2}E_{6}(2), 4.3), (E_{7}(2), 4.2), (E_{7}(3), 2.09), (E_{7}(2), 4.53).

Only the case of G_{2}(2) needs serious consideration. But here we actually have k_{0}(B_{0}) = 8, so our quotients are at most equal to 2. For the other cases it is immediate that the inequalities are always satisfied. □

4. Symmetric groups

Let’s turn to blocks of symmetric groups 𝔖_{n}. We denote by M(a, n) the set of a-partitions of n, that is, the set of a-tuples (π_{1}, …, π_{a}) of partitions with sum ∑ |π_{i}| = n. For a > 1 we have a natural map

ϕ(a, n) : M(a, n) ⟶ M(a − 1, n), (π_{1}, …, π_{a}) ↦ (π_{1}, …, π_{a−2}, π_{a−1}π_{a}),

where π_{a−1}π_{a} is the partition whose multi-set of parts is the union of the multi-sets of parts of π_{a−1} and of π_{a}.

Lemma 4.1. The fibers of ϕ(a, n), a ≥ 2, have size at most 1.871(e/2)^{n}. In particular, |M(a, n)| ≤ 1.871(e/2)^{n}|M(a − 1, n)|.

Proof. Two a-partitions (π_{1}, …, π_{a}) and (π′_{1}, …, π′_{a}) have the same image under ϕ(a, n) if and only if π_{i} = π′_{i} for 1 ≤ i ≤ a − 2 and π_{a−1}π_{a}, and π′_{a−1}π′_{a} coincide. Thus we have to count the number of ways a partition π of m := |π_{a−1}π_{a}| = |π_{a−1}||π_{a}| can be split into two parts. Assume that π has b_{i} parts of length i, 1 ≤ i ≤ m. Then there are ∏_{i=1}^{m}(b_{i} + 1) ways of splitting π into two parts. Thus we need to maximize ∏_{i=1}^{m}(b_{i} + 1) under the side condition that ∑_{i=1}^{m} ib_{i} = m. Making the change of variables b_{i} = (x_{i} − i)/i this becomes (∏_{i=1}^{m} x_{i})/n! under the side condition that ∑_{i=1}^{m} x_{i} = m(m + 3)/2. This is maximal if all x_{i} are equal, so that the maximum of ∏_{i=1}^{m}(b_{i} + 1) (if the b_{i} are allowed to be arbitrary real numbers) equals

\frac{1}{m!} (\frac{m + 3}{2})^{m} .

But m ≤ n, so with Stirling’s estimate n! ≥ (n/e)^{n}\sqrt{2πn} this shows that all fibers have size at most

(\frac{e}{2})^{n} (\frac{n + 3}{n})^{n} \frac{1}{\sqrt{2πn}}.

Now ((n + 3)/n)^{n} tends to e^{3}, hence is bounded, and it is then easy to check that

\frac{1}{\sqrt{2πn}} (\frac{n + 3}{n})^{n} < 1.871

for all n. The claim follows. □

We need the following facts from the representation theory of symmetric groups (see Olsson [9]). The irreducible complex characters of 𝔖_{n} are in bijection with partitions λ of n; we’ll identify a character with the corresponding partition. Two characters lie in the same p-block B if and only if their associated partitions λ, λ′ have the same p-core μ. The integer w := w(B) := (|λ| − |μ|)/p is called the weight of the p-block B. The quantities k(B), l(B) and the structure of the defect group D of B only depend of w, and we’ll write k(p, w), l(p, w) and D(p, w) for them. For a fixed prime p we let P_{1} := C_{p} and, for i ≥ 2, P_{i} := P_{i−1} ≀ C_{p}, an iterated wreath product.

Lemma 4.2. Let p be a prime and w ≥ 1. Then:

(a) k(p, w) = |M(p, w)|,

(b) l(p, w) = |M(p − 1, w)|,

(c) D(p, w) ≅ P_{1}^{a_{0}} ×⋯×P_{r+1}^{a_{r}} if w = ∑_{i=0}^{r} a_{i}p^{i} is the p-adic decomposition of w.

Proof. Parts (a) and (b) are in [6, Thms. 6.2.1 and 6.2.2], part (c) is for example in [9, p. 37]. □

Lemma 4.3. We have

k(P_{i}) ≥ \begin{cases}p^{p^{i−1}−∑_{j=0}^{i−2} p^{j}} & p odd, \\ 2 · 10^{2^{i−3}} & p = 2, i ≥ 3, \end{cases}

k(P′_{i}) ≥ \begin{cases}p^{p^{i−1}−∑_{j=1}^{i−2} p^{j}−i+1} & p odd, \\ 2^{3−i}10^{2^{i−3}} & p = 2, i ≥ 3. \end{cases}

Proof. Clearly k(P_{i}) ≥ k(P_{i−1})^{p}/p. Starting with k(P_{1}) = p this shows the assertion for p > 2 by induction. For p = 2 we start the induction with k(P_{3}) = 20.

Clearly P′_{i} is contained in the base group P_{i−1}^{p} of the wreath product P_{i} = P_{i−1}≀C_{p}, of index p^{i−1} by [9, Lemma (1.4)], so k(P′_{i}) ≥ k(P_{i−1})^{p}/p^{i−1}. Using part (a), this gives the result. □

Proposition 4.4. We have k(B)/l(B) ≤ k(D) for all p-blocks of 𝔖_{n}.

Proof. We have to show that k(p, w)/l(p, w) ≤ k(D(p, w)) for all primes p and all weights w. By Lemmas 4.1 and 4.2 we have

k(p, w)/l(p, w) = |M(p, w)|/|M(p − 1, w)| ≤ 1.871(e/2)^{w}.

Write w = ∑_{i=0}^{r} a_{i}p^{i} for the p-adic decomposition of w. Then

k(p, w)/l(p, w) = |M(p, w)|/|M(p − 1, w)| ≤ 1.871 ∏_{i=0}^{r}(e/2)^{a_{i}p^{i}}.

By Lemma 4.2(c) we have k(D(p, w)) = ∏_{i=0}^{r} k(P_{i+1})^{a_{i}}. Thus it suffices to see that

1.871(e/2)^{p^{i}} ≤ k(P_{i+1}) for i ≥ 1.

For p > 2 this holds since k(P_{i+1}) ≥ p^{p^{i}/2} by Lemma 4.3. For p = 2 the estimate in Lemma 4.3 is sufficient for i ≥ 4; for smaller i the inequality can be checked directly. □

Note that the proof works since e/2 = 1.359 … is smaller than \sqrt{2} = 1.414 . . .. In view of Lemma 4.2(a) we define k(a, n) := |M(a, n)| for n ≥ 1 and not neces-

sarily prime a.

Lemma 4.5. Let p, w ≥ 1. Then

k(p, w) ≤ k(1, pw) ≤ \frac{e^{π\sqrt{2pw/3}}}{4\sqrt{3}pw} .

Proof. By what we said above, k(p, w) counts the number of characters in p-blocks of symmetric groups of weight w. Since the principal block of 𝔖_{pw} has weight p, we see that k(p, w) ≤ |Irr(𝔖_{pw})| = |M(1, pw)|, the number of partitions of pw. Furthermore, |M(1, pw)| = k(1, pw) by definition. On the other hand, it is wellknown that the number of partitions of pw is bounded above by e^{π\sqrt{2pw/3}}/(4\sqrt{3}pw) (see [10, Sect. 121]). This completes the proof. □

The following sub-multiplicativity will enable us to reduce to the case that the weight is a power of p.

Lemma 4.6. Let w_{1}, w_{2} ≥ 1. Then (a) k(p, w_{1} + w_{2}) ≤ k(p, w_{1}) k(p, w_{2}) for p ≥ 3,

(b) k(2, w_{1} + w_{2}) ≤ 2 k(2, w_{1}) k(2, w_{2}).

Proof. Note that k(p, w) = |M(p, w)|, so we can prove the first assertion by constructing an injective map

Ψ : M(p, w_{1} + w_{2}) ⟶ M(p, w_{1}) × M(p, w_{2}).

Set w := w_{1}+w_{2}. For (π_{1}, …, π_{p}) ∈ M(p, w) let r be minimal subject to ∑_{i=1}^{r} |π_{i}| ≥ w_{1} and write π_{r} = (l_{1} ≥ … ≥ l_{t}). Assume first that r > 1. Then let s be minimal subject to ∑_{i=1}^{r−1} |π_{i}| + ∑_{j=1}^{s} l_{j} := m ≥ w_{1}. Thus l_{s} > m − w_{1} := δ ≥ 0. Let

π′_{r} := (l_{1} ≥ … ≥ l_{s−1} ≥ l_{s} − δ) and π′′_{r} := (l_{s+1} ≥ … ≥ l_{t})

and define

Ψ(π) := ((π_{1}, …, π_{r−1}, π′_{r}, −, …, −), (δ, −, …, −, π′′_{r}, π_{r+1}, …, π_{t})) .

If r = 1 then let s be minimal subject to ∑_{i=2}^{p} |π_{i}| + ∑_{j=1}^{s} l_{j} := m ≥ w_{2}, so l_{s} > m−w_{2} := δ ≥ 0. Set π′_{1} := (l_{1} ≥ … ≥ l_{s−1} ≥ l_{s}−δ) and π′′_{1} := (l_{s+1} ≥ … ≥ l_{t}). If π ≠ (π_{1}, −, …, −, π_{p}) then we define

Ψ(π) := ((π′′_{1}, −, …, −, δ), (π′_{1}, π_{2}, …, π_{p})) .

Finally, if π = (π_{1}, −, …, −, π_{p}) then we let

Ψ(π) := ((π′′_{1}, −, …, −, δ), (−, π′_{1}, −, …, −, π_{p})) .

It is straightforward to verify that Ψ is injective when p > 2.

For p = 2, elements in M(p, w_{1}) × M(p, w_{2}) have at most two pre-images under Ψ; thus (b) follows. □

It seems that even k(2, w_{1} + w_{2}) ≤ k(2, w_{1})k(2, w_{2}) unless w_{1} = w_{2} = 1, but we don’t need this stronger statement.

Proposition 4.7. We have k(B)/k_{0}(B) ≤ k(D′) for all p-blocks of 𝔖_{n}.

Proof. Let B have weight w, so that k(B) = k(p, w). We first consider the case p ≥ 3. Assume for the moment that w = p^{i} is a power of p. Then B has defect group D = P_{i+1} and we have

\begin{aligned}k(B) & = k(p, w) ≤ \frac{e^{π\sqrt{2pw/3}}}{4\sqrt{3}pw} = \frac{e^{π\sqrt{2p^{i+1}/3}}}{4\sqrt{3}p^{i+1}} =: A(p, i), \\ k(D′) & ≥ p^{p^{i}−∑_{j=1}^{i−1} p^{j}−i} =: B(p, i), \end{aligned}

by Lemma 4.5. It is readily checked that A(p, i) ≤ B(p, i) whenever p ≥ 11 and i ≥ 1, or p = 5, 7 and i ≥ 2, or p = 3 and i ≥ 4. For p = 5, 7 and i = 1 we still have k(5, 5) = 506 < 5^{4} = B(5, 1) and k(7, 7) = 12405 < 7^{6} = B(7, 1). Finally, for p = 3 explicit computation shows that k(P_{2}) = 17, k(P′_{2}) = 9, hence k(P_{3}) ≥ ⌈17^{3}/3⌉ = 1638, and so k(P′_{3}) ≥ 545, k(P′_{4}) ≥ 162771336. Since k(3, 3) = 22, k(3, 9) = 1479, and k(3, 27) = 5590110, we deduce that k(3, 3^{i}) ≤ 3k(D(3, 3^{i})′) for i = 1, 2 and k(3, 27) ≤ k(D(3, 27)′). In conclusion, we have shown that

k(p, p^{i}) ≤ \begin{cases}k(D(p, p^{i})′) & p ≥ 5 or i ≥ 3 \\ 3k(D(p, p^{i})′) & p = 3, i = 1, 2. \end{cases}

Now let w be arbitrary with p-adic decomposition w = ∑_{i=0}^{r} a_{i}p^{i}. The defect group D of B satisfies D ≅ P_{1}^{a_{0}} × ⋯ × P_{r+1}^{a_{r}} by Lemma 4.2(c) and so by Lemma 4.3 we have k(D′) = ∏_{i=0}^{r} k(P′_{i+1})^{a_{i}}. Furthermore,

k(B) = k(p, w) = k(p, ∑_{i=0}^{r} a_{i}p^{i}) ≤ ∏_{i=0}^{r} k(p, p^{i})^{a_{i}}  by Lemma 4.6(a)

≤ ∏_{i=0}^{r} k(D(p, p^{i})′)^{a_{i}} = k(D′) by the case w = p^{i}

when p ≥ 5. When p = 3, the first inequality still holds, while the second has to be replaced by

k(B) ≤ ∏_{i=0}^{r} k(D(p, p^{i})′)^{a_{i}} ∏_{i=1}^{2} 3^{a_{i}}.

Now by [9, Cor. 3.7] we have

k_{0}(B) = ∏_{i=0}^{r} k(3^{i+1}, a_{i}) ≥ k(3^{2}, a_{1})k(3^{3}, a_{2}) ≥ 3^{a_{1}+a_{2}}

(note that a_{i} ≤ p − 1 = 2 in this case), so we conclude k(B) ≤ k(D′)k_{0}(B) when p = 3.

It remains to consider p = 2. Again, let first w = 2^{i}. Here, Lemma 4.5 gives

k(D′) ≥ 2^{2−i}10^{2^{i−2}} =: B(2, i)

for i ≥ 3. Again, A(2, i) ≤ B(2, i) when i ≥ 5. For i ≤ 4 explicit computation gives the following table

\begin{matrix}i & 1 & 2 & 3 & 4 \\ k(2, 2^{i}) & 5 & 20 & 185 & 5822 \\ k(D′) & 2 & 10 & 122 & 3306 \\ k_{0}(B) & 4 & 8 & 16 & 32 \end{matrix}

so in conclusion k(B) ≤ k(D′)k_{0}(B)/2^{i−1} for all i ≥ 1. If w is arbitrary with 2-adic decomposition w = ∑_{i=0}^{r} a_{i}2^{i}, we have as before

k(B) = k(2, ∑_{i=0}^{r} a_{i}2^{i}) ≤ ∏_{a_{i}≠0} k(2, 2^{i})/2  by Lemma 4.6(b)

≤ ∏_{a_{i}≠0} k(D(2, 2^{i})′)k_{0}(B(2, 2^{i})) by the case w = 2^{i}

= k(D′)k_{0}(B).

Note that the first estimate here is very rough, but it suffices for our purpose. □

Note the contrast to the situation for groups of Lie type in their defining characteristic. In that case, the quotients k(B)/k_{0}(B) and k(B)/l(B) are very close to 1, in particular bounded above independently of k(D), k(D′). On the other hand, for the symmetric groups the quotient k(B)/k(D′) is close to 1, in particular bounded above independently of k_{0}(B). Thus the two cases investigated here show quite different behaviour with respect to the inequalities (1).

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[7] F. Lübeck, Website at http://www.math.rwth-aachen.de/ Frank.Luebeck/chev/index.html [8] G. Navarro, Characters and Blocks of Finite Groups. Cambridge University Press, 1998.

[9] J. Olsson, McKay numbers and heights of characters. Math. Scand. 38 (1976), 25–42.

[10] H. Rademacher, Topics in analytic number theory. Grundlehren, Band 169. Springer-Verlag, New York-Heidelberg, 1973.

[11] W. F. Reynolds, Blocks and normal subgroups of finite groups. Nagoya Math. J. 22 (1963), 15–32.

G.M.: Fachbereich Mathematik, Universität Kaiserslautern, Postfach 3049, D– 67653 Kaiserslautern, Germany.

E-mail address: malle@mathematik.uni-kl.de

G.N.: Departament d’Àlgebra, Universitat de València, 46100 Burjassot, SPAIN.

E-mail address: gabriel.navarro@uv.es