PACIFIC JOURNAL OF MATHEMATICS Vol. 187, No. 2, 1999

ORBIT SIZES AND CHARACTER DEGREES

Thomas Michael Keller

The objective of this paper is to develop criteria that guarantee that a finite group G which acts faithfully on a vector space V possesses “many” orbits of different sizes. This has consequences on (classical and modular) character degrees of finite solvable groups.

Introduction.

The starting point of this investigation was the well-known problem of bounding the derived length dl(G) of a finite solvable group G by the number |cd(G)| of its irreducible complex character degrees. As well-known, in 1985 D. Gluck established the bound

dl(G) ≤ 2 |cd(G)|

for arbitrary solvable groups (cf. [2]).

I.M. Isaacs was the first to doubt that this bound asymptotically is the right bound, and there are indications that a much better bound should exist. For example, for the p-Sylow subgroups of some classical groups logarithmic bounds have been established (cf. [6], [12]), and on the other hand for A-groups, i.e. solvable groups all Sylow subgroups of which are abelian, we found a bound that is even stronger than logarithmic (cf. [9]).

Unfortunately, despite the results on some well-known p-groups mentioned above, the abundance of p-groups seems to make it impossible to attack the problem for p-groups with today’s methods in general; for instance, no reasonable induction argument is available (cf. [11]). So in order to avoid the problems occuring with p-groups, B. Huppert and I.M. Isaacs suggested to consider the Fitting height instead of the derived length. Note that, as Fitting height and derived length coincide in A–groups, the result in [9] is already a result on the Fitting height. So we aimed at getting a similar bound for more general groups.

Now by standard character theory, in minimal counterexamples one often has a minimal normal subgroup N of G that has a complement H in G, and then character degrees that are induced from the extension of a linear character of N to its inertia group correspond to orbit sizes in the action of H on Irr(N). This is where the question of the number of different orbit sizes comes into play, a question, that surely is of independent interest and that we are going to study in this paper as generally as possible.

In the first section we prove that any finite group G that possesses a normal subgroup N of a certain type (almost extra-special) possesses (almost) as many distinct orbit sizes in its action on a vector space V as N does, provided that the center of N acts fixed point freely on V (which is a natural and necessary hypothesis); and for N this number is well–known. This result is not far from being best possible and one key step for the proof of our results in Section 2; namely it enables us to find many orbit sizes in quasiprimitive group actions of solvable groups. To handle the imprimitive case, we have to restrict ourselves to groups of odd order (which then are solvable by Feit-Thompson), and we make use of a recent result of A. Seress (cf. [13]) stating that the minimal base size of a solvable primitive permutation group of odd order is at most 3. This is why our inductive argument only works for primes greater than 3 whence we have an additional hypthesis in Theorem 2.1, where we examine the relationship between the number of orbit sizes of G on a vector space over a finite field of odd characteristic and the number of distinct orbit sizes of a suitable normal subgroup of G. But still this is strong enough to deal with our original subject, the dependence of the Fitting height f(G) on |cd(G)|, at least for groups of odd order, and we find absolute constants C_{1}, C_{2} such that

f(G) ≤ C_{1} log |cd(G)| + C_{2}

for any group of odd order (see Corollary 2.4). This improves the former linear bound by Broline-Garrison (cf. [7, Corollary (12.21)]) and in fact is the asymptotically best possible bound. We conjecture that such a bound is true for arbitrary solvable groups.

Finally in the third section we discuss a result of I. M. Isaacs on bounding the derived length of a finite solvable group by the number of its irreducible Brauer characters in the light of our results obtained so far.

Results and Proofs.

Notation. All groups considered in this paper are finite. By f(G) we denote the Fitting height (or nilpotent length) of G, and dl(G) is the derived length of G. Irr(G) is the set of irreducible complex characters of G, and cd(G) := {χ(1)|χ ∈ Irr(G)}.

For any group G we denote by d(G) the minimal number of generators of G. If p is a prime and n ∈ ℕ, then n_{p} is the p-part of n. For x ∈ ℝ we let ⌈x⌉ = max{n ∈ ℕ | n − 1 < x}. All other notation is standard.

1. Orbit sizes of quasiprimitive groups.

Definition 1.1. Let G be a group acting on a vector space V. If 0 ≠ v ∈ V such that C_{G}(v) = 1, then we call v^{G} a regular orbit of G on V.

If no non-identity element of G fixes any non–zero vector, then we say that G acts fixed point freely on V.

For a prime p and any integer m ≥ 0 we say that a p-group E is of type E(p, m) if

either p > 2 and E is extra-special of exponent p and order p^{2m+1} or p = 2 and E = E_{1}Z(E) (central product) with an extra-special group

E_{1} of order 2^{2m+1} and Z(E) cyclic of order 2 or 4.

This notion differs from [1] only in that we also allow m = 0 (in which case E = Z(E) is cyclic). Note that the Lemmas 2, 3, 4, 5 in [1] remain true for m = 0, so that we may apply them also to our groups of type E(p, m).

The following lemma and its proof were brought to my attention by Thomas R. Wolf.

Lemma 1.2. Let G be a group acting on a vector space V over K := GF (q) for a prime q. Suppose that E ⊴ G is of type E(p, m) for a prime p and an integer m ≥ 0 such that Z(E) acts fixed point freely on V. If g ∈ G with g ∉ C_{G}(E), then

dim_{K} C_{V} (g) ≤ \frac{p + 1}{2p} dim_{K} V.

Proof. As Z(E) acts fixed point freely on V, E acts faithfully on V and p ≠ q. So V_{E} is completely reducible, i.e. there is an n ∈ ℕ such that

V_{E} = V_{1} ⊕ ⋯ ⊕ V_{n}

with irreducible and obviously also faithful E–modules V_{i}. Now if 1 ≠ x ∈ E, then by [1, Lemma 3] we have dim_{K} C_{V_{i}}(x) ≤ (1/p) dim_{K} V_{i} for i = 1, …, n, and thus we easily obtain

dim_{K} C_{V} (x) ≤ \frac{1}{p} dim_{K} V .

To prove the lemma, let g ∈ G with g ∉ C_{G}(E). Then there exists a 1 ≠ x ∈ E such that 1 ≠ [g, x] ∈ E whence we have dim_{K} C_{V} ([g, x]) ≤ (1/p) dim_{K} V. So as clearly C_{V} (g^{x}) ∩ C_{V} (g) ≤ C_{V} ([g, x]), we find that

dim_{K} V ≥ dim_{K} (C_{V} (g^{x}) + C_{V} (g))
= dim_{K} C_{V} (g^{x}) + dim_{K} C_{V} (g) − dim_{K} (C_{V} (g^{x}) ∩ C_{V} (g))
≥ 2 dim_{K} C_{V} (g) − dim_{K} C_{V} ([g, x])
≥ 2 dim_{K} C_{V} (g) − \frac{1}{p} dim_{K} V.

Hence the assertion follows. □

Next we prove a lemma on the existence of p–regular orbits. We do not try here to figure out the minimal hypothesis we need to establish our claim, but simply give a straightforward argument that suffices to achieve our asymptotic results later on.

Lemma 1.3. Let G be a group acting on a vector space V over K := GF (q) for a prime q. Suppose that E ⊴ G is of type E(p, m) for a prime p and an integer m ≥ 0 such that C_{G}(E) acts fixed point freely on V.

If m ≥ 10, then G has a regular orbit on V. If p is odd, then G has a p–regular orbit on V even for m ≥ 5.

Proof. Clearly E acts faithfully on V and so (|E|, |V |) = 1. Consequently V_{E} is completely reducible so that for an n ∈ ℕ we have

V_{E} = V_{1} ⊕ ⋯ ⊕ V_{n}

with irreducible and obviously also faithful E-modules V_{i}.

Now put δ = 0 if p is odd, and δ = 1 in case of p = 2. Then it is wellknown that there is an abelian A ≤ E with A ∩ Z(E) = 1 of order p^{m−δ} (cf. [5, III, §13]). Now with [1, Lemma 3] we see that |C_{V_{i}}(A)| = |V_{i}|^{(1/p^{m−δ})} for all i = 1, …, n and thus |C_{V} (A)| = |V |^{(1/p^{m−δ})}. Moreover C_{G}(E) normalizes (and so acts fixed point freely on) C_{V} (A) which implies |C_{G}(E)| | |C_{V} (A)|−1, in particular |C_{G}(E)| ≤ |C_{V} (A)|.

Furthermore, if we put B := C_{G}(Z(E)), by [10, Corollary 1.6] B/EC_{B}(E) = B/C_{B}(E/Z) is isomorphic to a subgroup of Sp(2m, p), so that we find the elementary estimation |B/EC_{G}(E)| ≤ p^{2m^{2}+m} (note that C_{B}(E) = C_{G}(E)). Finally observe that obviously |G/B| ≤ |Aut(Z(E))| ≤ p.

Now assume that G has no p–regular orbit on V. Then obviously

V = ⋃_{g∈G\{1}} C_{V} (g),

and thus as C_{G}(E) acts fixed point freely on V, we obtain with Lemma 1.2 that

|V | ≤ ∑_{g∈G with o(g)=p} |C_{V} (g)| ≤ |G| · |V |^{\frac{p+1}{2p}}

and hence

|V |^{\frac{p−1}{2p}} ≤ |G| = |G/B| · |B/EC_{G}(E)| · |E/Z(E)| · |C_{G}(E)|
≤ p · p^{2m^{2}+m} · p^{2m} · |V |^{(1/p^{m−δ})}.

Now by [1, Lemma 2] for i = 1, …, n we have dim_{K} V_{i} = s p^{m}, where s ≥ 1 is the smallest integer such that |Z(E)| divides q^{s}−1. Hence p ≤ |Z(E)| ≤ q^{s} and

|V | ≥ |V_{1}| = q^{sp^{m}} ≥ p^{p^{m}}.

So we conclude that

p^{2m^{2}+3m+1} ≥ |V |^{( \frac{p^{m−δ}−p^{m−1−δ}−2}{2p^{m−δ}} )} ≥ p^{( \frac{p^{m}−p^{m−1}−2p^{δ}}{2} )}

and further that

p^{m−1}(p − 1) ≤ 4m^{2} + 6m + 2 + 2p^{δ}.

If p = 2, then this is only true for m ≤ 9 contradicting our hypothesis, and if p ≥ 3, we have 3^{m−1} ≤ 2m^{2} + 3m + 2 which only holds for m ≤ 4, so that again we have a contradiction. Hence the lemma is shown. □

Lemma 1.4. Let G be a group acting on a vector space V over K := GF (q) for a prime q. Suppose that E ⊴ G is of type E(p, m) for a prime p and an integer m ≥ 0 such that Z(E) acts fixed point freely on V. Now define

m′ = \{ \begin{matrix}m & if p is odd or E is the central product of m \\ & dihedral groups of order 8 and Z(E) \\ m − 1 & if m ≥ 1 and E is the central product of m − 1 \\ & dihedral groups of order 8, a quaternion group \\ & of order 8 and Z(E). \end{matrix}

Fix i ∈ {0, …, m′}. Moreover let Z(E) ≤ E_{i}⊴E be of type E(p, i). In case of m′ = m−1 and i ≥ 1 we also assume that E_{i} contains a quaternion subgroup. Let A_{i} ≤ E be elementary abelian of order p^{m′−i} such that E_{i} ∩ A_{i} = 1. (Such E_{i}, A_{i} exist, as one can see with the proof of [1, Lemma 5].) Hence [E_{i}, A_{i}] = 1, and (by [1, Lemma 3]) C_{V} (A_{i}) > 0.

Now let

C_{i} = C_{G}(E_{i} A_{i}/A_{i})
:= {g ∈ G | (xA_{i})^{g} = x^{g}A_{i}^{g} = xA_{i} for all x ∈ E_{i} }

and let A_{i} ≤ N_{i} ≤ C_{i} be maximal subject to C_{V} (N_{i}) > 0.

Put V_{i} = C_{V} (N_{i}) and H_{i} = N_{G}(N_{i}), and for any U ≤ H_{i} we write \overline{U} = UN_{i}/N_{i}. Then the following holds:

a) \overline{H_{i}} acts on V_{i}, and E_{i} ≅ \overline{E_{i}} ⊴\overline{H_{i}} and E_{i}N_{i} ⊴H_{i}. Furthermore E ∩H_{i} = E_{i} × A_{i} ⊴ H_{i}, E ∩ N_{i} = A_{i} ⊴ H_{i}, and Z(\overline{E_{i}}) = \overline{Z(E_{i})} = \overline{Z(E)} acts fixed point freely on V_{i}; in particular, \overline{E_{i}} acts faithfully on V_{i}.

b) C\overline{H_{i}}(\overline{E_{i}}) acts fixed point freely on V_{i}.

c) Assume now that 0 ≤ i ≤ m′ −1 and that A_{i} > A_{i+1} and E_{i} < E_{i+1}. If N_{i+1} is given, then we can choose N_{i}, V_{i} such that N_{i} > N_{i+1} and V_{i} < V_{i+1}.

Proof. a) First observe that C_{i} ≤ N_{G}(A_{i}) and that E_{i} ∩ N_{i} = 1.

Now of course \overline{H_{i}} normalizes V_{i}. As N_{i} ≤ C_{i}, for any x ∈ N_{i}, y ∈ E_{i} there is an a ∈ A_{i} such that xyx^{−1} = y^{x^{−1}} = ya. Hence x^{y} = y^{−1}xy = ax ∈ N_{i}. This shows that E_{i} ≤ H_{i}. So Z(E) ≤ H_{i} acts fixed point freely on V_{i}, and so does Z(\overline{E_{i}}) = \overline{Z(E_{i})} ≅ Z(E). Hence clearly \overline{E_{i}} ≅ E_{i}.

Now we have A_{i} × E_{i} ≤ H_{i}. If y ∈ E\(A_{i} × E_{i}), then there is an x ∈ A_{i} with 1 ≠ [x, y] ∈ Z(E); hence the assumption y ∈ H_{i} yields [x, y] ∈ N_{i} which contradicts C_{V} (N_{i}) > 0. This shows A_{i} × E_{i} = E ∩ H_{i} ⊴ H_{i} and then \overline{E_{i}} = \overline{E_{i} × A_{i}} ⊴ \overline{H_{i}} and E_{i}N_{i} ⊴ H_{i}. Clearly \overline{E_{i}} acts faithfully on V_{i}.

Now finally we have A_{i} ≤ E ∩ N_{i} ≤ E ∩ H_{i} = A_{i} × E_{i}, so we have E ∩ N_{i} = A_{i}, because E_{i} ∩ N_{i} = 1. So a) is shown.

b) It suffices to show that all elements of prime order in C\overline{H_{i}}(\overline{E_{i}}) act fixed point freely on V_{i}. Let 1 ≠ gN_{i} ∈ C\overline{H_{i}}(\overline{E_{i}}) with g ∈ H_{i}. Then we have 1 = [⟨gN_{i}⟩, \overline{E_{i}}] = [⟨g⟩, E_{i}]N_{i}/N_{i}, and so a) yields [⟨g⟩, E_{i}] ≤ N_{i} ∩ E = A_{i}. As furthermore g normalizes A_{i}, it follows that g ∈ C_{i}.

Let N_{i}^{∗} = ⟨N_{i}, g⟩ = N_{i}⟨g⟩ > N_{i}. Then we have A_{i} ≤ N_{i}^{∗} ≤ C_{i}, so the maximal choice of N_{i} forces C_{V} (N_{i}^{∗}) = 0. Consequently also C_{V_{i}}(N_{i}^{∗}) = 0, and as N_{i} acts trivially on V_{i}, we find C_{V_{i}}(g) = 0. Hence gN_{i} acts fixed point freely on V_{i}, as desired.

c) Let A_{i} ∩E_{i+1} = ⟨a⟩. By a) we have A_{i+1} ⊴N_{i+1}. Now A_{i} = A_{i+1} ×⟨a⟩, and thus for h ∈ N_{i+1} we have a^{h} = ab with a suitable b ∈ A_{i+1}, as N_{i+1} ≤ C_{i+1}. So it follows that N_{i+1} ≤ N_{G}(A_{i}). As N_{i+1} ≤ C_{i+1}, moreover we have that N_{i+1} ≤ C_{G}(E_{i} × ⟨a⟩ A_{i+1}/A_{i+1}) = C_{G}(E_{i}A_{i}/A_{i+1}). Hence it follows that N_{i+1} ≤ C_{i}.

Now we build the semidirect product K_{i+1} := N_{i+1}A_{i} = N_{i+1}⟨a⟩ > N_{i+1}. Then we have A_{i} ≤ K_{i+1} ≤ C_{i}. Moreover by a) we know that E_{i+1} ≤ H_{i+1} acts faithfully on V_{i+1}. Hence again with [1] we see that V_{i+1} is completely reducible as E_{i+1}–module and then that C_{V_{i+1}}(a) > 0. Hence 0 < C_{V_{i+1}}(N_{i+1}⟨a⟩) = C_{V_{i+1}}(K_{i+1}) and so C_{V} (K_{i+1}) > 0. Thus K_{i+1} has all the properties that we demand of N_{i} (except maximality), i.e. we can choose N_{i} such that N_{i} ≥ K_{i+1} > N_{i+1}, as wanted. Then clearly V_{i} ≤ V_{i+1}, and as E_{i+1} acts faithfully on V_{i+1} whereas a centralizes V_{i}, it follows that V_{i} ≤ C_{V_{i+1}}(a) < V_{i+1}, and finally c) is proven. □

Theorem 1.5. Let G be a group acting on a vector space V over K := GF (q) for a prime q. Suppose that E ⊴ G is of type E(p, m) for a prime p and an integer m ≥ 0 such that Z(E) acts fixed point freely on V. Then the following holds:

a) Suppose that p is odd and that m ≥ 5. For i = 5, …, m take subgroups N_{i} of G as in Lemma 1.4 such that N_{5} > ⋯ > N_{m} (which is possible by Lemma 1.4c)). Then there exist v_{i} ∈ V (i = 5, …, m) such that C_{G}(v_{i}) = N_{i} > 1. Hence by Lemma 1.4 the |v_{i}^{G}|_{p} (i = 5, …, m) are strictly increasing; in particular, G has at least m − 4 different nontrivial orbit sizes on V.

b) Suppose that p = 2, let m′ be defined as in Lemma 1.4, and suppose that m′ ≥ 10. For i = 10, …, m′ take subgroups N_{i} of G as in Lemma 1.4 such that N_{10} > ⋯ > N_{m′}. Then there exist v_{i} ∈ V (i = 10, …, m′) such that C_{G}(v_{i}) = N_{i} > 1, whence the 2-parts of the corresponding orbit sizes are strictly increasing; in particular, G has at least m′ − 9 different nontrivial orbit sizes on V.

Proof. a) Fix i ∈ {5, …, m}, and let the A_{i}, E_{i}, H_{i} and V_{i} be defined as in Lemma 1.4. First note that V_{i} ≤ C_{V} (A_{i}). By Lemma 1.4a) E_{i} acts faithfully (and clearly completely reducibly) on V_{i}, so by [1, Lemma 4] it has a regular orbit on V_{i}. Now for every v ∈ V_{i} that lies in a regular orbit of E_{i} on V_{i} obviously we have C_{E}(v) = A_{i}. Consequently A_{i} = E ∩ C_{G}(v) ⊴ C_{G}(v), so with M_{i} := N_{G}(A_{i}) we have C_{G}(v) ≤ M_{i}. Hence we have shown that for every v ∈ V_{i} that lies in a regular orbit of E_{i} we have C_{G}(v) = C_{M_{i}}(v) (∗).

Now clearly E_{i}A_{i} = E_{i} × A_{i} = E ∩ M_{i} ⊴ M_{i} and C_{i} ≤ M_{i} and thus C_{i} = C_{M_{i}}(E_{i}A_{i}/A_{i}) ⊴ M_{i}, and so we have N_{i} ≤ C_{i} ⊴ M_{i}.

Next let v ∈ V_{i} be an element out of a regular orbit of E_{i} on V_{i} and put S_{i} = C_{M_{i}}(v) ∩ C_{i} ⊴ C_{M_{i}}(v). We claim that S_{i} = N_{i}. Obviously N_{i} ≤ S_{i}. On the other hand we have A_{i} ≤ S_{i} ≤ C_{i} and v ∈ C_{V} (S_{i}), in particular C_{V} (S_{i}) > 0. Hence our maximal choice of N_{i} forces S_{i} = N_{i}, establishing our claim.

So now by (∗) we have N_{i} = S_{i} ⊴ C_{M_{i}}(v) = C_{G}(v), whence C_{G}(v) ≤ N_{G}(N_{i}) = H_{i}, and as v was chosen arbitrarily, now we know that for every v ∈ V_{i} that lies in a regular orbit of E_{i} on V_{i} we have C_{G}(v) = C_{H_{i}}(v) (∗∗).

Now by Lemma 1.4a), b) the action of \overline{H_{i}} := H_{i}/N_{i} on V_{i} fulfills the hypotheses of Lemma 1.3, so that Lemma 1.3 yields a v_{i} ∈ V_{i} that lies in a regular orbit of \overline{H_{i}} on V_{i}, i.e. |C\overline{H_{i}}(v_{i})| = 1. In particular, as E_{i} ≅ \overline{E_{i}}, v_{i} lies in a regular orbit of E_{i} on V_{i}, so that with (∗∗) we obtain C_{G}(v_{i}) = C_{H_{i}}(v_{i}) = C_{N_{i}}(v_{i}) = N_{i}, as desired.

Finally as N_{i} ≥ N_{i+1} and as by Lemma 1.4a) we have A_{i} ≤ N_{i} and A_{i} ≰ N_{i+1}, it readily follows that |N_{i}|_{p} > |N_{i+1}|_{p}, and the proof of a) is complete.

b) is proved analogously with the help of the results in the Lemmata 1.3 and 1.4 for the case p = 2. □

Remember that if G is a group and V a G–module such that V_{N} is homogeneous for every N ⊴ G, then V is called a quasiprimitive G–module. In this case the structure of G is well–known (cf. [10, §1]).

As a special case of Theorem 1.5 we get a result on quasiprimitive actions.

Corollary 1.6. Let G be a solvable group acting faithfully and quasiprimitively on a vector space V over a finite field. Then there is an E ⊴ G of type E(p, m) for a prime p and an integer m ≥ 0, and the following hold:

a) If p is odd and m ≥ 5, then G has at least m − 4 different nontrivial orbit sizes on V.

b) Let p = 2 and m′ be defined as in Lemma 1.4. If m′ ≥ 10, then G has at least m′ − 9 different nontrivial orbit sizes on V.

Proof. The existence of E is clear from [10, Corollary 1.10]. Moreover as Z(E) ⊴ G, V_{Z(E)} is homogeneous, and so Z(E) acts fixed point freely on V. Thus by Theorem 1.5 the assertion follows. □

2. The main results for groups of odd order.

Theorem 2.1. Let G be a (solvable) group of odd order that acts faithfully on a vector space V over a finite field K of odd characteristic q. Suppose that N ⊴ G is a p-group of class 1 or 2 and of exponent p for a prime p ≥ 5 with p ≠ q. Let A ≤ N be an abelian subgroup of N, and put s = d(A). Then G has on V at least m := ⌈\frac{s}{6}⌉ orbits, whose sizes have mutually distinct nontrivial p-parts. In particular, G has at least m different nontrivial orbit sizes on V.

Proof. We prove the result by induction on |G| + |V |.

Note that A is elementary abelian. If s ≤ 5, the assertion is trivial, so let s ≥ 6.

First suppose that V_{N} is homogeneous. Then V is a multiple of a faithful and irreducible N-module. Hence (by [3, Theorem 3.2.2]) Z(N) is cyclic, and as N′ ≤ Z(N), we see that either N′ = 1 and N is cyclic of order p or |N′| = |Z(N)| = p. In the first case our assertion is trivially fulfilled, so we only have to consider the latter. As exp(N) = p, we also have N′ = ϕ(N) (which is the Frattini subgroup of N). Consequently N is extra–special and thus of type E(p, n) for an integer n. Thus 6 ≤ s ≤ n + 1, and by Theorem 1.5 G has on V at least n − 4 ≥ s − 5 ≥ ⌈\frac{s}{6}⌉ = m orbits, whose sizes have mutually distinct nontrivial p-parts. So we are done in this case.

It remains to consider the case that V_{N} is inhomogeneous.

First assume that V is not irreducible, i.e. V = V_{1} ⊕ V_{2} for nontrivial G-modules V_{i} (i = 1, 2). Let C = C_{G}(V_{1}), s_{1} = d(AC/C) and m_{1} = ⌈\frac{s_{1}}{6} ⌉. Applying induction to the action of G/C on V_{1} yields v_{1}, …, v_{m_{1}} ∈ V_{1} such that |G|_{p} > |C_{G}(v_{1})|_{p} > ⋯ > |C_{G}(v_{m_{1}})|_{p}. Now let H = C_{G}(v_{m_{1}}), D = C_{H}(V_{2}), s_{2} = d((A ∩ H)D/D) and m_{2} = ⌈\frac{s_{2}}{6} ⌉. This time induction, applied to the action of H/D on V_{2}, yields w_{1}, …, w_{m_{2}} ∈ V_{2} such that |H|_{p} > |C_{H}(w_{1})|_{p} > ⋯ > |C_{H}(w_{m_{2}})|_{p}. Now put v_{m_{1}+i} = v_{m_{1}} +w_{i} for i = 1, …, m_{2}. Then C_{G}(v_{m_{1}+i}) = C_{G}(v_{m_{1}}) ∩ C_{G}(w_{i}) = C_{H}(w_{i}) for i = 1, …, m_{2}. Thus we obtain |G|_{p} > |C_{G}(v_{1})|_{p} > ⋯ > |C_{G}(v_{m_{1}+m_{2}})|_{p}. Now as C_{A}(V_{1}) ≤ H surely acts faithfully on V_{2}, we see that s_{1} + s_{2} ≥ s and consequently m_{1} + m_{2} ≥ ⌈\frac{s_{1}}{6} + \frac{s_{2}}{6} ⌉ ≥ ⌈\frac{s}{6}⌉ which shows that we have established our assertion.

So now we may assume that V is an irreducible imprimitive G–module. By [10, Proposition 0.2(ii)] there exists a N ≤ C ⊴ G such that V_{C} = V_{1} ⊕ ⋯ ⊕ V_{n} (with n > 1) with C-invariant V_{i} (i = 1, …, n) that are faithfully and primitively permuted by G/C.

If for any M ⊆ {V_{1}, …, V_{n}} we put stab_{G/C}(M) = {hC ∈ G/C | M^{hC} = M}, then by [13, Theorem 1.3] we may assume that

(∗)  ⋂_{j=1}^{3} stab_{G/C}({V_{j}}) = 1, i.e. ⋂_{j=1}^{3} N_{G}(V_{i}) = C.

Now H = N_{G}(V_{1}) is known to be a maximal subgroup of G, and V is induced

from the H-module V_{1}, i.e. if G = ⋃_{j=1}^{|G:H|} Hg_{j} with suitable g_{j} ∈ G (and g_{1} :=

1) is the decomposition of G into right cosets of H, then V = V_{1} ⊗_{KH} KG, and thus (by identifying V_{1} and V_{1} ⊗1) we have V_{i} = V_{1} ⊗g_{i} for i = 1, …, n.

Next observe that (by [10, Lemma 4.1]) v^{H} ≠ (−v)^{H} for all v ∈ V_{1}. Hence there is an X ⊆ V_{1} such that with Y := −X we have V_{1}\{0} = X ∪ Y and

X ∩ Y = ∅. Now we define B ⊆ V as follows: If v ∈ V and v = ∑_{i=1}^{n} x_{i} ⊗ g_{i}

with x_{i} ∈ V_{1} for all i, then v ∈ B if and only if (1) x_{i} ∈ X for i = 1, …, min{3, n} and x_{i} ∈ Y for i = 4, …, n, and

(2) whenever x_{i} = 0 for an i < n, then x_{i+1} = 0.

Now let v = ∑_{i=1}^{n} x_{i} ⊗ g_{i} ∈ B, and define S(v) := ⋂_{i=1}^{n} C_{G}(x_{i} ⊗ g_{i}). Then we

claim that S(v) ⊴ C_{G}(v) such that

(∗∗)  |C_{G}(v)/S(v)| ∈ {1, 3}.

To see this, first let j ∈ {1, …, n} be maximal subject to x_{j} ≠ 0. If j = 1, clearly C_{G}(v) = S(v), and this is also true for j = 2, as |G| is odd. So let

n ≥ j ≥ 3. Now note that for any g ∈ G we have v^{g} ∈ ⨁_{i=1}^{j} x_{i}^{H} ⊗ g_{σ(i)},

where σ ∈ S_{n} is the permutation defined by the permution action of gC on the V_{i}, i.e. V_{i}^{gC} = V_{σ(i)} for i = 1, …, n. So as X and Y are H-invariant, by (∗) and our definition of B we find that if g ∈ C_{G}(v), then gC ∈ T/C := stab_{G/C}({V_{1}, V_{2}, V_{3}}). Now by (∗) we have that T/C is isomorphic to a subgroup of S_{3}, so as |G| is odd, either T = C or |T/C| = 3. Furthermore as j ≥ 3 and as naturally C_{G}(x_{i} ⊗ g_{i}) ≤ H^{g_{i}} = N_{G}(V_{i}) for all i, (∗) also implies that S(v) ≤ C. So we conclude that S(v) = C ∩ C_{G}(v) ⊴ C_{G}(v) and further that |C_{G}(v)/S(v)| = |C_{G}(v)C/C| | |T/C| ∈ {1, 3} whence (∗∗) is shown.

Now we can construct representatives of the orbits that we are looking for. We will do this by an inductive process, namely we put v_{0} = 0 ∈ V, t_{0} = 0 and for all i = 1, …, n proceed as follows:

(∗∗∗) Suppose that we have already t_{i−1} ∈ ℕ and v_{0}, …, v_{t_{i−1}} ∈ B with |G|_{p} > |C_{G}(v_{1})|_{p} > ⋯ > |C_{G}(v_{t_{i−1}})|_{p}, then let H_{i} = C_{G}(v_{t_{i−1}}) and C_{i} = C_{H_{i}}(V_{i}). Furthermore let s_{i} = d((A ∩ H_{i})C_{i}/C_{i}) and m_{i} = ⌈\frac{s_{i}}{6} ⌉. First suppose that m_{i} ≥ 1. Apply induction to the action of N_{H_{i}}(V_{i})/C_{i} on V_{i}. As obviously C_{H_{i}}(w) ≤ N_{H_{i}}(V_{i}) for all w ∈ V_{i}, this yields elements w_{i,1}, …, w_{i,m_{i}} ∈ V_{i} such that |H_{i}|_{p} > |C_{H_{i}}(w_{i,1})|_{p} > ⋯ > |C_{H_{i}}(w_{i,m_{i}})|_{p}. If i ≤ 3, we choose the w_{i,j} (j = 1, …, m_{i}) to be out of X ⊗ g_{i}, if i ≥ 4, we take them out of Y ⊗ g_{i} (which is obviously possible). Then put v_{t_{i−1}+j} = v_{t_{i−1}} + w_{i,j} ∈ B (j = 1, …, m_{i}) and put t_{i} := t_{i−1} + m_{i}. Hence as p ≥ 5 and because of (∗∗) we conclude |C_{G}(v_{t_{i−1}+j})|_{p} = |S(v_{t_{i−1}+j})|_{p} = |S(v_{t_{i−1}}) ∩ C_{G}(w_{i,j})|_{p} = |C_{G}(v_{t_{i−1}}) ∩ C_{G}(w_{i,j})|_{p} = |C_{H_{i}}(w_{i,j})|_{p} < |H_{i}|_{p} for j = 1, …, m_{i}. Thus we have |G|_{p} > |C_{G}(v_{1})|_{p} > ⋯ > |C_{G}(v_{t_{i}})|_{p}. Now suppose that m_{i} = 0. If i ≤ 3, take an arbitrary v ∈ X ⊗ g_{i}, if i ≥ 4, then take an arbitrary v ∈ Y ⊗ g_{i}. Put t_{i} = t_{i−1} and simply replace v_{t_{i}} by v_{t_{i}} + v. Then still v_{t_{i}} ∈ B and |G|_{p} > |C_{G}(v_{1})|_{p} > ⋯ > |C_{G}(v_{t_{i}})|_{p}.

If i = n, then we are done, else we repeat the procedure (∗∗∗).

So this process yields t_{n} orbits on V the sizes of which have mutually distinct p–parts > 1. It remains to show that t_{n} ≥ m. If we show that

t := ∑_{i=1}^{n} s_{i} ≥ s, then we obtain t_{n} = ∑_{i=1}^{n} m_{i} = ∑_{i=1}^{n}⌈\frac{s_{i}}{6} ⌉ ≥ ⌈∑_{i=1}^{n} \frac{s_{i}}{6} ⌉ ≥ ⌈\frac{s}{6}⌉ = m,

as wanted.

So we have to prove t ≥ s. For this let 1 ≠ a ∈ A, and surely it suffices to show that there is an i ∈ {1, …, n} such that a ∈ H_{i} and a ∉ C_{i}, i.e. 1 ≠ aC_{i} ∈ (A ∩ H_{i})C_{i}/C_{i}. To see this note first that if a ∉ C_{1}, then we are done with i = 1, as a ∈ A ≤ N ≤ C ≤ H = H_{1}. So let a ∈ C_{1}, and let j ∈ {1, …, n} be maximal such that a ∈ C_{l} for l = 1, …, j. Surely j ≤ n−1

because else we would get the contradiction 1 ≠ a ∈ ⋂_{k=1}^{n} C_{G}(V_{k}) = 1. Then

clearly a ∈ H_{j+1} and a ∉ C_{j+1}. So we choose i := j + 1. This completes the proof of the theorem. □

Notice that Theorem 2.1 is also true for p = 3 if K is not too small, more precisely, if besides −1 there is another a ∈ K^{×} whose order does not divide |G|. Namely then in an obvious way we can define B in the proof of Theorem 2.1 even such that |C_{G}(v)/S(v)| = 1 for all v ∈ B. Details are left to the reader.

Notice furthermore that obviously with the proof of Theorem 2.1 one even can get elements w_{1}, …, w_{m} ∈ V whose centralizers have orders with increasing 3′–parts (but this is not relevant for our purposes).

In the following for any solvable group G we shall make use of subgroups K_{i}(G) (i ∈ ℕ ∪ {0}) which are defined as follows:

K_{0}(G) = G, and for i ≥ 1 and given K_{i−1}(G) let K_{i}(G) be the smallest normal subgroup of K_{i−1}(G) such that K_{i−1}(G)/K_{i}(G) is nilpotent.

Then by [5, III, Satz 4.6] we know that the K_{i}(G) are characteristic subgroups of G and that if n is the minimal number such that K_{n}(G) = 1, then n = f(G). Furthermore, by the definition of the K_{i} in [5] it is clear that for any N ⊴ G and for all i ∈ ℕ we have

K_{i}(G/N) = K_{i}(G)N/N.

Hence for any N ⊴ G with K_{n−1}(G)N/N > 1 we have f(G/N) = f(G).

Lemma 2.2. Let G be a solvable group, p a prime and H a Hall-p′-subgroup of G. Put n = f(G) and K = K_{n−1}(G). Then

dl(H) ≥ \{ \begin{matrix}\frac{f(G)}{2} & if K is not a p-group \\ \frac{f(G)−1}{2} & if K is a p-group. \end{matrix}

Proof. We prove the result by induction on |G|. It is obviously true if G is nilpotent. So let f(G) ≥ 2.

First suppose that K is a p-group. Then, as K_{n−2}(G) is not nilpotent, K_{n−2}(G/K) = K_{n−2}(G)/K is not a p-group, and thus induction yields dl(H) = dl(HK/K) ≥ \frac{f(G/K)}{2} = \frac{f(G)−1}{2}, as wanted.

Hence it remains that K is not a p-group. If p | |K|, then let P be a Sylowp-subgroup of K. As K is nilpotent, we have P ⊴ G, and as K/P > 1, we have f(G/P) = n = f(G) and K_{n−1}(G/P) = K/P is not a p-group, whence induction yields dl(H) = dl(HP/P) ≥ \frac{f(G/P)}{2} = \frac{f(G)}{2}. Hence we may assume that K is a p′-group.

Now if there is an N ⊴ G with 1 < N < K, then induction yields dl(H) ≥ dl(H/N) ≥ \frac{f(G/N)}{2} = \frac{f(G)}{2}. Thus we may assume that K is a minimal normal subgroup of G, and as such it is an elementary abelian q-group for a prime q ≠ p.

Let k = dl(H) and T = H^{(k−1)} the (k − 1)-th commutator subgroup of H, then T > 1 and T is abelian. Put C := C_{G}(K). Clearly K_{n−2}(G) ≰ C and thus K_{n−2}(G/C) = K_{n−2}(G)C/C > 1 and so f(G/C) = n − 1 = f(G)−1. Thus if T ≤ C, by induction we conclude dl(H) ≥ dl(HC/C)+1 ≥ \frac{f(G/C)−1}{2} + 1 = \frac{f(G)−2}{2} + 1 = \frac{f(G)}{2}, as desired.

Hence we may assume that \overline{T} := T C/C > 1. We will lead this to a contradiction. Now as K may be regarded as a faithful and irreducible G/Cmodule over GF (q), we have that \overline{T} is a q′-group of automorphisms of K (cf. [5, V, Satz 5.17]). Hence by [3, Theorem 5.3.6] for 1 < R := [\overline{T} , K] ≤ K we have [\overline{T} , R] = R. Going back to G, this obviously means that R = [T, K] and [T, R] = R (∗).

Next we claim that R ≤ H^{(i)} for i = 0, …, k − 1. For i = 0 this is trivial, and if we already have R ≤ H^{(i)} for an i ∈ {0, …, k − 2}, then, as also T ≤ H^{(i)}, by (∗) we have H^{(i+1)} ≥ [T, R] = R, which proves our claim. Hence we have R ≤ H^{(k−1)} = T whence (by (∗)) we get the contradiction 1 = [T, T] ≥ [T, R] = R > 1. We are done. □

Theorem 2.3. Let G be a (solvable) group of odd order that acts faithfully and irreducibly on a vector space V over a finite field of odd characteristic q. Put n = f(G) and let m be the number of different orbit sizes of G on V. Assume further that K := K_{n−1}(G) is not a 3-group. Then

f(G) ≤ 8 log_{2} m + 30.

Proof. We may assume that n ≥ 2. Let p ≥ 5 be a prime dividing |K|. Then p ≠ q (by [5, V, Satz 5.17]). Let P ∈ Syl_{p}(K) and put C = C_{G}(P). Then P ⊴G and K_{n−2}(G) ≰ C, because else K_{n−2}(G)/O_{p′}(K) would be nilpotent contradicting the definition of K. Hence we have f(G/C) ≥ f(G) − 1. Now let H be a Hall-p′-subgroup of G/C. Then H acts faithfully on P with (|H|, |P |) = 1. Hence by [3, Theorem 5.3.13] there is a characteristic subgroup N of P of class at most 2 and of exponent p such that H acts faithfully on N. Note that N ⊴ G. By [3, Theorem 5.1.4] H acts also faithfully on N/ϕ(N) (which is of order p^{d(N)}) whence by [10, Corollary 3.12(b)] we have dl(H) ≤ 2 log_{2} d(N) + 2. Now by Lemma 2.2 we know that dl(H) ≥ \frac{f(G/C)−1}{2} ≥ \frac{f(G)}{2} − 1, so that we obtain f(G) ≤ 2dl(H) + 2 ≤ 4 log_{2} d(N) + 6. As p is odd, by [5, III, Satz 12.3] there is an abelian A ⊴ N with d(N) ≤ \frac{d(A)(d(A)+1)}{2} ≤ d(A)^{2}, so that f(G) ≤ 8 log_{2} d(A) + 6. By Theorem 2.1 we have m ≥ \frac{d(A)}{6}, whence we finally conclude f(G) ≤ 8 log_{2}(6m) + 6 ≤ 8 log_{2} m + 30, as desired. □

Actually we have not tried to optimize the constants in Theorem 2.3.

Corollary 2.4. Let G be a group of odd order. Then

f(G) ≤ 8 log_{2} |cd(G)| + 32.

Proof. Put n = f(G), K = K_{n−1}(G) and (in case of n ≥ 2) L = K_{n−2}(G) and F = F (L), the Fitting subgroup of L. First we show:

(∗) If n ≤ 3 or if n ≥ 4 and L/F is not a 3-group, then f(G) ≤ 8 log_{2} |cd(G)| + 31.

To prove (∗), we argue by induction on |G|. For n ≤ 3 there is nothing to prove. Let n ≥ 4 and let p ≥ 5 be a prime dividing |L/F |, and let P ∈ Syl_{p}(L). Surely P F/F > 1, and there is a prime q ≠ p such that for Q_{0} ∈ Syl_{q}(K) we have [P, Q_{0}] > 1 because else P would centralize O_{p′}(K) and thus P × O_{p′}(K) = P K ≤ F against our choice of P. Clearly Q_{0} ⊴ G, and with [3, Theorem 5.3.6] we conclude that for Q := [P, Q_{0}] ≤ Q_{0} we have [P, Q] = Q > 1.

Now if there is a 1 < N ⊴ G such that QN/N > 1, then surely we have f(G/N) = f(G) and QN/N ≤ Q_{0}N/N ≤ F (LN/N) = F (K_{n−2}(G/N)), and as [P N/N, QN/N] = [P, Q]N/N = QN/N > 1, it follows that 1 < P N/N ≤ LN/N = K_{n−2}(G/N) and P N/N ≰ F (K_{n−2}(G/N)). Consequently we see that K_{n−2}(G/N)/F (K_{n−2}(G/N)) is not a 3–group whence induction yields f(G) = f(G/N) ≤ 8 log_{2} |cd(G/N)| + 31 ≤ 8 log_{2} |cd(G)| + 31, as desired. Hence we may assume that no such N exists (+); in particular, O_{q′}(G) = 1, thus K = Q_{0} and F (G) is a q-group. Now as P K/K ≤ F (G/K) and F (G)/K ≤ F (G/K), we see that P F (G)/K is nilpotent and hence 1 = [P K/K, F (G)/K] = [P K, F (G)]K/K and therefore [P, F (G)] ≤ K. Thus with [3, Theorem 5.3.6] we see that Q ≤ [P, F (G)] = [P, [P, F (G)]] ≤ [P, K] = Q and hence 1 < Q = [P, F (G)] ≰ ϕ(F (G)) (by [3, Theorem 5.3.5]). Thus Qϕ(F (G))/ϕ(F (G)) > 1, and by (+) we conclude that ϕ(F (G)) = 1, i.e., F (G) is elementary abelian, and now with (+) we easily see that F (G) = K = Q is a minimal normal subgroup of G.

Hence by [5, III, Hilfssatz 4.4] let H be a complement of K in G; then K is an irreducible and faithful H-module. Now let V = Irr(K). By [10, Proposition 12.1] V is a faithful and irreducible H–module. As by [7, Problem (6.18)] every λ ∈ V can be extended to its inertia group in G, we conclude by standard character theory that every orbit size in the action of H on V is an irreducible complex character degree of G = HK. So if m is the number of different orbit sizes of H on V, then we have m ≤ |cd(G)|. Now as K_{n−2}(H) ≅ K_{n−2}(G)/K = K_{n−2}(G)/F (K_{n−2}(G)) is not a 3–group and f(H) = n − 1, by Theorem 2.3 we have f(H) ≤ 8 log_{2} m + 30 ≤ 8 log_{2} |cd(G)| + 30, and we get f(G) ≤ 8 log_{2} |cd(G)| + 31. So (∗) is shown.

To prove the corollary, note first that by (∗) we may assume that n ≥ 4 and that L/F is a 3–group. Let \overline{G} = G/F. Then f(\overline{G}) = n − 1, and surely K_{n−3}(\overline{G})/F (K_{n−3}(\overline{G})) ≅ (K_{n−3}(G)/F)/F (K_{n−3}(G)/F) is not a 3group, because else K_{n−3}(\overline{G})/O_{3′}(K_{n−3}(\overline{G})) would be a 3-group, hence K_{n−3}(\overline{G}) would be isomorphic to a subgroup of K_{n−3}(\overline{G})/O_{3′}(K_{n−3}(\overline{G})) × K_{n−3}(\overline{G})/K_{n−2}(\overline{G}) and as such be nilpotent against f(\overline{G}) = n − 1. Hence application of (∗) to G/F yields f(G) = f(G/F) + 1 ≤ 8 log_{2} |cd(G/F)| + 31 + 1 ≤ 8 log_{2} |cd(G)| + 32, and we are done. □

The result in Corollary 2.4 is asymptotically best possible, as the following trivial example shows (which also can be used to see that Theorem 2.3 is asymptotically best possible, too).

Example 2.5. Let A, B be groups of order 3 and 5, respectively. For i ∈ ℕ we let Z_{i} = A if i is even, and Z_{i} = B if i is odd, and for n ∈ ℕ we define G_{n} = Z_{1} ≀ ⋯ ≀ Z_{n} to be iterated (regular) wreath products of the Z_{i}. Then obviously f(G_{n}) = n, and if t is the number of divisors of |G_{n}|, then clearly |cd(G_{n})| ≤ t. Now if B_{n} = B ≀ ⋯ ≀ B (with n factors), then it is well–known that |B_{n}| = 5^{s} with s = ∑_{i=0}^{n−1} 5^{i} = \frac{1}{4}(5^{n} − 1) ≤ 5^{n} − 1.

Now |G_{n}| ≤ |B_{n}| ≤ 5^{5^{n}−1} and thus obviously |G_{n}| | (3 · 5)^{5^{n}−1}. This implies that t ≤ 5^{2n}. Thus we have

f(G_{n}) = n ≥ \frac{1}{2} log_{5} t ≥ \frac{1}{2} log_{5} |cd(G_{n})|,

whence asymptotically Corollary 2.4 is the right bound.

3. An application to modular character theory.

In this section we want to discuss briefly a modification of results obtained by Isaacs in [8] on orbit sizes and modular character degrees. Because of the hypotheses in Theorem 2.1, we have to restrict ourselves to groups of odd order and to primes greater than 3. In this case, for groups whose p-Sylow subgroups have bounded exponent, we can asymptotically improve the bounds obtained in [8] considerably. The reason why we need that bounded exponent is that with Theorem 2.1 we only get many different orbit sizes of a p-group P acting on a vector space, if d(A) is large for some abelian subgroup of P. But, as well-known, there are p-groups of arbitrarily large derived length all of whose abelian normal subgroups are generated by only three elements (cf. [4]).

Theorem 3.1. Let G be a (solvable) group of odd order and p ≥ 5 be a prime. Suppose that G acts on an abelian p′-group V of odd order, and let b be the number of different p-parts among the orbit sizes of G on V. Let P = O_{p}(G) and exp(P) = p^{e}. Then

dl(P/C_{P} (V)) ≤ 2 log_{2} b + 2 log_{2} e + 8.

Proof. As in the proof of [8, Theorem (2.1)] we see that we may assume that V is elementary abelian and a faithful and irreducible G–module. Now let k = dl(P). If P is abelian, there is nothing to prove, hence let k ≥ 2.

Now let P^{(k−1)} ≤ D be a maximal characteristic abelian subgroup of P^{(k−2)}. Then we define subgroups C, A, M of G as follows:

If D is also a maximal normal abelian subgroup of P^{(k−2)}, then let C = D = M and A = Ω_{1}(M) ≤ C; in particular d(A) = d(M).

If not, then let D < M ≤ P^{(k−2)} be a maximal normal abelian subgroup of P^{(k−2)}. Thus M ≤ C_{P^{(k−2)}}(D) =: H. By the choice of D clearly H/D > 1 is abelian, so let D ≤ C ≤ P^{(k−2)} such that C/D = H/D ∩ Ω_{1}(P^{(k−2)}/D) = Ω_{1}(H/D). Then Ω_{1}(M/D) ≤ C/D and hence obviously A := Ω_{1}(M) ≤ C and d(A) = d(M).

This shows that in any case there is a subgroup C of P^{(k−2)} such that Ω_{1}(C) contains an abelian subgroup A with d(A) = d(M) for some maximal abelian normal subgroup M of P^{(k−2)}. Now observe that by our construction of C and the proof of [3, Theorem 5.3.12] we see that C is a critical subgroup of P^{(k−2)} in the sense of [3]; so by the proof of [3, Theorem 5.3.13] we know that Ω_{1}(C)⊴G is a p-group of class 1 or 2 and of exponent p which contains the abelian subgroup A. Hence we may apply 2.1 to the action of G on V which yields that

b ≥ \frac{1}{6} d(A).

Now by the proof of [5, III, Satz 7.11] we have |P^{(k−2)}| = p^{n} with n ≥ 2^{k−2} +2, so that with [5, III, Satz 7.3b)] we have |M| = p^{a} with 2a^{2} ≥ a(a+ 1) ≥ 2n ≥ 2^{k−1} + 4 whence we conclude that a ≥ 2^{(k−2)/2} and consequently d(A) ≥ a/e ≥ e^{−1} 2^{(k−2)/2}. Thus altogether we have

b ≥ \frac{1}{6e} 2^{\frac{k−2}{2}}

from which we easily deduce the assertion of the theorem. □

The following corollary is the counterpart of Theorem B in [8].

Corollary 3.2. Let p ≥ 5 be a prime and P be a p-group that acts faithfully on an abelian p′-group V of odd order. Let exp(P) = p^{e} and b be the number of different orbit sizes in the action of P on V. Then

dl(P) ≤ 2 log_{2} b + 2 log_{2} e + 8.

Proof. This follows from Theorem 3.1 with G = P. □

Now we can also derive a counterpart of Theorem A of [8].

Theorem 3.3. Let G be a (solvable) group of odd order and p ≥ 5 be a prime. Let cd_{p}(G) denote the set of the degrees of the irreducible Brauer characters at the prime p. Put c = |cd_{p}(G)| and let p^{e} be the exponent of a p–Sylow subgroup of G. Then

dl(G/O_{p}(G)) ≤ 3 (c − 1) (2 log_{2} c + 2 log_{2} e + 8) + 1.

Proof. The proof runs completely the same as the proof of Theorem A in [8], i.e. first one proves a counterpart of [8, Theorem (3.2)] by using Theorem 3.1 instead of [8, Theorem (2.1)], and then one completes the proof with the help of [8, Theorem (3.3)]. □

Acknowledgments. This paper was written while I had a visiting position and later was a Research Fellow, supported by a DFG-fellowship, at Ohio University. I would like to thank the Department of Mathematics for its hospitality and T.R. Wolf, G. Navarro and D.S. Passman for stimulating discussions on the subject. I am also grateful to the Deutsche Forschungsgemeinschaft for granting me the research fellowship.

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Received August 27, 1997.

University of Leicester Leicester, LE1 7RH, England E-mail address: tmk99@mcs.le.ac.uk This paper is available via http://nyjm.albany.edu:8000/PacJ/1999/187-2-8.html.