ON BRAUER’S k(B)-PROBLEM

by

GEOFFREY R. ROBINSON DEPARTMENT OF MATHEMATICS &

COMPUTER SCIENCE UNIVERSITY OF LEICESTER

LEICESTER LE1 7RH ENGLAND

and

JOHN G. THOMPSON DEPARTMENT OF MATHEMATICS

UNIVERSITY OF FLORIDA PO BOX 118000 GAINESVILLE FL 32611-8000 USA INTRODUCTION: A long-standing problem of R. Brauer is to prove that whenever B is a block with defect group D of a finite group, then k(B), the number of ordinary irreducible characters in B, is at most |D|. In this paper, we provide an affirmative solution to the problem for blocks of p-solvable groups, for all but an explicit finite number of primes. This result has implications for the structure of blocks of general finite groups, in the light of other current conjectures.

By the results of Fong [1] and Nagao [9], Brauer’s problem reduces in the p-solvable case to the “k(GV)-problem”, which is to prove that whenever p is a prime, G is a finite p′-group, and V is a faithful GF (p)G-module, then k(GV), the number of conjugacy classes of the semi-direct product GV, is at most |V |. In fact, it suffices to consider the case that G acts irreducibly on V.

The k(GV)-problem has some interest in its own right, but has so far proved to be extremely stubborn in its full generality. Partial results have been obtained by Gluck [2], Gow [3] and Knörr ([5], [6]), with Knörr providing the fundamental idea underpinning most attacks on the problem to date. The character theory of this paper is influenced to a large extent by the work of Gow and Knörr.

Our main result is: THEOREM 1: Let p be a prime with p > 5^{30}, B be a p-block with defect group D of a p-solvable finite group. Then k(B) ≤ |D|.

This relies on the solution (for primes p > 5^{30}) of the k(GV)-problem, which is provided by the following two results. THEOREM 2: Let p be a prime, G be a finite p′-group, V be a faithful irreducible GF (p)G-module. Suppose that there is a vector v ∈ V such that Res_{C_{G}(v)}^{G}(V) has a faithful self-dual submodule. Then k(GV) ≤ |V |. THEOREM 3: Let p be a prime with p > 5^{30}, G be a finite p′-group, V be a faithful irreducible GF (p)G-module. Then there is a vector v ∈ V such that Res_{C_{G}(v)}^{G}(V) has a faithful submodule which is a permutation module (so is certainly self-dual).

The proof of Theorem 2 is subsumed within the proof of a rather more technical, and somewhat stronger, Theorem 4, whose statement we defer for the moment. The proof of Theorem 3 relies ultimately on the classification of the finite simple groups, through a recent improvement by M.W. Liebeck [8] of a result of Hall, Liebeck and Seitz [4]. If the bound 5^{30} appearing in Liebeck’s theorem could be improved to some smaller integer N, then our bound in Theorems 1 and 3 could be improved to max(5^{6}, N).

1. THE CHARACTER THEORY: NOTATION AND GENERAL

REMARKS

We fix a prime p, a finite p′-group G, and a faithful irreducible GF (p)Gmodule V. We set ω = exp(2πi/|G|), and let σ denote that element of Gal (ℚ[ω]/ℚ) with ω^{σ} = ω^{p}. We let Φ denote the Brauer character of G afforded by V, and we fix an irreducible complex constituent χ of Φ. We let τ denote a generator of Stab_{<σ>}(χ). Notice that Φ = ∑_{γ∈<σ>/<τ>} χ^{γ}.

Suppose that there is an irreducible GF (p)G-module W with W ≅ W^{∗} and let W afford Brauer character ϕ. Then ϕ = ϕ̄. Since ϕ is the sum of irreducible complex characters of G which are transitively permuted by < σ >, we see that when μ is a complex irreducible constituent of ϕ, μ and μ̄ are in the same < σ >-orbit. If U is a GF (p)G-module with U ≅ U^{∗} and U has an irreducible submodule W with W ≅ W̸^{∗}, then U has a submodule isomorphic to W^{∗}. In that case, the irreducible constituents of the Brauer character of W are transitively permuted by < σ >, and if μ is one of them, then μ and μ̄ are in different < σ >-orbits.

Since G is a p′-group, we can go somewhat further:

Let μ be a complex irreducible character of G, and suppose that there is some γ ∈ Gal (ℚ[ω]/ℚ) such that μ and μ^{γ} are in different < σ >-orbits. Then ∑_{λ} μ^{λ} is the Brauer character of an irreducible GF (p)G-module, U say (where λ runs through a transversal T to Stab_{<σ>}(μ) in < σ >). Similarly, ∑_{λ∈T} μ^{γλ} is the Brauer character of another irreducible GF (p)G-module, which we label U^{γ}. Although U and U^{γ} are not isomorphic GF (p)G-modules, we still have |C_{U}(g)| = |C_{U^{γ}} (g)| for all g ∈ G, a fact which we will exploit below.

We recall the well-known fact that induction of class functions yields a bijection between generalized characters of G and generalized characters of GV which vanish off p-regular elements. Furthermore, for any generalized character, θ, of G, we have Ind_{G}^{GV} (θ)[g] = |C_{V} (g)|θ(g) for all g ∈ G. A consequence of these facts which we will frequently use is that whenever ψ is a generalized character of GV which vanishes off p-regular elements, the class function, θ, of G with θ(g) = |C_{V} (g)|^{−1}ψ(g) for all g ∈ G is a generalized character of G.

2. PROOFS OF THE CHARACTER-THEORETIC THEOREMS

The proof of Theorem 2 is included in the proof of the somewhat more technical: THEOREM 4: Let G, V, Φ, χ, σ, τ be as in section 1. Suppose that k(GV) > |V |. Choose v ∈ V, and set H = C_{G}(v). Let β be the sum of those irreducible constituents ϕ of Res_{H}^{G}(χ) such that:

(i) The algebraic conjugates of ϕ which occur as irreducible constituents of Res_{H}^{G}(χ) are transitively permuted by < τ >.

(ii) Stab_{<σ>}(ϕ) ⊆< τ >.

(iii) Res_{H}^{G}(χ) contains ϕ with multiplicity 1.

(iv) The character ϕ̄ is in a different < σ >-orbit from ϕ. Then Res_{H}^{G}(χ) − β is not faithful.

PROOF: Let ψ = Res_{H}^{G}(Φ). We will prove that it is not possible to find irreducible constituents ϕ_{1}, ϕ_{2}, …, ϕ_{t} of ψ such that:

(i) ∑_{i=1}^{t} ϕ_{i} is a faithful character of H.

(ii) No two of the ϕ_{i} are algebraically conjugate.

(iii) For each i, at least one of the following conditions is satisfied:

(a) ϕ_{i} occurs with multiplicity greater than one in ψ.

(b) An algebraic conjugate of ϕ_{i} in a different < σ >-orbit from ϕ_{i} occurs as a constituent of ψ.

(c) ϕ̄_{i} is in the same < σ >-orbit as ϕ_{i}.

We first explain how Theorems 2 and 4 both follow, once we have established this. Suppose that Res_{H}^{G}(V) has a faithful submodule U for which U ≅ U^{∗}. Let μ be the Brauer character afforded by U. Whenever ϕ is an irreducible constituent of μ, ϕ̄ is also a constituent. If ϕ̄ is in the same < σ >-orbit as ϕ, then ϕ satisfies condition c) above. Otherwise, ϕ satisfies condition b) above. In this case, then, we may take {ϕ_{i} : 1 ≤ i ≤ t} to be a set of irreducible constituents of μ which is maximal subject to the property that no two of the ϕ_{i} are algebraically conjugate, and we obtain a contradiction.

Suppose next that Res_{H}^{G}(χ) − β_{1} is faithful, where β_{1} is the sum of those irreducible constituents ϕ of Res_{H}^{G}(χ) for which none of conditions a), b), c) above hold. Then we can find {ϕ_{i} : 1 ≤ i ≤ t}, a set of irreducible constituents of ψ, with the properties outlined above, a contradiction. Thus Res_{H}^{G}(χ) − β_{1} is not faithful. Let ϕ be an irreducible constituent of β_{1}. Then ϕ certainly occurs with multiplicity one in Res_{H}^{G}(χ). If there is some α ∈ Gal (ℚ[ω]/ℚ) such that ϕ^{α} is in a different < τ >-orbit from ϕ, but ϕ^{α} occurs with non-zero multiplicity in Res_{H}^{G}(χ), then since condition b) fails for ϕ, we have ϕ^{α} = ϕ^{σ^{i}} for some integer i. Then χ^{σ^{i}} ≠ χ, since σ^{i} ∉< τ > by assumption. Hence ϕ occurs with multiplicity greater than one in Res_{H}^{G}(χ + χ^{σ^{−i}}), a contradiction, as both χ and χ^{σ^{−i}} are constituents of Φ.

Similarly, if ϕ^{γ} = ϕ for some γ ∈< σ >, then γ ∈< τ >, for otherwise ϕ occurs with multiplicity greater than one in Res_{H}^{G}(χ + χ^{γ}), while χ and χ^{γ} are distinct constituents of Φ, a contradiction. By definition of β_{1}, ϕ and ϕ̄ are in different < σ >-orbits, so that ϕ is an irreducible constituent of β. Since Res_{H}^{G}(χ) − β_{1} is not faithful, Res_{H}^{G}(χ) − β certainly is not, thus completing the proof of Theorem 4.

We suppose, then, that {ϕ_{i} : 1 ≤ i ≤ t} satisfying properties i), ii) and iii) above can be found. We will produce a faithful submodule U of Res_{H}^{G}(V), an integer-valued generalized character, α′, of H and a subgroup K of H with [H : K] ≤ 2 such that:

(i) If p = 2, then K = H.

(ii) α′(h)^{2} = [U : C_{U}(h)] for h ∈ K, α′(h)^{2} = \frac{[U:C_{U} (h)]}{p} for h ∈ H\K.

Once we have established that α′ exists, we may complete the proof of Theorem 4 as follows:

Notice that for h ∈ H^{#}, α′(h) is divisible by p unless p is odd, h ∈ H\K, and |C_{U}(h)| = \frac{|U|}{p}. If K ≠ H, we let λ denote the linear character of H whose kernel is K. For any h ∈ H, |UC_{V} (h)| ≤ |V |, so that [U : C_{U}(h)] ≤ [V : C_{V} (h)] ( with equality if and only if [V, h] ≤ U). Hence α′(h)^{2} ≤ [V : C_{V} (h)] for each h ∈ H. Since p |̸α′(1), it follows from Lemma 1 of [12] (and the remarks following) that we have k(GV) ≤ |V | (contrary to hypothesis) unless perhaps p is odd and there is some h ∈ H\K with |C_{U}(h)| = \frac{|U|}{p}. Hence we may assume that p is odd, and that |C_{U}(h)| = \frac{|U|}{p} for some h ∈ H\K.

Let γ = α′+λα′. Then γ(h) = 2α′(h) for h ∈ K, while γ vanishes identically on H\K. Hence γ(h) is an integer multiple of p for each h ∈ H^{#}, while γ(1) = 2α′(1), so p |̸γ(1).

Evidently, HV = C_{GV} (v). For each generator, w, of < v > we define a class function Γ^{(w)} of GV via:

Γ^{(w)}(x) = |C_{V} (h)|γ(h) if x is conjugate to hw for some h ∈ H, 0 otherwise. We note that Γ^{(w)} = Ind_{HV}^{GV} (μ^{(w)}), where μ^{(w)} = ∑_{ϕ∈Irr(HV)} m_{ϕ} \frac{ϕ(w^{−1})}{ϕ(1)}ϕ

and Ind_{H}^{HV} (γ) = ∑_{ϕ∈Irr(HV)} m_{ϕ}ϕ. In particular, Γ^{(w)} is an algebraic integer combination of characters of GV.

The above description of Γ^{(w)} allows us to deduce that for each irreducible character ϕ of GV, < Γ^{(w)}, ϕ >= \frac{1}{|H|} ∑_{h∈H} γ(h)ϕ(h^{−1}w^{−1}). Since p |̸|H|ϕ(1)γ(1), and since p|γ(h) for all h ∈ H^{#}, this inner product is certainly not zero. We also note that for each such ϕ, {< Γ^{(w)}, ϕ >:< w >=< v >} is closed under algebraic conjugation.

For each generator w, of < v > we may write Γ^{(w)} = Γ_{0}^{(w)} + Γ_{1}^{(w)}, where Γ_{i}^{(w)}(x) = λ^{i}(h)|C_{V} (h)|α′(h) if x is conjugate to hw for some h ∈ H, 0 otherwise.

For each such w, every irreducible character, ϕ, of GV occurs with non-zero multiplicity in at least one of the Γ_{i}^{(w)}, and by remarks similar to those above, each < Γ_{i}^{(w)}, ϕ > is an algebraic integer. Furthermore, for each i, and each generator, w, of < v >, < Γ_{i}^{(w)}, ϕ > is an algebraic conjugate of < Γ_{i}^{(v)}, ϕ >, so that for each i, < Γ_{i}^{(v)}, ϕ > is non-zero if any only if each < Γ_{i}^{(w)}, ϕ > is non-zero.

Let S_{i} denote the set of irreducible characters of GV which occur with nonzero multiplicity in Γ_{i}^{(v)}. For μ ∈ S_{i}, {< Γ_{i}^{(w)}, μ >:< w >=< v >} is a set of algebraic integers closed under algebraic conjugation, so the arithmetic - geometric mean inequality yields ∑_{{w:<w>=<v>}} | < Γ_{i}^{(w)}, μ > |^{2} ≥ ϕ(| < v > |), where ϕ denotes Euler’s function.

For each i, and each generator, w, of < v >, we see that:

< Γ_{i}^{(w)}, Γ_{i}^{(w)} > = |H|^{−1} ∑_{h∈H} |C_{V} (h)|α′(h)^{2}
= |H|^{−1} ∑_{h∈K} |C_{V} (h)[U : C_{U}(h)] + |H|^{−1} ∑_{h∈H\K} |C_{V} (h)|\frac{[U : C_{U}(h)]}{p}
≤ \frac{|V |}{2} + \frac{|V |}{2p}
= (\frac{p + 1}{2p}) |V |

Hence we obtain (for each i):

(\frac{p + 1}{2p}) |V |ϕ(| < v > |) ≥ ∑_{\{w:<w>=<v>\}} ∑_{μ∈S_{i}} | < Γ_{i}^{(w)}, μ > |^{2} ≥ |S_{i}|ϕ(| < v > |),

so that |S_{i}| ≤ (\frac{p+1}{2p}) |V | for each i.

For each generator, w, of < v >, we also see that

⟨Γ_{0}^{(w)} − Γ_{1}^{(w)}, Γ_{0}^{(w)} − Γ_{1}^{(w)}⟩ = |H|^{−1} ∑_{h∈H\K} 4|C_{V} (h)|\frac{[U : C_{U}(h)]}{p} ≤ \frac{2|V |}{p}.

Arguing as above, but with {Γ_{0}^{(w)} − Γ_{1}^{(w)} :< w >=< v >}, we deduce that ∑_{\{w:<w>=<v>\}} ⟨Γ_{0}^{(w)} − Γ_{1}^{(w)}, Γ_{0}^{(w)} − Γ_{1}^{(w)}⟩ ≤ \frac{2|V |}{p}ϕ(| < v > |), and then that at most \frac{2|V |}{p} irreducible characters of GV occur with non-zero multiplicity in Γ_{0}^{(v)} − Γ_{1}^{(v)}. In particular, we conclude that (|S_{1}\S_{0}| + |S_{0}\S_{1}|) ≤ \frac{2|V |}{p}.

Now we have:

k(GV) = |S_{0} ∪ S_{1}| = \frac{1}{2} [|S_{0}| + |S_{1}| + |S_{1}\S_{0}| + |S_{0}\S_{1}|]
≤ \frac{1}{2} [(\frac{p + 1}{2p}) |V | + (\frac{p + 1}{2p}) |V | + \frac{2|V |}{p}]
= (\frac{p + 3}{2p}) |V |.

Since p is odd, we have k(GV) ≤ |V |, a contradiction.

It remains, then, to construct the submodule U, and the generalized character α′, given the existence of ϕ_{1}, ϕ_{2}, …, ϕ_{t} satisfying conditions i) - iii).

We label so that for 1 ≤ i ≤ r, ϕ_{i} satisfies condition iii) a), for r+1 ≤ i ≤ s, ϕ_{i} satisfies condition iii) b), but not iii) a), and for s + 1 ≤ i ≤ t, ϕ_{i} satisfies condition iii) c), but neither condition iii) a) nor condition iii) b).

For 1 ≤ i ≤ r, let V_{i} be an irreducible GF (p)H-module whose Brauer character has ϕ_{i} as a constituent. Then Res_{H}^{G}(V) has a submodule U_{i} which is isomorphic to V_{i} ⊕ V_{i}. By the results of Knörr [4], there is a generalized character δ_{i} of H defined via δ_{i}(h) = [V_{i} : C_{V_{i}}(h)] for each h ∈ H. We note that δ_{i}(h)^{2} = [U_{i} : C_{U_{i}}(h)] for each h ∈ H, δ_{i} is integer valued, and that δ_{i}(h) is divisible by p unless h acts trivially on U_{i}.

For r + 1 ≤ i ≤ s, we choose γ_{i} ∈ Gal (ℚ[ω]/ℚ) so that ϕ_{i}^{γ_{i}} is an irreducible constituent of ψ in a different < σ >-orbit from ϕ_{i}. We let V_{i} be an irreducible GF (p)H-module whose Brauer character has ϕ_{i} as a constituent, V_{i}^{γ_{i}} be an irreducible GF (p)H-module whose Brauer character has ϕ_{i}^{γ_{i}} as a constituent. Then Res_{H}^{G}(V) has a submodule U_{i} which is isomorphic to V_{i} ⊕ V_{i}^{γ_{i}}. Again, we define the generalized character δ_{i} of H via δ_{i}(h) = [V_{i} : C_{V_{i}}(h)] for each h ∈ H. Again we see that δ_{i}(h)^{2} = [U_{i} : C_{U_{i}}(h)] for each h ∈ H, that δ_{i} is integer-valued, and that δ_{i}(h) is divisible by p unless h acts trivially on U_{i}.

For s + 1 ≤ i ≤ t, Res_{H}^{G}(V) has an irreducible self-dual submodule U_{i} such that the Brauer character afforded by U_{i} has ϕ_{i} as a constituent. We note that when i ≠ j, U_{i} and U_{j} have no common composition factor, so that U = ⊕_{i=1}^{t}U_{i} is a submodule of Res_{H}^{G}(V). Furthermore, U is a faithful GF (p)H-module. We set W = ⊕_{i=s+1}^{t}U_{i}, and we write W = X ⊕ Y where X admits an H-invariant, non-degenerate, alternating form and Y admits an H-invariant, non-degenerate, symmetric form. If p = 2, then we assume that H acts trivially on Y, as we may.

Now X has even-dimension, say 2m, over GF (p). From Gow[3], there is a rational-valued generalized character, α, of H such that α(h)^{2} = |C_{X}(h)| for each h ∈ H, and the class function γ of HX defined by γ(u) = p^{m}α(h) if u is conjugate to h ∈ H, γ(u) = 0 otherwise, is a generalized character of HX. Corresponding to γ, there is a generalized character, ζ, of H with ζ(h) = \frac{γ(h)}{|C_{X}(h)|} for each h in H.

Then ζ(h)^{2} = [X : C_{X}(h)] for each h in H. Furthermore, p |̸ζ(1), and for any h ∈ H which acts non-trivially on X, ζ(h) is divisible by p.

If p = 2, or if W = X, we let θ be the trivial character of H, and take K = H. Otherwise, we let λ be the linear character of H afforded by the determinant of its action on Y, K = ker λ. Then [H : K] ≤ 2, as Y ≅ Y^{∗}. We construct an integer-valued generalized character, θ, of H with θ(h)^{2} = [Y : C_{Y} (h)] for h ∈ K, θ(h)^{2} = \frac{[Y:C_{Y} (h)]}{p} for h ∈ H\K. From Gow [3] (pages 448-449), we know that when Y is even-dimensional over GF (p), there is an integer-valued generalized character, ξ, of H with ξ(h)^{2} = |C_{Y} (h)| for h ∈ K, pξ(h)^{2} = |C_{Y} (h)| for h ∈ H\K. When Y is odd-dimensional over GF (p), there is an integer-valued generalized character, ξ, of H with ξ(h)^{2} = p|C_{Y} (h)| for h ∈ K, ξ(h)^{2} = |C_{Y} (h)| for h ∈ H\K.

In either case, if Y has dimension 2m or 2m + 1 respectively, we define the integer-valued generalized character, γ of HY via γ(u) = p^{m}ξ(h) if u is conjugate to h ∈ H, 0 otherwise. There is a corresponding generalized character, θ, of H with θ(h) = \frac{γ(h)}{|C_{Y} (h)|} for h ∈ H. We see that θ(h)^{2} = [Y : C_{Y} (h)] for h ∈ K, θ(h)^{2} = \frac{[Y:C_{Y} (h)]}{p} for h ∈ H\K, so that θ is the required generalized character.

Now we are ready to complete the proof of Theorem 4. We let α′ be the generalized character θζΠ_{i=1}^{s}δ_{i} of H. Then α′ is integer-valued, and α′(h)^{2} = [U : C_{U}(h)] for h ∈ K, α′(h)^{2} = \frac{[U:C_{U} (h)]}{p} for h ∈ H\K. Furthermore, p |̸α′(1), and for h ∈ H^{#}, α′(h) is divisible by p unless p is odd and h ∈ H\K with |C_{U}(h)| = \frac{|U|}{p}. Thus α′ is the generalized character required to complete the proof of Theorem 4. Careful examination of the proof above shows that theorem 2 can be sharpened: COROLLARY 5: Let p be a prime, G be a finite p′-group, V be a faithful irreducible GF (p)G-module. Suppose that for some v ∈ V, Res_{H}^{G}(V) has a faithful submodule U with U ≅ U^{∗}, where H = C_{G}(v). Then k(GV) ≤ |V | and, if p is odd, then k(GV) ≤ (\frac{p+3}{2p}) |V |. If equality holds in either case, then V ≅ V^{∗} as GF (p)H-module and [V, H] ≤ U; furthermore, if p is odd in this eventuality, then some element of H acts as a reflection on U. If U is a permutation module, then equality forces Res_{H}^{G}(V) to be a permutation module and, if p is odd, also forces some element of H to act as a tranposition on the distinguished basis of U. PROOF: If [V, H] ≰ U, then there is some h ∈ H with [V, h] ≰ U, so that (as we remarked in the proof of Theorem 4), [U : C_{U}(h)] < [V : C_{V} (h)]. Hence for each Γ_{i}^{(w)} constructed as in the proof of Theorem 4, we have ⟨Γ_{i}^{(w)}, Γ_{i}^{(w)}⟩ < (\frac{p+1}{2}) |V | (also, in the earlier argument relying on lemma 1 of [12] we have ⟨α′, α′⟩ < |V |). In that case we obtain k(GV) < |V |, and k(GV) < (\frac{p+3}{2p}) |V | if p is odd. If p is odd, and no element of H acts as a reflection on U, then p|α′(h) for each h ∈ H^{#}, and each irreducible character of GV occurs with non-zero multiplicity in each of Γ_{0}^{(v)}, Γ_{1}^{(v)}, so that k(GV) = |S_{0}| ≤ (\frac{p+1}{2}) |V |.

If [V, H] ≤ U, then Res_{H}^{G}(V) = U ⊕ T where H acts trivially on T. In that case V ≅ V^{∗} as GF (p)H-module, and if U is a permutation module, then so is Res_{H}^{G}(V). In the latter case, if p is odd, any element of H which acts as a reflection on U acts as a transposition on the distinguished basis of U. This suffices to complete the proof of Corollary 5. REMARK: We consider that Theorem 4 indicates that the conditions under which the desired inequality k(GV) ≤ |V | can be violated are extremely restrictive. However, if we take G be the semi-direct product of an extra-special group of order 27 and exponent 3 with SL(2, 3) (acting in the obvious manner), then G has an absolutely irreducible 3-dimensional representation over GF (7). Letting V denote the associated GF (7)G-module, we find (using the notation of Theorem 4) that Res_{C_{G}(v)}^{G}(χ) − β is faithful for each v ∈ V. However, the inequality k(GV) ≤ |V | is not violated in this example, using (for example), the results of Knörr [5], since there is a vector v ∈ V such that C_{G}(v) is Abelian. 3 REGULAR ORBITS AND PERMUTATION SUMMANDS

We will be concerned in later sections with primes which satisfy: HYPOTHESIS 1: p is a prime with the property that whenever H is a finite p′-group, F is a finite field,of characteristic p and U is a faithful F H-module such that either

(a) For some prime q, O_{q}(H) acts absolutely irreducibly on U, and every characteristic Abelian subgroup of O_{q}(H) is central, or

(b) E(H) is quasi-simple and acts absolutely irreducibly on U, then there is a vector u ∈ U such that Res_{C_{H}(u)}^{H}(U) has a faithful permutation module as a summand.

In this section we will prove that all primes p with p > 5^{30} satisfy hypothesis 1. To deal with case b) above, we rely on a recent improvement by M.W. Liebeck [8] of Theorem 6 of Hall, Liebeck and Seitz [4]: THEOREM (M.W. Liebeck, [8]): Let p be a prime with p > 5^{30}, G be a finite p′-group, V be a faithful GF (p)G-module such that G has no regular orbit on V and G has a quasi-simple normal subgroup H which acts irreducibly on V. Then H ≅ A_{n} for some integer n with n < p, and V has dimension (n − 1). Furthermore, Res_{H}^{G}(V) is isomorphic to the quotient of the “natural” n-dimensional permutation module for H by its trivial submodule.

We draw a corollary to this result which explains its relevance for us. COROLLARY 6: Let p be a prime greater than 5^{30}, F be a finite field of characteristic p. Let G be a finite p′-group such that E = E(G) is quasi-simple. Let V be a faithful F G-module such that Res_{E}^{G}(V) is absolutely irreducible. Then there is a vector v ∈ V such that Res_{C_{G}(v)}^{G}(V) has a faithful permutation module as a summand. PROOF: Let σ be the matrix representation of G afforded by V. Let k be the extension field of GF (p) generated by {trace (gσ) : g ∈ G}. There is a kGmodule U such that V ≅ U ⊗_{k} F, and Res_{E}^{G}(U) is also absolutely irreducible. Hence we may suppose that k = F, and do so. Then there is an irreducible GF (p)G-module W such that V is isomorphic to a summand of W ⊗_{GF (p)} F and F ≅ End_{GF (p)G}(W). Furthermore, Res_{E}^{G}(W) is irreducible, and the permutation actions of G on vectors of V, W respectively are equivalent. We may suppose that G has no regular orbit on V, hence has no regular orbit on W. Then Liebeck’s theorem yields E ≅ A_{n} for some integer n < p, and W = V is absolutely irreducible of dimension n − 1. Let Z = Z(G)(= C_{G}(E) under the hypotheses of the corollary), and let e be the exponent of G/Z. For any g ∈ G with |C_{V} (g)| > 1 we must have g^{e} = 1_{G}. Hence no coset of Z in G can contain more than e elements g for which |C_{V} (g)| > 1. Since G has no regular orbit on V, we have V = ∪_{g∈G^{#}}C_{V} (g), so that p^{n−1} < e[G : Z]p^{n−2}, and p < e[G : Z]. Certainly, then, n > 6, as p > 5^{30}. Thus Aut(E) ≅ S_{n}, and we may identify V with {∑_{i=1}^{n} α_{i}v_{i} : α_{i} ∈ GF (p), ∑_{i=1}^{n} α_{i} = 0} where {v_{i} : 1 ≤ i ≤ n} is linearly independent, and for each g ∈ G there is some λ ∈ GF (p)^{#} such that λg permutes {v_{i} : 1 ≤ i ≤ n}.

Let u = (n − 1)v_{1} − ∑_{i=2}^{n} v_{i}. If ug = u for some g ∈ G, then g must fix v_{1} and permute {v_{1} − v_{i} : 2 ≤ i ≤ n}. Hence C_{G}(u) ≅ A_{n−1} or S_{n−1}, and Res_{C_{G}(u)}^{G}(V) is isomorphic to the natural permutation module.

To deal with case a) of hypothesis 1, we require some preparation. LEMMA 7: Let q be a prime, Q be a central product of the form ZE where Z is a finite cyclic q-group and E is an extra-special q-group (of exponent q if q is odd). Suppose that Q◁P with P = Q < x > and | < x > | = q. Suppose that P has a faithful complex irreducible character χ of degree q^{n}, where |E| = q^{2n+1}. Then ⟨Res_{<x>}^{P}(χ), 1⟩ ≤ \frac{q^{n}+q^{n−1}}{2} . PROOF: We may, and do, suppose that Q = Ω_{1}(Q) if q is odd, Q = Ω_{2}(Q) if q = 2. We first claim that it suffices to deal with the case that C_{Q}(x) is Abelian. Suppose that that case has been dealt with, but that C_{Q}(x) is non-Abelian. Then C_{Q}(x) contains an extra-special subgroup T of order q^{3}, and Q = T C_{Q}(T). Let P_{o} =< x > C_{Q}(T). Then χ(x) = qμ(x) for some irreducible character μ of degree q^{n−1} of P_{o}, and ⟨Res_{<x>}^{P}(χ), 1⟩ = q ⟨Res_{<x>}^{P_{o}}(μ), 1⟩. Since P_{o} = Q_{o} < x >, where Q_{o} = C_{Q}(T) ◁ P_{o}, and since Q_{o} has structure analogous to that of Q, our first claim is established. Hence we assume that C_{Q}(x) is Abelian from now on. We may also suppose that χ(x) ≠ 0, so that, in particular, x ∉ Q.

The maximum order of an Abelian subgroup of E is q^{n+1}, so that |C_{Q}(x)| ≤ q^{n+1}, |C_{P} (x)| ≤ q^{n+2} when |Z(Q)| = q, while if |Z(Q)| = 4 we obtain |C_{Q}(x)| ≤ q^{n+2}, |C_{P} (x)| ≤ q^{n+3}. Suppose that Z(Q) has order q.

Then there are (at least) q − 1 algebraic conjugates of χ, each of which may be multiplied by the q linear characters of P/Q, so we deduce that ∑_{i=1}^{q−1} |χ(x^{i})|^{2} < q^{n+1}. Hence ∥Res_{<x>}^{P}(χ)∥^{2} < \frac{1}{q}(q^{2n} + q^{n+1}) = q^{2n−1} + q^{n}. If ⟨Res_{<x>}^{P}(χ), 1⟩ > \frac{q^{n}+q^{n−1}}{2} , we obtain (\frac{q^{n}+q^{n−1}}{2} )^{2} < q^{2n−1} + q^{n} which yields q = 2, n ≤ 3 or q = 3, n = 1. If n = 1 we have ⟨Res_{<x>}^{P}(χ), 1⟩ ≤ q − 1 ≤ \frac{q+1}{2}. If q = n = 2, we have ⟨Res_{<x>}^{P}(χ), 1⟩ ≤ 3 = \frac{2^{2}+2^{1}}{2} . If n = 3 and q = 2, then ⟨Res_{<x>}^{P}(χ), 1⟩ > \frac{2^{3}+2^{2}}{2} forces χ(x) = 6, a contradiction as |C_{P} (x)| ≤ 32.

Suppose then that Z(Q) has order 4. Then we have |C_{P} (x) ≤ 2^{n+3}. Now there are (at least) two algebraic conjugates of χ which each may be multiplied by the two linear characters of P/Q, so we may conclude that χ(x)^{2} < 2^{n+1}. We may argue to a contradiction as above, except that we must also note that χ̄ ≠ χ when n = 3, while |C_{P} (x)| ≤ 64. THEOREM 8: Let p be a prime, F be a finite field of characteristic p with |F | ≥ 2^{27}, G be a finite p′-group, V be a faithful F G-module. Suppose that, for some prime q, G has a normal q-subgroup Q which acts absolutely irreducibly on V, and all of whose characteristic Abelian subgroups are central. Then G has a regular orbit on V. PROOF: Let r denote |F |. Since Q acts absolutely irreducibly on V, every characteristic Abelian subgroup of Q is central in G. We may, and do, suppose that G contains all non-zero scalar transformations. Then F (G) = ZQ, where Z = Z(G) ≅ F^{#}, and E(G) = 1_{G}. There is an integer n such that dim_{F} (V) = q^{n} and G/F (G) is isomorphic to a subgroup of Sp(2n, q). We may, and do suppose that Q = Ω_{1}(Q) if q is odd, Q = Ω_{2}(Q) if q = 2.

Suppose that G has no regular orbit on V. Then V = ∪_{g∈G^{#}}C_{V} (g). We claim that dim_{F} (C_{V} (g)) ≤ \frac{q^{n}+q^{n−1}}{2} for each g ∈ G^{#}. Let χ be the Brauer character of G afforded by V, so by hypothesis χ is a complex irreducible character of G. To establish the inequality in the claim, it suffices to consider the case that g has prime order. By Lemma 7, we may suppose that g has prime order s ≠ q. Applying Glauberman correspondence within the group Q < g >, there is an irreducible character, μ, of C_{Q}(g), there is a sign ϵ, and there is an s-th root of unity, ω, such that χ(g) = ϵωμ(1). We may suppose that g ∉ Z(G), so that μ(1) ≤ q^{n−1}. Hence ⟨Res_{<g>}^{G}(χ), 1⟩ ≤ \frac{1}{s} [q^{n} + (s − 1)q^{n−1}] ≤ \frac{q^{n}+q^{n−1}}{2} .

Using the elementary estimate |Sp(2n, q)| < q^{2n^{2}+n}, and the fact that |Z(G)| = r − 1, we obtain:

|V | = r^{q^{n}} < ∑_{g∈G^{#}} |C_{V} (g)| < |G|r^{\frac{q^{n}+q^{n−1}}{2}} < r · q^{2n^{2}+3n}r^{\frac{q^{n}+q^{n−1}}{2}} ,

so that (except in the cases n = 1, q < 4 or q = n = 2), we obtain log(r) < (\frac{4n^{2}+6n}{q^{n}−q^{n−1}−2}) log(q), which leads easily (via elementary calculus) to r < 2^{27}, contrary to hypothesis.

In the exceptional cases, the more delicate argument used during the proof of Lemma 7 leads this time to r^{q^{n}} < e[G : Z]r \frac{q^{n}+q^{n−1}}{2} where Z = Z(G) and e is the exponent of G/Z. Thus r^{\frac{q^{n}−q^{n−1}}{2}} < e[G : Z]. In the cases n = 1, q = 2, n = 1, q = 3 and n = q = 2 we respectively obtain [G : Z] < 32, [G : Z] < 243 and [G : Z] < 2^{14}, so in each case we easily obtain r < 2^{27}, contrary to hypothesis.

Corollary 6 and Theorem 8 establish that all primes p with p > 5^{30} satisfy hypothesis 1.

4 CLIFFORD THEORY AND THE VECTOR v

In this section, we use Clifford-theoretic reductions to complete the proof of Theorem 3. We collect some straightforward lemmas together for the convenience of the reader. All are well-known and easily proved. LEMMA 9: Let p be a prime, F be a finite field of characteristic p, H be a finite p′-group, U and V be F H-modules, k be a finite extension field of F. Then if V ⊗_{F} k has a summand isomorphic to U ⊗_{F} k, V has a summand isomorphic to U. LEMMA 10: Let p be a prime, F be a finite field of characteristic p, H be a finite p′-group, V be a faithful irreducible F H-module, k =End_{F H}(V). Then the irreducible summands of V ⊗_{F} k are absolutely irreducible, and if U is any one of them, the permutation actions of H on vectors of V, U repectively are equivalent. LEMMA 11: Let p be a prime, F be a finite field of characteristic p, k be a finite extension of F, H be a finite p′-group. Let V be a finite-dimensional kHmodule such that trace_{V} (h) ∈ F for all h ∈ H. Then there is an F H-module U with V ≅ U ⊗_{F} k. The main result of this section is somewhat stronger than theorem 3. THEOREM 12: Let p be a prime which satisfies hypothesis 1 of section 3. Let k be a finite field of characteristic p, G be a finite p′-group, U be an absolutely irreducible faithful kG-module. Then there is a vector u ∈ U such that Res_{C_{G}(u)}^{G}(U) has a summand which is a faithful permutation module. PROOF: Suppose that the theorem is false, and choose (G, U, k) to violate the theorem with dim_{k}(U) as small as possible. We assume, as we may, that G contains all non-zero scalar transformations of U, regarding G as a subgroup of GL(U).

We first claim that U is primitive. For if U ≅ Ind_{H}^{G}(W) where H < G and W is a kH-module, then W is absolutely irreducible. The minimality of dim_{k}(U) allows us to conclude that there is some w ∈ W such that Res_{C_{H}(w)}^{H}(W) has a summand which is a permutation module, say X, with C_{H}(w) ∩ C_{H}(X) = C_{H}(W). Let T be a right transversal to H in G. Then, as k-vector space, U = ⊕_{t∈T} W t. Let u = ∑_{t∈T} wt. Then it is routine to verify that ⊕_{t∈T} Xt is a faithful permutation module for C_{G}(u), and is a summand of Res_{C_{G}(u)}^{G}(U), contrary to hypothesis.

We next claim that whenever N ◁ G, all irreducible summands of Res_{N}^{G}(U) are absolutely irreducible. Suppose otherwise, and choose N ◁ G to be maximal subject to : Res_{N}^{G}(U) has an irreducible summand, W, which is not absolutely irreducible. As U is a primitive kG-module, we see easily that End_{kN}(U) ≅ M_{d} (End_{kN}(W)), where dim_{k}(U) = d dim_{k}(W), and that Z (End_{kN}(U)) ≅ End_{kN}(W), a finite extension of k. Let L = Z (End_{kN}(U)), M = C_{G}(L). Then M ◁ G and N ≤ M. Let t = [L : k].

Now W ⊗_{k} L is a direct sum of t pairwise non-isomorphic but Galois conjugate, absolutely irreducible LN-modules, and G/M is isomorphic to a subgroup of Gal(L/k). Let W ⊗_{k} L = ⊕_{i=1}^{t}W_{i}, where the W_{i} are irreducible LN-modules (each of which is faithful). By definition of M, Z(End_{kN}(U)) ⊆ End_{kM}(U) ⊆ End_{kN}(U), so that Z(End_{kN}(U)) ⊆ Z(End_{kM}(U)). Hence Z (End_{kM}(U)) ≅ k̸, so the argument used above for N tells us that Res_{M}^{G}(U) has an irreducible summand which is not absolutely irreducible. The maximal choice of N forces N = M.

Since [G : M] ≤ t, and Res_{M}^{G}(U ⊗_{k} L) ≅ d⊕_{i=1}^{t} W_{i}, we deduce from Clifford’s Theorem and Lemma 11 that [G : M] = t, and that Ind_{M}^{G}(W_{1}) ≅ U ⊗_{k} L. Thus Res_{M}^{G}(U) = W. By the minimality of dim(U), there is a vector w_{1} ∈ W_{1} such that Res_{C_{M} (w_{1})}^{M}(W_{1}) has a summand, X_{1} say, which is faithful permutation module. By Lemma 10, there is a vector w ∈ W with C_{M}(w_{1}) = C_{M}(w).

Now U ⊗_{k} L ≅ Ind_{M}^{G}(W_{1}), so by Mackey decomposition, Res_{C_{G}(w)}^{G}(U ⊗_{k} L) ≅ Ind_{C_{M} (w)}^{C_{G}(w)}(Res_{C_{M} (w)}^{M}(W_{1})) ⊕ Y for some LC_{G}(w)-module Y. Thus Res_{C_{G}(w)}^{G}(U ⊗_{k} L) ≅ Ind_{C_{M} (w)}^{C_{G}(w)}(X_{1}) ⊕ Y′ for some LC_{G}(w)-module Y′. Using Lemma 9, Res_{C_{G}(w)}^{G}(U) has a summand which is a faithful permutation module, contrary to hypothesis.

We next claim that whenever N ◁G with N ≰ Z(G), Res_{N}^{G}(U) is irreducible. If possible, choose N ◁ G with N ≰ Z(G) such that Res_{N}^{G}(U) has an irreducible constituent, W say, with dim_{k}(W) < dim_{k}(U). Let σ : N → GL(W) be the representation afforded by W. Since U is primitive and W is absolutely irreducible, there is (as usual) for each g ∈ G an element ζ_{g} ∈ GL(W) (unique up to non-zero scalar multiples) such that for all n ∈ N, ζ_{g}^{−1}(nσ)ζ_{g} = (g^{−1}ng)σ. Let T be a transversal to N in G (with 1_{G} ∈ T), and arrange so that ζ_{tn} = ζ_{t}(nσ) for all t ∈ T, n ∈ N. For each g, h ∈ G there is α(g, h) ∈ k^{#} with ζ_{g}ζ_{h} = α(g, h)ζ_{gh}. We set G̃ = {(g, z) : g ∈ G, z ∈ k^{#}} with product (g_{1}, z_{1})(g_{2}, z_{2}) = (g_{1}g_{2}, α(g_{1}, g_{2})z_{1}z_{2}). Then G̃ is a finite group with G̃/Z̃ ≅ G, where Z̃ = {(1, z) : z ∈ k^{#}} ≤ Z(G̃). Let Ñ = {(n, 1) : n ∈ N}. Then Ñ ◁ G̃, and Ñ ≅ N.

We may endow W with the structure of a kG̃-module by defining σ̃ : G̃ → GL(W) via (g, z)σ̃ = zζ_{g} for all g ∈ G, z ∈ k^{#}. Identifying N with Ñ, this extends the original action of N on W. In particular, W is an absolutely irreducible kG̃-module. We now view U as a kG̃-module on which Z̃ acts trivially, using the action of G.

There is a finite extension, L, of k such that U ⊗_{k}L “factorizes” as (U ⊗_{k}L) ≅ (W ⊗_{k} L) ⊗ X̃ for some (absolutely irreducible) LG̃-module X̃, and Ñ acts trivially on X̃. Now {nσ : n ∈ N} spans End_{k}(W) as W is an absolutely irreducible kN-module. For any g̃ ∈ G̃, we may choose ñ ∈ Ñ so that trace_{W ⊗_{k}L}(g̃ñ) ≠ 0. Then traceX̃(g̃) = \frac{trace_{U⊗_{k}L}(g̃ñ)}{trace_{W⊗_{k}L}(g̃ñ)} ∈ k. By Lemma 11, there is a kG̃-module X with X̃ ≅ X ⊗_{k} L. Hence U ≅ W ⊗ X as kG̃-module, and X is absolutely irreducible.

By the minimality of dim(U), there are vectors w ∈ W, x ∈ X such that ResG̃ ; CG̃(w)(W) has a summand Y_{1} which is a permutation module with CG̃(Y_{1}) ∩ CG̃(w) = CG̃(W) and ResG̃ ; CG̃(x)(X) has a summand Y_{2} which is a permutation module with CG̃(Y_{2}) ∩ CG̃(x) = CG̃(X). Let u = w ⊗ x.

If (g, 1) ∈ G̃ fixes w ⊗ x then there is some λ ∈ k^{#} such that w(g, 1) = λw, x(g, 1) = λ^{−1}x. Hence (λ^{−1}g, λ^{−1}) ∈ CG̃(w), (λg, λ) ∈ CG̃(x), so that (λ^{−1}g, λ^{−1}) permutes the distinguised basis of Y_{1}, (λg, λ) permutes the distinguished basis of Y_{2}. Thus (g, 1) permutes the distinguished basis of Y_{1} ⊗ Y_{2}. Furthermore, if (g, 1) fixes (elementwise) the distinguished basis of Y_{1} ⊗ Y_{2}, then (λ^{−1}g, λ^{−1}) fixes (elementwise) the distinguished basis of Y_{1}, (λg, λ) fixes (elementwise) the distinguished basis of Y_{2}. Thus (λ^{−1}g, λ^{−1}) ∈ CG̃(W) and (λg, λ) ∈ CG̃(X) so that (g, 1) ∈ CG̃(W ⊗ X). Hence Y_{1} ⊗ Y_{2} is a summand of Res_{C_{G}(u)}^{G}(W ⊗ X) which is a faithful permutation module for C_{G}(u), contrary to hypothesis, as U ≅ W ⊗ X.

Next, we claim that F (G) ≤ Z(G), and that E(G) = E is a central product of a single G-conjugacy class of components, but E is not quasi-simple.

If there is a prime q such that O_{q}(G) ≰ Z(G), then O_{q}(G) acts absolutely irreducibly on U and every characteristic Abelian subgroup of O_{q}(G) is central in G, so hypothesis 1 of section 3 implies that there is a vector u ∈ U such that Res_{C_{G}(u)}^{G}(U) has a faithful submodule which is a permutation module, a contradiction. Hence F (G) ≤ Z(G), so that E = E(G) ≰ Z(G). Now E acts absolutely irreducibly on U, so hypothesis 1 of section 3 ensures that E can’t be quasi-simple. Since every non-central normal subgroup of G acts absolutely irreducibly on U, E must be the central product of a single G-conjugacy class of components.

Now we proceed to a final contradiction. We may write E = L_{1}L_{2} … L_{t} where t > 1 and the L_{i} are distinct G-conjugate components. We set M_{1} = N_{G}(L_{1}), and let {g_{1} : 1 ≤ i ≤ t} be a right transversal to M_{1} in G, labelled so that L_{1}^{g_{i}} = L_{i} for each i (and we take g_{1} = 1_{G}). We reproduce the now standard argument that the given representation, σ say, of G on U is tensor induced (as a projective representation) from a projective representation of M_{1}. Since Res_{E}^{G}(U) is absolutely irreducible, we see easily that Res_{E}^{G}(U) ≅ ⊗_{1=t}^{t}U_{i}, where U_{1} is an absolutely irreducible submodule of Res_{L_{1}}^{G}(U) and where U_{i} = U_{1}g_{i} is an absolutely irreducible kL_{i}-module for 2 ≤ i ≤ t. Since Res_{E}^{G}(U) is absolutely irreducible, we note that gσ is the unique (up to non-zero scalar multiples) linear transformation T ∈ End_{k}(U) with the property that T^{−1}(xσ)T = (g^{−1}xg)σ for all x ∈ E.

Since Res_{L_{1}}^{G}(U) is homogeneous, M_{1} preserves the isomorphism type of U_{1}, and for each m ∈ M_{1} there is a unique (up to non-zero scalar multiples) element ζ(m) ∈ End_{k}(U_{1}) with ζ(m)yτζ(m)^{−1} = (mym^{−1})τ for all y ∈ L_{1}, where τ is the given representation of L_{1} on U_{1}.

Choose h ∈ G. Let γ ∈ S_{t} be defined by M_{1}g_{iγ} = M_{1}g_{i}h for 1 ≤ i ≤ t. There is a unique linear transformation T_{h} ∈ End_{k}(U) specified by: for 1 ≤ i ≤ t, and for all u_{1} ∈ U_{1}, (u_{1}g_{i})T_{h} = u_{1}ζ(g_{i}hg_{iγ}^{−1})g_{iγ} and extending this action in the obvious fashion to tensors. A routine computation verifies that (T_{h})^{−1}xσT_{h} = (h^{−1}xh)σ for all x ∈ E, so that T_{h} is a non-zero scalar multiple of hσ. This establishes that σ is tensor induced (as a projective representation) from the above projective representation of M_{1} on U_{1}.

As before, we may construct a central extension M̃_{1} of M_{1}, having a central subgroup Z̃ ≅ k^{#} with M̃_{1}/Z̃ ≅ M_{1}, and having a normal subgroup L̃_{1} ≅ L_{1} (which may be identified with L_{1}) such that (with this identification), τ extends to a genuine representation of M̃_{1} on U_{1}.

Since L̃_{1} acts absolutely irreducibly on U_{1}, hypothesis 1 of section 3 allows us to conclude that there is a vector u_{1} ∈ U_{1} such that ResM̃ ; CM̃_{1} (u_{1})(U_{1}) has a summand, X_{1} say, which is a permutation module with CM̃_{1}(u_{1}) ∩ CM̃_{1}(X_{1}) = CM̃_{1}(U_{1}). Even if CM̃_{1}(u_{1}) = CM̃_{1}(U_{1}), we may, and do assume that dim_{k}(X_{1}) ≥ 2.

For 2 ≤ i ≤ t, we set u_{i} = u_{1}g_{i} ∈ U_{i} and X_{i} = X_{1}g_{i} ≤ U_{i}. We let u = u_{1} ⊗ u_{2} ⊗ … ⊗ u_{t}. We claim that X_{1} ⊗ X_{2} … ⊗ X_{t} is a submodule of Res_{C_{G}(u)}^{G}(U) which is a faithful permutation module.

If h ∈ G satisfies uh = u, then there are scalars λ_{1}, …, λ_{t} ∈ k^{#} and there is a permutation γ ∈ S_{t} such that Π_{i=1}^{t}λ_{i} = 1 and u_{1}g_{i}h = λ_{i}u_{1}g_{iγ} for each i. Then for each i, (λ_{i}^{−1}g_{i}hg_{iγ}^{−1}, λ_{i}^{−1}) ∈ CM̃_{1}(u_{1}), so permutes the distinguished basis of X_{1}. Hence λ_{i}^{−1}h sends the distinguished basis of X_{i} to the distinguished basis of X_{iγ}. Since Π_{i=1}^{t}λ_{i} = 1, h permutes the distinguished basis of X_{1}⊗X_{2}⊗…⊗X_{t}. If h fixes this basis elementwise, we first note that for each i, (λ_{i}^{−1}g_{i}hg_{iγ}^{−1}, λ_{i}^{−1}) acts trivially on U_{1}. For otherwise there is a value of i such that the distinguished basis of X_{1} contains a vector y which is not fixed by (λ_{i}^{−1}g_{i}hg_{iγ}^{−1}, λ_{i}^{−1}) (hence is not sent to a scalar multiple of itself either). In that case, the distinguished basis vector yg_{1} ⊗ yg_{2} … ⊗ yg_{t} of X_{1} ⊗ X_{2} … ⊗ X_{t} is not fixed by h, contrary to assumption. Next we claim that γ is the identity permutation of S_{t}. For if iγ ≠ i for some i, we choose distinct vectors x, y from the distinguished basis of X_{1} and we let w = yg_{1} ⊗ … ⊗ yg_{i−1} ⊗ xg_{i} ⊗ yg_{i+1} … ⊗ yg_{t}, a distinguished basis vector for X_{1} ⊗ X_{2} … ⊗ X_{t}. Then wh is a distinguished basis vector for X_{1} ⊗ X_{2} … ⊗ X_{t} with xg_{iγ} in position iγ, so certainly wh ≠ w, contrary to assumption. Now we know that for 1 ≤ i ≤ t, (λ_{i}^{−1}g_{i}hg_{i}^{−1}, λ_{i}^{−1}) acts trivally on U_{1}, so that (h, 1) acts as multiplication by λ_{i} on U_{i} = U_{1}g_{i} for each i. Since Π_{i=1}^{t}λ_{i} = 1, h acts trivially on U, as required to complete the proof of Theorem 12.

Now we can state a slightly sharper version of Theorem 3 (which implies the version stated in the introduction by the results of section 3). THEOREM 3: Let p be a prime which satisfies hypothesis 1 of section 3. Let G be a finite group of order prime to p, and V be faithful irreducible GF (p)Gmodule. Then there is a vector v ∈ V such that Res_{C_{G}(v)}^{G}(V) has a summand which is a faithful permutation module. PROOF: Let F = GF (p), k = End_{F G}(V). Let U be an irreducible summand of V ⊗_{F} k. Then U is a faithful, absolutely irreducible, kG-module, so by Theorem 12 there is a vector u ∈ U such that Res_{C_{G}(u)}^{G}(U) has a faithful permutation module as a summand. By Lemma 10 there is a vector v ∈ V with C_{G}(v) = C_{G}(u), so the result follows by Lemma 9. REMARK: Similarly, the analogue of Theorem 1 of the introduction is valid for all primes p which satisfy hypothesis 1 of section 3. 5 CONSEQUENCES FOR GENERAL BLOCKS AND RELATION-SHIPS TO OTHER CONJECTURES

We recall two long-standing conjectures in modular representation theory. THE ALPERIN-McKAY CONJECTURE: Let B be a p-block with defect group D of a finite group G, and let b be the Brauer correspondent of B for N_{G}(D). Then k_{0}(B) = k_{0}(b). (Recall that k_{0}(B) is the number of complex irreducible characters in B whose degrees are not divisible by p[G : D]_{p}). BRAUER’S HEIGHT ZERO CONJECTURE: Let B be a p-block with defect group D of a finite group G. Then k_{0}(B) = k(B) if and only if D is Abelian.

The Alperin-McKay conjecture and one half of Brauer’s height zero conjecture are known to have the following consequence, which would also be a consequence of Alperin’s weight conjecture, or of conjectures of Broué on the structure of blocks with Abelian defect groups. We state the consequence as another conjecture which would be implied by those above. CONJECTURE: Let B be a p-block with Abelian defect group D of a finite group G and let b be the Brauer correspondent of b for N_{G}(D). Then k(B) = k(b).

J.B. Olsson [10] has recently proposed: CONJECTURE: Let B be a p-block with defect group D of a finite group G. Then k_{0}(B) ≤ [D : D′].

B. Külshammer proved in [7] (relying ultimately on results of W.F. Reynolds [11]) that Olsson’s conjecture would be a consequence of the Alperin-McKay conjecture together with a solution of the k(GV)-problem.

We may (using [11] again for part ii)) state: THEOREM 13: Let B be a p-block with defect group D of a finite group G, and let b be the Brauer correspondent for B of N_{G}(D). Suppose that p satisfies hypothesis 1 of section 3 (in particular this will hold if p > 5^{30}). Then:

(i) If k_{0}(B) = k_{0}(b), we have k_{0}(B) ≤ [D : D′].

(ii) If D is Abelian and k(B) = k(b), then k(B) ≤ |D|.

PROOF:

(i) Follows from the hypotheses and our Theorem 1, using Külshammer’s

arguments in [7].

(ii) Follows from the hypotheses, our Theorem 1, and the results of Reynolds [11] which show that there is a block b̃ of a p-solvable group H such that k(b̃) = k(b) and b̃ has defect group isomorphic to D.

References

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[8] M. Liebeck, Regular orbits for linear groups, preprint, 1995.

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[10] J.B. Olsson, On the number of characters in blocks Math. Zeitschrift, 186, (1984), 41-47.

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

[12] G.R. Robinson, Some remarks on the k(GV)-problem, J. Algebra, 172, (1995), 159-166.