Regular orbits and the k(GV)-problem

Martin W. Liebeck

January 16, 1997

An old question of Richard Brauer from the 1950’s asks whether it is true that if B is a block of a finite group, with defect group D, then the number of ordinary irreducible characters in B is at most |D|. By results of Fong [Fo] and Nagao [Na], for p-solvable groups Brauer’s question reduces to the following:

k(GV)-problem. Let p be a prime, G a finite p′-group, and V a faithful finite-dimensional 𝔽_{p}G-module. Denote by k(GV) the number of conjugacy classes of the semidirect product GV. Show that k(GV) ≤ |V |.

Note that equality can be achieved, for any prime p and any dimension n: for example, let q = p^{n} and regard V = 𝔽_{q} as an n-dimensional space over 𝔽_{p}. Let G = 𝔽_{q}^{∗}, with natural scalar action on 𝔽_{q}. Then k(GV) = k(G) + 1 = q = |V |. Observe also that an affirmative answer to the problem would imply that any p′-subgroup of GL_{n}(p) has at most p^{n} − 1 conjugacy classes, which seems far from obvious (at least to me).

A number of authors have worked on the k(GV)-problem, particular highlights being its solution in the following cases:

(1) if G is supersolvable or |G| is odd [Kn1, Gl];

(2) if V ≅ V^{∗} as 𝔽_{p}G-modules [Go];

(3) if there exists v ∈ V such that V ↓ C_{G}(v) is a permutation module [Kn2].

Nevertheless the problem has proved rather stubborn, and it is only recently that it has been solved, for large primes p, by Robinson and Thompson [RT].

We discuss briefly the approach used in [RT]. First, character theory is used to solve the problem in the following case, extending (2) and (3) above:

(4) if there exists v ∈ V such that V ↓ C_{G}(v) has a faithful self-dual submodule.

In particular, this will be satisfied if there exists v ∈ V such that C_{G}(v) = 1, in which case the orbit v^{G} is a regular orbit of G on V.

After this, Robinson and Thompson reduce the proof of an affirmative solution of the k(GV)-problem to the following two situations:

(A) G has a normal irreducible q-subgroup Q of symplectic type (q prime, q ≠ p): that is, all characteristic abelian subgroups of Q are central. (Such q-groups are determined by a well known result of Philip Hall.)

(B) G has a normal irreducible quasisimple subgroup H.

In case (A) it is proved in [RT, Theorem 8] that, provided p > 2^{27}, such a group G has a regular orbit on V, solving the problem by (4) above.

Now consider case (B). One might hope that here too, G always has a regular orbit provided p is sufficiently large. However, this is not the case, as the following example shows.

Example Let H = A_{c}, and let p be a prime with p > c. Define

V = {(α_{1}, …, α_{c}) : α_{i} ∈ 𝔽_{p}, ∑ α_{i} = 0}

and take S_{c} to act on V by permuting coordinates. As is well known, V is an irreducible 𝔽_{p}S_{c}-module; we call V the deleted permutation module for S_{c} over 𝔽_{p}.

Now suppose that p = c + 1, and take G = ⟨−1⟩ × S_{c}. We claim that G has no regular orbit on V. To see this, let v = (α_{1}, …, α_{c}) ∈ V. If α_{i} = α_{j} for some i ≠ j, then (i j) ∈ C_{G}(v). Otherwise, {α_{1}, …, α_{c}} = 𝔽_{p}\{α} for some α, and so ∑ α_{i} = (∑_{β∈𝔽_{p}} β) − α = −α, whence α = 0. Reordering coordinates, we can therefore take

v = (1, −1, 2, −2, …, \frac{p − 1}{2}, \frac{p + 1}{2}).

Defining x = (1 2)(3 4) …, we then have vx = −v, so −1.x ∈ C_{G}(v). Consequently C_{G}(v) ≠ 1 for all v ∈ V, as claimed.

The following theorem, taken from [Li], shows that, at least for large primes, the above examples are the only obstacles to a regular orbit.

Theorem Assume p > 5^{30}, and let G be a p′-subgroup of GL_{n}(p) containing a normal irreducible quasisimple subgroup H. Then one of the following holds:

(i) G has a regular orbit on V;

(ii) H = A_{n+1}, p > n + 1, and V_{n}(p) is the deleted permutation module for H as described above.

Remark Thompson (private communication) has shown that the only cases in conclusion (ii) in which the group 𝔽_{p}^{∗} ×S_{n+1} has no regular orbit are those with n + 2 ≤ p ≤ n + 4.

It follows from the Theorem that for p > 5^{30}, the vector v required for the application of (4) above exists in case (B) also. Thus the k(GV)-problem is solved for primes p > 5^{30}.

It should be pointed out that the proof of the Theorem relies on the classification of finite simple groups, and hence so does the solution of the k(GV)-problem.

Recently, Robinson [Ro] has taken further the work on case (A), and has shown that the required vector v exists, assuming only that p > 157; and D. Goodwin, in his PhD work at Imperial College, London, has done the same for case (B). Consequently the k(GV)-problem is now in fact solved for primes p > 157.

We close with a few words on the proof of the above Theorem in [Li]. Suppose G is as in the hypothesis, and G has no regular orbit on V = V_{n}(p). Then C_{G}(v) ≠ 1 for all v ∈ V, whence V = ⋃_{g∈G^{♯}} C_{V} (g) (where G^{♯} = G\{1}). Thus

(1) p^{n} = |V | ≤ ∑_{g∈G^{♯}} |C_{V} (g)|.

To illustrate the argument for groups of Lie type, consider the case where G = H is quasisimple, and G/Z(G) ≅ L_{d}(r). Observe

(2) |G| < r^{d^{2}},

(3) n ≥ r^{d−1} − 1 by [LS].

For g ∈ G\Z(G), let r(g) be the minimum number of conjugates of g required to generate G, and define r(G) to be the maximal value of r(g). By [HLS, Theorem 2], we have r(G) ≤ 8(2d − 1). Therefore (4) dim C_{V} (g) ≤ n(1 − \frac{1}{8(2d−1)}) for all g ∈ G\Z(G).

By (1) and (4), we have

p^{n} < |G|p^{n(1−1/(8(2d−1)))}.

Hence, using (2) and (3), we obtain

p < |G|^{8(2d−1)/n} < r^{d^{2}.8(2d−1)/(r^{d−1}−1)}.

It is readily checked that the right hand side of this inequality is less than 5^{30}, contradicting the hypothesis that p > 5^{30}.

Similar arguments handle the other groups of Lie type (although in general matters are complicated by the fact that one only has H ◁ G rather than H = G). The sporadic groups succumb easily, in view of the hypothesis p > 5^{30}. The alternating groups require a somewhat more delicate argument.

Finally, we offer some remarks on the above-mentioned improvement of the Theorem by D. Goodwin. For his proof, Goodwin obtains some new bounds for the numbers r(G) defined above, when G is sporadic; and for G of Lie type he uses a recent improvement [GS] of the bounds for r(G) given by [HLS].

References

[Fo] P. Fong, “On the characters of p-solvable groups”, Trans. Amer. Math. Soc. 98 (1961), 263-284.

[Gl] D. Gluck, “On the k(GV)-problem”, J. Algebra 89 (1984), 46-55.

[Go] R. Gow, “On the number of characteris in a block and the k(GV)- problem for self-dual V ”, J. London Math. Soc. 48 (1993), 441-451.

[GS] R. Guralnick and J. Saxl, “Generation of classical groups by conjugates”, to appear.

[HLS] J. Hall, M.W. Liebeck and G.M. Seitz, “Generators for finite simple groups, with applications to linear groups”, Quart. J. Math. 43 (1992), 441-458.

[Kn1] R. Knorr, “On the number of characters in a block of a p-solvable group”, Illinois J. Math. 28 (1984), 181-210.

[Kn2] R. Knorr, “On Brauer’s k(B)-conjecture”, J. Algebra 131 (1990), 444-450.

[LS] V. Landazuri and G.M. Seitz, “On the minimal degrees of projective representations of the finite Chevalley groups”, J. Algebra 32 (1974), 418–443.

[Li] M.W. Liebeck, “Regular orbits of linear groups”, J. Algebra 184 (1996), 1136-1142.

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

[Ro] G.R. Robinson, “Further reductions for the k(GV)-problem”, preprint, Univ. of Leicester.

[RT] G.R. Robinson and J.G. Thompson, “On Brauer’s k(B)-problem”, J. Algebra 184 (1996), 1143-1160.

Dept. of Mathematics, Imperial College, London SW7 2BZ, UK