On Brauer’s k(B)-problem for blocks of p-solvable groups with

non-Abelian defect groups Geoffrey R. Robinson, School of Mathematics and Statistics, University of Birmingham, Birmingham B15 2TT e-mail: g.r.robinson@bham.ac.uk

INTRODUCTION: The proof of Brauer’s k(B)-problem for p-solvable groups has recently been completed by Gluck,Magaard, Riese and Schmid [2]. Their latest work settled the only outstanding case ( p = 5) and represents the completion of the work of a series of authors dating back over 40 years ( see, for example, earlier work by various combinations of these authors for small primes and papers such as [1],[3],[5],[6],[7]).

Brauer’s k(B)-problem is to prove that if the p-block B has defect group D, then k(B) ≤ |D|, where k(B) denotes the number of ordinary irreducible characters of B. Two of the earliest published reductions of this problem (for B a p-block of a p-solvable group) were achieved by H. Nagao, who showed that it was sufficient to prove that when H is a p-solvable group with O_{p′}(H) = 1, then k(H) ≤ |H|_{p}, and that furthermore, it was sufficient to consider the case that H = GV for G a p′-group acting faithfully and irreducibly on the elementary Abelian p-group V. We will prove here that the inequality of Brauer’s k(B) problem is strict when B is a p-block of a p-solvable group with a non-Abelian defect group D. Our proof makes use of the affirmative answer to the k(B)- problem for p-solvable groups and does not provide an alternative proof of that result. Our main result is:

THEOREM 1 : Let B be a p-block of a p-solvable group which has a non-Abelian defect group D. Then k(B) < |D|.

REMARKS: By the Fong reductions, it suffices to prove that if H is a finite p-solvable group with a non-Abelian Sylow p-subgroup D and we have O_{p′}(H) ≤ Z(H), then we have k(H, λ) < |D| for each irreducible character, λ, of O_{p′}(H). Here, as usual, k(H, λ) denotes the number of complex irreducible characters of H which lie over λ. In fact, Nagao’s further reduction from [6] even shows that it suffices to consider the case that O_{p′}(H) = 1. In other words, it is more that sufficient to prove:

THEOREM 1′: Let H be a p-solvable group with O_{p′}(H) = 1 and with a non-Abelian Sylow p-subgroup D. Then k(H) < |D|.

In fact, even when H as above has an Abelian Sylow p-subgroup D, we can say much more when k(H) = |D|. The next result shows that, in that case, H can be constructed from the “obvious” building blocks.

THEOREM 2: Let H be a p-solvable group with O_{p′}(H) = 1 and with k(H) = |D| for D ∈ Syl_{p}(H). Then H = Z(H) × L, where L is either trivial, or else

L = V_{1}G_{1} × … × V_{n}G_{n}

where, for each i, V_{i} is an elementary Abelian p-group, G_{i} is a non-trivial p′- group acting faithfully and irreducibly on V_{i}, and k(G_{i}V_{i}) = |V_{i}|.

PREPARATORY LEMMAS

When N is a normal subgroup of a finite group G, we let k_{G}(N) denote the number of G conjugacy classes of N.

LEMMA 1: Let H be a finite group of the form GV, where G is a p′-group acting faithfully on the elementary Abelian p-group V. Then we have k(L) ≤ |V | for each subgroup L of H, and, furthermore, we have k(L) < |V | whenever L does not contain V.

PROOF: We may write L = W T where W = O_{p}(L) = L ∩ V and T is a Hall p′-subgroup of L. By Maschke’s theorem, there is a T -invariant complement to W in V, say U ( we note that L normalizes U). Then V T = UL, and the latter product is semi-direct. Hence k(L) ≤ k(UL) = k(V T) ≤ |V |. Furthermore, if L does not contain V, then we have U ≠ 1, so that k(L) < k(UL) ≤ |V |.

LEMMA 2: Let N be a normal subgroup of the finite group H such that H/N has the form GV, where G is a p′-group acting faithfully on the elementary Abelian p-group V. Then we have k(H) ≤ |V |k_{H}(N), where k_{H}(N) denotes the number of H-conjugacy classes of N. In particular, k(H) < |V |k(N), except possibly when every irreducible character of N is H-stable.

PROOF: This follows from remark A2′ on page 447 of Kovács-Robinson [4], since by Lemma 1 above, every subgroup of H/N has at most |V | conjugacy classes.

LEMMA 3: Let N be a normal subgroup of the finite group H. Suppose that O_{p′}(H) = 1, that O_{p}(H) ∈ Syl_{p}(N) and that H/N has the form GV, where G is a p′-group acting faithfully on the non-trivial elementary Abelian p-group V. Then we have k(H) < |H|_{p}.

PROOF: We note that O_{p′}(N) = O_{p′}(H/N) = 1. Let W = O_{p}(N). We have

k(H) ≤ k(N)k(H/N) ≤ |W ||V | = |H|_{p}.

Furthermore, by Lemma 2, equality forces every irreducible character of N to be H-stable. In that case, every irreducible character of N/W is H-stable. This in turn implies that a Sylow p-subgroup V^{∗} of H/W fixes every irreducible character (and hence every conjugacy class) of the p′-group N/W. Hence V^{∗} acts trivially on N/W. But N/W = O_{p′}(H/W), so that N/W contains its own centralizer in H/W. This contradicts the fact that V^{∗}(≅ V) is a non-trivial p-group.

The following Lemma is presumably well-known, but we provide a proof for the convenience of the reader. We thank L.G. Kovács for providing a short group-theoretic proof to replace our earlier character-theoretic proof of the special case we needed.

LEMMA 4: Let Z be a central subgroup of the finite group H. Then we have k(H) ≤ |Z|k(H/Z) with equality if and only if no non-identity element of Z is a commutator in H.

PROOF: Let {x_{1}, x_{2}, …, x_{n}} denote a set of n elements of H whose images in H/Z provide a full set of conjugacy class representatives for H/Z, so that n = k(H/Z). Then every element of H is conjugate to x_{i}z for some i and some z ∈ Z. Furthermore, if x_{i}z and x_{j}z′ are H-conjugate for z, z′ ∈ Z, then clearly i = j. Hence H has at most n|Z| conjugacy classes. Furthermore, if x_{i}z and x_{i}z′ are H-conjugate for z, z′ ∈ Z, then z(z′)^{−1} is a commutator in H. Conversely, if 1 ≠ z(z′)^{−1} = x^{−1}x^{h} for some x, h ∈ H, and some z, z′ ∈ Z, then there is some w ∈ Z and some i such that x^{h′} = x_{i}w for some h′ ∈ H. We may replace x by x^{h′} without disturbing z(z′)^{−1}, so we may suppose that x = x_{i}w, in which case x_{i}z and x_{i}z′ are conjugate. Hence k(H) < n|Z| in that case.

LEMMA 5: Let H be a p-solvable finite group with O_{p′}(H) = 1. Let M be a minimal normal subgroup of H contained in Φ(O_{p}(H)). Then k(H) ≤ k_{H}(M)[H : M]_{p}. In particular, if M ≰ Z(H), then k(H) < |H|_{p}.

PROOF: We first recall the standard fact that under the current hypotheses, every minimal normal subgroup of H is contained in the centre of O_{p}(H), as any such subgroup intersects Z(O_{p}(H)) non-trivially. Let λ be an irreducible character of M, and let I be its inertial subgroup in H. Then O_{p}(H) ≤ O_{p}(I). Furthermore, Φ(O_{p}(H)) ≤ Φ(O_{p}(I)), so that O_{p′}(I/M) = 1 (the last claim follows because a Hall p′-subgroup of a pre-image in I of O_{p′}(I/M) acts trivially on O_{p}(I)/Φ(O_{p}(I)), so centralizes O_{p}(I)).

Hence

k(H, λ) = k(I, λ) ≤ k(I/M) ≤ [I : M]_{p}.

Since λ was an arbitrary irreducible character of M, we conclude that k(H) ≤ k_{H}(M)[H : M]_{p}, and the result follows.

LEMMA 6: Let N be a normal subgroup of the finite group H. Suppose that every irreducible character of N is H-stable, and that k(H) = k(N). Then N = H.

PROOF: For each irreducible character μ of N, there is at least one irreducible character of H lying over μ. If N is proper, then the trivial character of N is covered by at least two irreducible characters of H, so the result follows.

PROOF OF THEOREM 1′: Let H be a minimal counterexample to the Theorem, so that H is p-solvable with a non-Abelian Sylow p-subgroup D, and O_{p′}(H) = 1, yet k(H) = |D|. Let W = O_{p}(H) and let N = O_{p,p′}(H). We first claim that W = D. For suppose otherwise. Let V be a minimal normal subgroup of H/N with full pre-image T in H, and let X be the (normal) subgroup of H defined by X/T = O_{p′}(H/T). We note that V = O_{p}(X/N) and that O_{p′}(X/N) = 1. Hence X fulfils the hypotheses of Lemma 3, so that k(X) < |X|_{p}. But O_{p′}(H/X) = 1, so that k(H/X) ≤ [H : X]_{p}. Since k(H) ≤ k(X)k(H/X), we obtain k(H) < |H|_{p}, contrary to assumption.

Hence we have W = D ⊲ H and W′ ≠ 1. By Lemma 5 (using the minimality of H), every minimal normal subgroup of H which is contained in W′ is central in H. In particular, Z(H) ≠ 1. Let Y be a central subgroup of H of order p. Then O_{p′}(H/Y) = 1, so that k(H/Y) ≤ |W/Y |. But k(H) ≤ pk(H/Y), so k(H) = |W | forces k(H/Y) = |W/Y | and k(H) = |Y |k(H/Y). Hence W/Y is Abelian (by minimality of H) and no non-identity element of Y is a commutator ( by Lemma 4). It follows that W′ = Y has order p and that no non-identity element of W′ is a commutator, evidently a contradiction. This completes the proof of Theorem 1′.

PROOF OF THEOREM 2: We now prove Theorem 2, assuming Theorem 1′. Let H be a finite group with an Abelian normal Sylow p-subgroup D, with O_{p′}(H) = 1, and with k(H) = |D|. Let G be a Hall p′-subgroup of H, and set U = Φ(D).

Since we certainly have U ≤ Z(D), the argument used in Lemma 5 may be adapted to show that k(H) ≤ k_{H}(U)[D : U] and hence that U ≤ Z(H) under the current hypotheses.

Now D may be expressed as a direct product of G-invariant homocyclic factors ( each of which is clearly normal in H). If D_{i} is one such factor, but is not elementary Abelian, then

Ω_{1}(D_{i}) ≤ Φ(D_{i}) ≤ Φ(D) ≤ Z(H).

Hence G centralizes D_{i}, as G is a p′-group. Since D = C_{D}(G) × [D, G], we conclude that [D, G] is elementary Abelian. Now we have H = C_{D}(G)×[D, G]G, and C_{D}(G) = Z(H), so it suffices to deal with the case that H = [D, G]G. Hence we may suppose that D is elementary Abelian, and that Z(H) = 1.

Now we may write D = V_{1} ×…×V_{n}, where each V_{i} is a minimal G-invariant subgroup of D, ( hence also a minimal normal subgroup of H). For each i, set V_{i′} = ∏_{j≠i} V_{j}, set G_{i} = C_{G}(V_{i′}) and set H_{i} = V_{i}G.

For each i, we have H = V_{i′}H_{i} and

C_{H_{i}}(V_{i′}) = V_{i}G_{i} ⊲ H.

Now H/V_{i}G_{i} is (isomorphic to) a semi-direct product of V_{i′} acted on faithfully by G/G_{i}, so that by Lemma 1, all of its subgroups have at most |V_{i′}| conjugacy classes. From Lemma 2, we obtain k(H) ≤ |V_{i′}|k_{H}(V_{i}G_{i}). But G_{i} clearly acts faithfully on V_{i}, so that we have k(V_{i}G_{i}) ≤ |V_{i}|. Since k(H) = |D|, we conclude that k(V_{i}G_{i}) = |V_{i}| and that every conjugacy class of G_{i}V_{i} is H-stable. Set N = G_{1}V_{1} × … × G_{n}V_{n} ( notice that the product is indeed direct). Then we have N ⊲ H, k(N) = |D| = k(H), and we have seen above that every conjugacy class of N is H-stable. Hence N = H by Lemma 6. This completes the proof of Theorem 2.

REFERENCES

[1] R.Brauer and W.Feit, On the number of irreducible characters of finite groups in a given block, Proc. Nat. Acad. Sciences USA 45, (1959),361-365.

[2] D.Gluck, K.Magaard, U.Riese & P.Schmid, The Solution of the k(GV)- problem, preprint, 2003.

[3] R. Gow, On the number of characters in a block and the k(GV)-problem for self-dual V, J. London Math. Soc., (2), 48, (199), 441-451.

[4] L.G. Kovács & G.R. Robinson, On the number of conjugacy classes of a finite group, Journal of Algebra, 160, 441-460 (1993).

[5] R.Knörr, On the number of characters in a block of a p-solvable group, Ill. J. Math, 28, (1984), 181-210.

[6] H. Nagao, On a conjecture of Brauer for p-solvable groups, J. Math. Osaka City University, 13, (1962), 35-38.

[7] G. Robinson and J.G.Thompson, On Brauer’s k(B) Problem, Journal of Algebra,184,(1996),1143-1160.