Strongly p-solvable blocks^{∗}

Charles W. Eaton

June 21st, 2004

Abstract

We define strongly p-solvable blocks, which generalize blocks of p-solvable groups, and show that a version of the Fong-Swan theorem holds for irreducible Brauer characters in such blocks. We also show that the height of an irreducible character in a strongly p-solvable block is bounded by the exponent of the central quotient of a defect group, which in particular implies that if further the defect groups are abelian, then every irreducible character in the block has height zero.

1 Introduction

Let G be a finite group, p a prime. Let Q be a p-subgroup of G and

G : 1 = G_{0} ◁ G_{1} ◁ ⋯ ◁ G_{n−1} ◁ G_{n} = G

a normal series for G.

We say that G is compatible with Q if for each i > 0, G_{i}/G_{i−1} is a p-group or Q_{i} = Q_{i−1}, where Q_{i} = Q ∩ G_{i}. Note that if G is compatible with Q, then every refinement of G is compatible with Q, and in particular some composition series must be compatible with Q. Another important property is that if G is compatible with Q, then G is compatible with every subgroup of Q.

In general this definition is problematic. In particular, it is not always the case that if one composition series is compatible with Q, then every composition series must be. To see this, consider G = H × ⟨y⟩, where |⟨y⟩| = p and H is nonabelian simple with x ∈ H of order p. Let Q = ⟨xy⟩. Then 1◁H ◁G is compatible with Q whilst 1◁⟨y⟩◁G is not. Notice however that such a Q cannot occur as a defect group of a p-block of G, since a defect group must contain O_{p}(G) = ⟨y⟩. Remarkably, if D is a defect group for some p-block of G, then it is the case that if some composition series is compatible with D, then so is every composition series for G.

Definition 1.1 We say that a p-block B of G is strongly p-solvable if there is a composition series for G which is compatible with a defect group of B.

∗The author is supported by a Royal Society University Research Fellowship

Examples of such blocks are p-blocks whose defect groups are contained in a normal p-solvable group (as studied in [5] and [3]) and blocks of defect zero. Observe that a group is p-solvable if and only if the principal block is strongly p-solvable. It should be noted however that we are not attempting to make a general definition of a ‘p-solvable block’ in the same way that a nilpotent block is a generalization of a block of a pnilpotent group. Indeed, there are numerous examples of nilpotent blocks of positive defect of non-abelian simple groups, so defect groups of nilpotent blocks are not in general strongly p-solvable.

The rationale for our definition is as follows: One reason blocks of p-solvable groups are of interest is because we may perform the Fong reductions using O_{p′}(G), effectively replacing it by a central p′-subgroup. This allows induction to be used to prove that a desired property holds for p-solvable groups whenever that property is compatible with the Fong reductions. A classic example of this is in the proof of the Fong-Swan theorem. It is now well-known that similar reductions may be applied when we consider blocks covering blocks of defect zero of a normal subgroup. It is therefore sensible to consider p-blocks B of groups such that either O_{p}(G) ≠ 1 or there is a non-trivial normal subgroup N such that B covers blocks of defect zero of N. We will see that strongly p-solvable blocks satisfy such a condition.

Throughout, a p-block (or block if p is obvious from the context) B of G is defined with respect to a complete local discrete valuation ring O whose residue field O/J(O) has characteristic p and is algebraically closed and whose field of fractions K has characteristic zero. We assume that O contains a primitive |G|^{3} root of unity. Write Irr(G, B) for the set of irreducible K-characters of G in B. If N ◁ G and b is a block of N, then write I_{G}(b) for the stabilizer of b under conjugation in G. If μ ∈ Irr(N), then write I_{G}(μ) for the inertia subgroup.

For convenience, when writing composition series we allow repeated terms.

The paper is structured as follows: In Section 2 we present some useful properties of composition series compatible with a defect group, including the crucial property that defect groups of strongly p-solvable blocks are compatible with every composition series of the group. In Section 2 we summarize the Clifford theory necessary for the proof of the version of the Fong-Swan theorem, which we give in Section 4. In Section 5 we give a bound on the height of an irreducible character in a strongly p-solvable block.

2 Composition series compatible with a defect group

Lemma 2.1 Let B be a strongly p-solvable p-block of a finite group G with defect group D. Then there is a composition series for G which is compatible with D.

Proof. There is a composition series G : G_{0} ◁ G_{1} ◁ ⋯ ◁ G_{n−1} ◁ G_{n} compatible with some defect group P of B. Then D = P^{g} for some g ∈ G, and the composition series G^{g} : G_{0}^{g} ◁ G_{1}^{g} ◁ ⋯ ◁ G_{n−1}^{g} ◁ G_{n}^{g} is compatible with D. □

Lemma 2.2 (a) If N ◁ G and b is a block of N covered by a strongly p-solvable block of G, then b is strongly p-solvable.

(b) If H ≤ G and b is a block of H with strongly p-solvable Brauer correspondent in G, then b is strongly p-solvable.

Proof. (a) and (b) follow from Lemma 2.1 and the fact that a defect group of a block of a subgroup is contained in a defect group of a covering block or Brauer correspondent respectively, and are left to the reader. □

The following are crucial for inductive arguments concerning strongly p-solvable blocks, since they enable us to move to quotient groups by allowing us to take a composition series which has the relevant normal subgroup as one of its terms.

Proposition 2.3 Let B be a p-block of a finite group G with defect group D. Let G : 1 = G_{0} ◁ G_{1} ◁ ⋯ ◁ G_{n−1} ◁ G_{n} = G be a composition series for G. If G is compatible with D, then every composition series for G is compatible with every defect group of B.

Proof. Suppose that G : G_{0} ◁ G_{1} ◁ ⋯ ◁ G_{n−1} ◁ G_{n} is compatible with D, and that H : 1 = H_{0} ◁ ⋯ ◁ H_{n} = G is another composition series for G. Write H = H_{n−1}, K = G_{n−1} and N = H ∩ K. Then H/N ≅ G/K and G/H ≅ K/N.

We use induction on |G| to show that H is also compatible with D. The result then follows by Lemma 2.1. So assume that the result holds for every p-block of every group of order strictly less than |G|. We may assume that n ≥ 2.

By [1, 15.1] there is a block of N covered by B with defect group D∩N and a block of K covered by B with defect group D∩K. Let N : 1 = N_{0}◁⋯◁N_{n−2} be a composition series for N. By Lemma 2.2, the first part of the proof and induction N_{0}◁⋯◁N_{n−2}◁K is a composition series compatible with D ∩ K. Hence N_{0} ◁ ⋯ ◁ N_{n−2} ◁ K ◁ G is compatible with D. Similarly N_{0} ◁ ⋯ ◁ N_{n−2} ◁ H and H_{0} ◁ ⋯ ◁ H_{n−2} ◁ H are compatible with D ∩ H. We must show that G/H is a p-group or D ≤ H.

If G/N is a p-group or D ≤ N, then we are done.

If D ≤ K but D ≰ N, then K/N is a p-group and again we are done.

Hence we may assume that D ≰ K and D∩K ≤ N. Then |G/K| = |H/N| = p, and |D| = p|D ∩ N|. We may also assume that G/H ≅ K/N = X, where X is non-abelian simple. Write H/N = Q and \overline{G} = G/N. Then \overline{G} = X × Q.

The blocks of N covered by B form a single G-conjugacy class {b_{1}, …, b_{t}}, which we identify with \{1, …, t\}. The conjugation action gives a homomorphism φ : \overline{G} → S_{t}. By [1, 15.1], the stabilizer of each b_{i} contains a defect group of B, hence \overline{D} is contained in some \overline{G}_{i}. Without loss of generality assume \overline{D} ≤ \overline{G}_{1}.

Since H/N is a p-group, by [4, V.3.5] there is a unique block B_{H} of H covering b_{1}. By [1, 15.1], there is a defect group P of B_{H} such that \overline{P} is a Sylow p-subgroup of \overline{H}_{1}. But B_{H} is covered by B, so P is contained in a defect group of B. Hence if Q ≤ \overline{G}_{1}, then \overline{P} = Q and D ≤ H by consideration of the order of D, and we are done. The same argument shows that we are done if Q fixes any point of {1, …, t}. We assume otherwise, i.e., that Q acts regularly, and derive a contradiction.

Suppose that X ≤ \overline{G}_{1}. Then \overline{D} ≤ \overline{G}_{1} = X, so D ≤ K, a contradiction.

We have shown that φ is faithful and that Q acts regularly (as a product of t/p disjoint p-cycles). Further, φ(Q) ≤ Z(φ(\overline{G})). Hence φ(X) is isomorphic to a subgroup of C_{p} ≀ S_{t/p} ≤ S_{t}.

Now \overline{D} has order p and is not Q, so there are x ∈ X and y ∈ Q, both of order p, such that \overline{D} = ⟨xy⟩. Recall that we have \overline{D} ≤ \overline{G}_{1}, so φ(x^{−1}) has a p-cycle in common with φ(y). Hence φ(X) must be a transitive subgroup of S_{t}, a contradiction since every transitive subgroup of C_{p} ≀ S_{t/p} in S_{t} has a non-trivial proper normal subgroup. □

Lemma 2.4 If B is a strongly p-solvable block of a nonabelian group G, and O_{p}(G) ≤ Z(G), then there is a non-trivial normal subgroup N of G, strictly containing Z(G), with D ∩ N = O_{p}(G).

Proof. Let N be a normal subgroup of G minimal subject to it strictly containing Z(G). Then N/Z(G) is a direct product of simple groups. N/Z(G) cannot be a pgroup, otherwise O_{p}(G) ≰ Z(G). If N/Z(G) is a p′-group, then we are done. Otherwise N/Z(G) is a direct product of isomorphic nonabelian simple groups. By Lemma 2.2 if b is a block of N covered by B, then b is strongly p-solvable and hence has defect group O_{p}(G). □

As a consequence, if O_{p}(G) = 1 and B is strongly p-solvable, then with N as in the above lemma, B covers blocks of defect zero of N.

3 Clifford theory

Lemma 3.1 Let B be a strongly p-solvable block of G with defect group D. If \overline{B} is a p-block of G/O_{p}(G) possessing an irreducible Brauer character which belongs to B when lifted to G, then \overline{B} is strongly p-solvable.

Proof. Choose a composition series G : G_{0} ◁ ⋯ ◁ G_{n} for G which refines 1 ◁ O_{p}(G) ◁ G. By Proposition 2.3 G is compatible with D. By [4, V.4.2] \overline{B} has a defect group P contained in D/O_{p}(G). But \overline{G} : G_{0}/O_{p}(G) ◁ ⋯ ◁ G_{n}/O_{p}(G) is compatible with D/O_{p}(G), and so is also compatible with P. □

Following [2, 3.1] we have the following (see also [6]):

Theorem 3.2 Let B be a block of positive defect of G and N ◁ G. Suppose that B covers a p-block b of defect zero of N. Write I = I_{G}(b). Then there is a central extension Ĩ of I/N by a cyclic p′-group W̃ and a block b̃ of defect zero of W̃ such that B is Morita equivalent to a block B̃ ∈ Blk(Ĩ) covering b̃, with defect groups isomorphic to those of B.

Further, if B is strongly p-solvable, then B̃ is strongly p-solvable.

Proof. Throughout we make use of the proof of [2, 3.1], which gives the first part of the result. Now suppose that B is strongly p-solvable.

There is a unique block B_{I} of I with B_{I}^{G} = B, and further there is a defect group D of B_{I} which is also a defect group of B. By Lemma 2.2 B_{I} is strongly p-solvable. By Proposition 2.3 we may choose a composition series I : I_{0} ◁ ⋯ ◁ I_{n} which refines 1 ◁ N ◁ I and is compatible with D. Then \overline{I} : I_{0}/N ◁ ⋯ ◁ I_{n}/N is compatible with DN/N.

Choose a defect group D̃ ≤ Ĩ of B̃ such that D̃W̃ /W̃ = DN/N and also for each i, Ĩ_{i} containing W̃ such that Ĩ_{i}/W̃ = I_{i}/N. Then Ĩ_{0} ◁ ⋯ ◁ Ĩ_{n} is a composition series for Ĩ which is compatible with D̃, and we are done. □

It is well known that if a p-solvable group has trivial intersection (TI) Sylow psubgroups, then these Sylow p-subgroups must be cyclic or generalized quaternion. Using the classification of finite simple groups, a similar result holds for strongly psolvable blocks:

Proposition 3.3 Let B be a strongly p-solvable block of a finite group G, and suppose that B has TI defect groups which are not normal in G. Then the defect groups are cyclic or generalized quaternion.

Proof. Suppose that B has a defect group which is neither cyclic nor generalized quaternion, i.e., contains a subgroup of the form C_{p} × C_{p}. Let b be a block of O_{p′}(G) covered by B. By the proof of [2, 3.1] there is defect group of B which is a non-normal TI subgroup of I_{G}(b). It then follows from [2, 4.1] that Ĩ in Theorem 3.2 has nonnormal TI defect groups isomorphic to D. Hence we may assume that O_{p′}(G) ≤ Z(G). Since B is strongly p-solvable we have D ∩ F^{∗}(G) = 1 (where F^{∗}(G) is the generalized Fitting subgroup), and the result follows by [2, 2.8]. □

4 Lifting Brauer characters

We show that a version of the Fong-Swan theorem holds for strongly p-solvable blocks.

Theorem 4.1 Let B be a strongly p-solvable block of G and let φ ∈ IBr(G, B). Then there is χ ∈ Irr(G, B) such that φ = χ_{p′}, the restriction of χ to the p-regular elements of G.

Proof. We use induction first on [G : Z(G)], then on |G|. Let φ be an irreducible Brauer character in B. Then φ may be regarded as an irreducible Brauer character belonging to a block, say \overline{B}, of G/O_{p}(G). By Lemma 3.1 \overline{B} is strongly p-solvable. If O_{p}(G) ≠ 1, then by induction there is an irreducible character χ of G/O_{p}(G) with χ_{p′} = φ. When regarded as a character for G, we again have χ_{p′} = φ, so we may assume that O_{p}(G) = 1.

Let D be a defect group for B. By Lemma 2.4 there is a normal subgroup N ≠ 1 of G such that D ∩ N = 1 and N strictly contains Z(G) = O_{p′}(Z(G)). By Theorem 3.2 there is a strongly p-solvable block B̃ of a group Ĩ which is Morita equivalent to B, and [Ĩ : Z(Ĩ)] ≤ [G : N] < [G : Z(G)]. By induction, every irreducible Brauer character in B̃ is given by the restriction of an irreducible character to p-regular elements. Since Morita equivalent blocks have the same decomposition matrices (up to reordering of characters), the result follows. □

Of course as a consequence of this, every irreducible Brauer character in a strongly p-solvable block of G has degree dividing |G| (in fact dividing [G : A] for every abelian normal subgroup A of G).

5 Heights of irreducible characters in strongly psolvable blocks

The defect of an irreducible character χ is the integer d(χ) such that p^{d(χ)}χ(1)_{p} = |G|_{p} the height of χ is the difference between d(χ) and the defect of the block containing χ.

Note from [7] that χ ∈ Irr(G) is called N-projective for N ◁G if there is a relatively N-projective OG-module affording χ, and that this is equivalent to χ(1)_{p} = [G : N]_{p}μ(1)_{p} for each μ ∈ Irr(N) covered by χ.

Theorem 5.1 Let B be a strongly p-solvable block with defect group D, and let [D : Z(D)] = p^{c}. If χ ∈ Irr(G, B), then χ has height at most c.

Proof. Let χ ∈ Irr(G, B) be a counterexample with |G| minimized. Then |G| > 1. Write h for the height of χ. Let N be a maximal normal subgroup of G. We choose a block b of N covered by B with defect group D ∩ N and μ ∈ Irr(N, b) covered by χ. Then D ≤ N or [G : N] = p. By Lemma 2.2 b is strongly p-solvable.

Suppose first that D ≤ N. Then χ is N-projective, and χ(1)_{p} = [G : N]_{p}μ(1)_{p}. Hence μ has height h in b. This contradicts minimality since D = D ∩ N is also a defect group for b.

Hence D ≰ N, and [G : N] = p. Note that |D| = p|D ∩ N|. Then I_{G}(μ) = N or G. If I_{G}(μ) = N, then χ = μ^{G} and μ has height h − 1. By [4, V.1.2] b has Brauer correspondent B, and so by [4, V.1.6] Z(D) ≤ D ∩ N. Then [D ∩ N : Z(D ∩ N)] = \frac{1}{p}[D : Z(D ∩ N)] < [D : Z(D)], contradicting minimality. So I_{G}(μ) = G. Since G/N is cyclic, μ = Res_{N}^{G}(χ), and μ has height h. But [D ∩ N : Z(D ∩ N)] ≤ [D : Z(D)], again contradicting minimality. □

We immediately have half of Brauer’s abelian defect group conjecture for strongly p-solvable blocks:

Corollary 5.2 Let B be a strongly p-solvable block with abelian defect group D. Then every irreducible character in B has height zero.

ACKNOWLEDGEMENTS

Some of this work was done whilst visiting RWTH Aachen, and the author is grateful for their hospitality.

References

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[2] J. An and C. W. Eaton, Blocks with trivial intersection defect groups, Math. Z. 247(2004), 461-486.

[3] C. W. Eaton, Vertices for a class of blocks, preprint.

[4] W. Feit, The Representation Theory of Finite Groups, (North-Holland, Amsterdam, New York, Oxford, 1982).

[5] S. M. Gagola, Jr., A note on lifting Brauer characters, Proc. Amer. Math. Soc. 53(1975), 295-300.

[6] B. Külshammer and L. Puig, Extensions of nilpotent blocks, Invent. Math 102 (1990), 17-71.

[7] B. Külshammer and G. R. Robinson, Characters of relatively projective modules, II, J. London Math. Soc. (2) 36(1987), 59-67.

Department of Mathematics UMIST P.O.Box 88 Manchester M60 1QD United Kingdom email: c.eaton@man.ac.uk