On a minimal counterexample to Dade’s Projective Conjecture Charles W. Eaton and Geoffrey R. Robinson ^{1} School of Mathematics and Statistics University of Birmingham Edgbaston Birmingham B15 2TT England

INTRODUCTION: In this paper, we show that the cancellation theorem proved by the second author in [4] can be used in conjunction with some of the Clifford theory for blocks developed by E.C. Dade in [1] to further restrict the structure of a minimal counterexample to Dade’s Projective Conjecture ( henceforth referred to as DPC, see [1]). One of the points of the approach we adopt here is that we wish to pursue as far as possible the reduction of the ( relatively easily stated) basic form of DPC to a group as close to simple as possible. In the announcements made so far in Dade’s own programme of reductions, it appears that the conjectures involved have become increasingly complex. We feel therefore that the direct approach mentioned above has some value. This is not just for aesthetic reasons, but also for the convenience of those who wish to contribute to the programme of verifying the conjectures, so that :

a) They will know precisely what would be required to be checked to complete an inductive proof of the conjecture.

b) The simplest possible form of the conjecture is left to be verified.

We prove here that a (putative) minimal counterexample G to DPC has a unique conjugacy class of components ( a component being a quasi-simple subnormal subgroup). It already follows from the main theorem of [3] that such a G has F (G) ≤ Z(G). The ultimate objective of any programme of Clifford-theoretic reductions is to reduce to the case where a purported counterexample G has F^{∗}(G)/Z(G) a non-Abelian simple group. From the point of view of the logical structure of performing such a reduction, it seems that our main result should afford a simplification, since, even if it were to prove necessary to make stronger assumptions for inductive reasons to complete the final step of the reduction, it would not be necessary to prove the stronger statements for all finite groups: at worst, it would only be necessary to prove such stronger statements for finite groups H with F (H) ≤ Z(H) and E(H) a central product of a single conjugacy class of components.

1This research is part of ARC project 1166

The notation used here will be that established in Section 1 of [3]. Our main Theorem here is :

THEOREM 1: Suppose that the formula appearing in DPC fails to hold ( for some defect d and some linear character, λ, of the central subgroup Z = O_{p}(G)) for the block B of RG, and that first [G : Z(G)], then |G|, have been minimized subject to such a failure occurring. Then whenever M is a non-central normal subgroup of G containing Z(G), we have C_{G}(M) = Z(G) and F^{∗}(M) > F (M). Equivalently, G has central Fitting subgroup and a unique conjugacy class of components. REMARK: We note that the choice of G forces λ to be a faithful linear character of Z ( otherwise, DPC would still fail for a block of RG/kerλ).

SECTION 1 : DADE CORRESPONDENCE

Let us consider a finite group X with a normal subgroup Y and an irreducible character ζ of Y. Let I be the inertial subgroup of ζ in X. Let B be a block of RX. Let Irr(B, ζ) denote the set of irreducible characters in B which lie over (X-conjugates of) ζ, and assume that Irr(B, ζ) is non-empty. Then each element of Irr(B, ζ) is induced from a (unique) irreducible character of Irr(I, ζ).

Let β be an irreducible character which induces to an irreducible character in Irr(B, ζ). Let B′ be the block of RI containing ζ. Then (as is well-known) all irreducible characters in B′ which lie over ζ induce to irreducible characters in Irr(B, ζ). For let γ = Ind_{I}^{X}(β). Then a routine ( and instructive) computation shows that for each a ∈ X, we have γ(a^{X})/γ(1) = β(a^{X} ∩ I)/β(1) ( where, by a slight abuse, we let a^{X} denote the sum in RX of the elements of the conjugacy class of a in X).It follows that characters in Irr(I, ζ) which lie in the same p-block induce to irreducible characters in the same p-block of X.

Now let us consider the case that X = I, that is to say, ζ is X-stable. By standard Clifford Theory, there is a finite central extension X̃ of X/Y having a cyclic central subgroup W̃ , which in turn has a linear character λ̃ such that there is a bijection between Irr(X, ζ) and Irr(X̃, λ̃). Furthermore, this bijection may be explicitly realised as follows : there is a finite central extension X̂ of X, having a cyclic central subgroup Ŵ and a normal subgroup Ŷ ( isomorphic to Y and canonically identifiable with Y) such that Ŷ ∩ Ŵ = 1X̂. The irreducible character of Ŷ canonically identified with ζ may be extended to an irreducible character of X̂, denoted by ζ̂. Then ζ̂ lies over a unique (faithful) linear character of Ŵ , say μ̂. The group X̃ is X̂/Ŷ , which has a cyclic central subgroup W̃ , canonically isomorphic to Ŵ ( in fact the image of Ŵ under the natural homomorphism). The linear character λ̃ is the complex conjugate of the linear character of W̃ canonically identified with μ̂. Given an irreducible character α of X̃ which lies over λ̃, we may regard it as an irreducible character of X̂ which lies over ¯ μ̂, and then α̂ = αζ̂ may be viewed as an irreducible character of X which lies over ζ. All elements of Irr(X, ζ) arise uniquely in this fashion.

We now describe briefly how Dade correspondent blocks of X̃ may be constructed. Strictly speaking, we are usually interested not in the blocks themselves, but in those irreducible characters in them which lie over the linear character ResW̃ ; O_{p}(W̃)(λ̃) of O_{p}(W̃). This reconciles our usage with the use of blocks of twisted group algebras in Dade [1]. Once this fact is noted, what follows below is just a more elementary description of the constructions in Dade [1].

Let αζ̂ be as above. Choose an element a ∈ X̂ whose image in X̃ is denoted by ã. Let Ĥ(a) be the full pre-image in X̂ of CX̃(ã). Notice that Ĥ(a) contains Ŷ Ŵ, so that ζ̂ restricts to an irreducible character of Ĥ(a). In particular, \frac{[Ĥ(a):CX̂(a)]ζ̂(a)}{ζ̂(1)} is an element of R. We note that the Ŷ -conjugacy class sum of a is already represented by a scalar matrix in the representation affording ζ̂. It follows that \frac{[Ĥ(a):CX̂(a)]ζ̂(a)}{ζ̂(1)} is an element of J(R) unless Ŷ CX̂(a) contains a Sylow p-subgroup of Ĥ(a).

Notice that we now have

\frac{[X̂ : CX̂(a)]α̂(a)}{α̂(1)} = \frac{[X̃ : CX̃(ã)]α(ã)}{α(1)} \frac{[Ĥ(a) : CX̂(a)]ζ̂(a)}{ζ̂(1)}.

It follows from this equation that if α and β are irreducible characters of Irr(X̃, λ) which belong to the same p-block, then the corresponding irreducible characters α̂ and β̂ both lie in the same p-block of X and lie over ζ. ( The observant reader might be concerned by our passage from X̂ to X. If the element a is conjugate to az for a non-identity element z of Ŵ , then ζ̂(a) = 0, so that α̂(a) = 0. Otherwise, a has the same number of conjugates in X̂ as its image does in X. In either case, the left-hand side of the equation above tells us how to evaluate the corresponding central character of RX).

Given the block B of RX such that Irr(B, ζ) is non-empty, consider the set of all irreducible characters α ∈ Irr(X̃, λ̃) such that α̂ ∈ Irr(B, ζ). This forms the set of irreducible characters lying over ResW̃ ; O_{p}(W̃)(λ̃) in a sum of blocks of RX̃ lying over ResW̃ ; O_{p′} (W̃)(λ̃). We call the blocks which occur in this sum the set of Dade correspondents of B associated to ζ. The elements of Irr(B, ζ) are in bijection with the irreducible characters lying over ResW̃ ; O_{p}(W̃)(λ̃) in the Dade correspondents of B associated to ζ. The elements of defect d in the former set are in bijection with the elements of defect d̃ in the latter set, where d̃= d + log_{p}(ζ(1)_{p}|W̃ |_{p}/|Y |_{p}). Each block of RX̃ which lies over ResW̃ ; O_{p′} (W̃)(λ̃) occurs as the Dade correspondent of a unique block of RX.

We give an elementary proof that Dade correspondence commutes with Brauer correspondence in the situation that we consider. This may also be deduced from 14.3 of [1]. As usual, if χ is an irreducible character of a finite group M, we let ω_{χ} denote the central character of RM defined by ω_{χ}(c) = \frac{χ(c)}{χ(1)} for all c ∈ Z(RM). We let ω_{B} denote the central character of F M associated to the block B of RM, which may be computed by calculating the residue ( mod J(R)) of ω_{χ} for any χ ∈ Irr(B). For ease of notation, we retain the abuse introduced earlier that a^{M} denotes the class sum of a ∈ M in RM. When L is a subgroup of M, we will let π_{L} denote the projection from RM onto RL with kernel R[M\L].

Let T be a subgroup of X which contains Y. Then, still using ζ, we may repeat the earlier arguments with T in place of X, defining T̂ and T̃ in the obvious way. Let b̃ be a block of RT̃ with Irr(b̃, λ̃) nonempty, and suppose that the Brauer correspondent b̃X̃ = B̃ is defined. Let b and B be the Dade correspondents of b̃ and B̃ respectively. To show that Dade correspondence commutes with Brauer correspondence in this situation, it suffices to demonstrate that the Brauer correspondent of b for X is defined and is B.

Let γ ∈ Irr(b̃, λ̃) and α ∈ Irr(B̃, λ̃). Choose a ∈ X̂. We must show that ωb̂ ∘πT̂(aX̂) ≡ ωB̂(aX̂) ( mod J(R)).

An easy computation shows that ( with congruences taken (mod J(R))),

ωb̂ ∘ πT̂(aX̂) ≡ ωγ̂ ∘ πT̂(aX̂) = \frac{IndX̂ ; T̂(γ̂)[aX̂]}{IndX̂ ; T̂(γ̂)[1]}

= \frac{IndX̂ ; T̂(γζ̂)[aX̂]}{IndX̂ ; T̂(γζ̂)[1]} = [X̂ : CX̂(a)]\frac{IndX̂ ; T̂(γζ̂)[a]}{IndX̂ ; T̂(γζ̂)[1]}.

This is equal to

\frac{1}{[X̃ : CX̃(ã)]}.\frac{[X̂ : CX̂(a)]ζ̂(a)}{ζ̂(1)}.\frac{IndX̃ ; T̃(γ)[ãX̃]}{IndX̃ ; T̃(γ)[1]}(∗).

We recall that Ĥ(a) is the full inverse image in X̂ of CX̃(ã), so that we have

[Ĥ(a) : CX̂(a)] = \frac{[X̂ : CX̂(a)]}{[X̃ : CX̃(ã)]}.

Furthermore, we recall that \frac{[Ĥ(a):CX̂(a)]ζ̂(a)}{ζ̂(1)} ∈ R.

Since γ ∈ b̃, we now see that (∗) is congruent (mod J(R)) to

\frac{[Ĥ(a) : CX̂(a)]ζ̂(a)}{ζ̂(1)}.ωb̃ ∘ πT̃(ãX̃)(∗∗).

Since b̃X̃ = B̃, we see that (**) is congruent ( mod J(R)) to

\frac{[Ĥ(a) : CX̂(a)]ζ̂(a)}{ζ̂(1)}ωB̃(ãX̃) ≡ \frac{[Ĥ(a) : CX̂(a)]ζ̂(a)}{ζ̂(1)}ω_{α}(ãX̃).

This last expression is equal to ωα̂(aX̂), as we have already seen. We now have ωb̂ ∘ πT̂(aX̂) ≡ ωB̂(aX̂) ( mod J(R)), which suffices to show that b^{X} = B.

SECTION 2: THE CENTRALIZER OF A NORMAL SUBGROUP

Let G be our purported minimal counterexample to DPC. We know from [3] that F (G) ≤ Z(G). Let M be the full pre-image in G of a minimal normal subgroup of G/Z(G). Then M = Z(G)E, where E is a central product of a single G-conjugacy class of components of G ( and E = M′). Let N = C_{G}(E). Then N is a normal subgroup of G such that M ∩ N = Z(G).

Let B be a block of RG for which the equality predicted by DP C fails to hold ( with respect to the linear character λ of the central p-subgroup Z = O_{p}(G), for some defect d). We suppose for the moment that N > Z(G). We will show presently that this assumption leads to a contradiction. By theorem 1 of [4], we know that B covers a G-stable block of RM, say b, whose defect group strictly contains Z.

Also from Theorem 1 of [4], we know that

∑_{σ∈N (M,Z)/G}(−1)^{|σ|}k_{d}(B_{σ}, λ) ≠ 0

( using our assumption of failure of DPC). Notice that N ≤ G_{σ} for each σ ∈ N (M, Z). For ζ ∈ Irr(N, λ), we count the contribution to the above alternating sum from irreducible characters of defect d which lie over G-conjugates of ζ. Let ζ^{G} denote the G-orbit of ζ.

Then we may choose ζ so that

∑_{σ∈N (M,Z)/G}(−1)^{|σ|} ∑_{γ∈ζ^{G}/G_{σ}} k_{d}(B_{σ}, γ) ≠ 0.

Notice that in the double sum, we are really summing over G-orbits of ordered pairs (σ, γ), where σ ∈ N (M, Z) and γ is a G-conjugate of ζ. We may change the order of summation to obtain

∑_{σ∈N (M,Z)/I_{G}(ζ)}(−1)^{|σ|}k_{d}(B_{σ}, ζ) ≠ 0.

Let B′ denote the sum of blocks of RI_{G}(ζ) whose irreducible characters lying over ζ induce to characters in B. Then the central character of F G associated to B may be evaluated at a class sum by deleting elements outside I_{G}(ζ), then applying any of the central characters of F I_{G}(ζ) associated to B′. This central character may also be evaluated at a class sum for each σ ∈ N (M, Z) by the usual Brauer construction of deleting terms outside G_{σ}, then applying any of the central characters associated to B_{σ}. The same argument works inside I_{G}(ζ), so we see that

∑_{σ∈N (M,Z)/I_{G}(ζ)}(−1)^{|σ|}k_{d}(B_{σ}, ζ) = ∑_{σ∈N (M,Z)/I_{G}(ζ)}(−1)^{|σ|}k_{d}(B′_{σ}, ζ) ≠ 0

( it may be helpful to recall that 1_{B_{σ}} = Br_{V^{σ}}(1_{B}), and that induction of irreducible characters is defect preserving).

There must be a block b′ which is an indecomposable summand of B′ such that

∑_{σ∈N (M,Z)/I_{G}(ζ)}(−1)^{|σ|}k_{d}(b′_{σ}, ζ) ≠ 0.

It remains to show that we may replace b′ by the sum of the Dade correspondent blocks associated to ζ of an appropriate central extension of I_{G}(μ)/N and obtain a similar non-vanishing alternating sum. Once this is done, we will easily obtain a contradiction. We will be more precise later, but letting Ĩ denote this central extension, the non-vanishing of the alternating sum means that O_{p}(Ĩ) ≤ Z(Ĩ), and that there is a block of Ĩ such that DPC fails (for some defect and some linear character of some central p-subgroup of Ĩ). On the other hand, we certainly have [Ĩ: Z(Ĩ)] < [G : Z(G)], contrary to the minimality of G.

Let I denote I_{G}(μ). As before, we construct an appropriate central extension of I, say Î by a cyclic central subgroup Ŵ so that ζ extends to an irreducible character ζ̂ of Î. Now N = C_{G}(M) and Z(I) centralizes NM ≥ F^{∗}(G), so that Z(I) ≤ Z(F (G)) = Z(G). Also [O_{p}(I), M] = [O_{p}(I), F^{∗}(N)] = 1, so that [O_{p}(I), F^{∗}(G)] = 1 and O_{p}(I) ≤ Z(G). Hence Z = O_{p}(I). We note also that O_{p}(M̃) = O_{p}(W̃), as MN/N is a direct product of non-Abelian simple groups (where M̃ is the full preimage in Ĩ of MN/N). Now if L is a subgroup of G such that [L, M] ≤ N, then we have [L, M] ≤ M ∩ N = Z(G), so that [L, M′] = 1 by the three subgroups lemma. Hence [L, M] = 1, as M = M′Z(G). It follows that MN/N has trivial centralizer in G/N. Consequently, we conclude that F (Ĩ) = W̃ .

Since M∩N = Z(G), it is elementary to check that there is a natural bijection between N (M, Z) and N (MN/N). Consequently, there is a natural bijection between N (M, Z) and N (M̃, O_{p}(W̃)). Furthermore, these bijections respect chain stabilizers and conjugacy in the obvious fashion ( for example, the full pre-image of a p-subgroup of MN/N in MN has the form QN for some p-subgroup Q of M ( containing Z). If P and Q are p-subgroups ( containing Z) of M such that P N/N and QN/N are conjugate via an element xN of I/N, then P^{x} ≤ (QN ∩ M) = Q(M ∩ N) = QZ(G), so that ( as P is a p-group), P^{x} ≤ QZ = Q. Similarly, we deduce that Q^{x^{−1}} ≤ P, so that P^{x} = Q). It is also worth noting that if Q is a p-subgroup of M containing Z, and xN ∈ C_{I/N}(QN/N), then for each u ∈ Q, we have u^{x} ∈ uN ∩ M = uZ(G), so that u^{x} ∈ uZ as u is a p-element. If x is p-regular, this forces u^{x} = u by a standard and well-known argument. Hence the p-regular elements in C_{I/N}(QN/N) are precisely the images of the p-regular elements of C_{I}(Q).

For a chain σ ∈ N (M, Z), let σ̃ be the corresponding chain in N (M̃, O_{p}(W̃)). We recall our equation

∑_{σ∈N (M,Z)/I_{G}(ζ)}(−1)^{|σ|}k_{d}(b′_{σ}, ζ) ≠ 0.

Let b̃ denote the set of Dade correspondents of b′ associated to ζ. Then the fact that Dade correspondence and Brauer correspondence commute allows us to conclude that b̃σ̃ is the sum of the Dade correspondents of (b′)_{σ}. Hence we obtain

∑_{σ∈N (M̃,O_{p}(W̃))/Ĩ}(−1)^{|σ̃|}kd̃(b̃σ̃, ResW̃ ; O_{p}(W̃))(λ̃)) ≠ 0,

where d̃= d + log_{p}(ζ(1)|W̃ |/|N|).

It follows from Theorem 1 of [4] that one of the block summands of b̃ must cover a block of RM̃ with defect group O_{p}(W̃). By the results of Külshammer-Robinson [2], this means that some block in b′ contains an irreducible character lying over ζ which also lies over an N-projective irreducible character of MN. Since M ∩ N = Z, this means in turn that some block in b′ covers a block of RM with defect group Z. Hence B covers a block of M with central defect group, contrary to hypothesis.

References

[1] Dade, E.C., Counting Characters in Blocks,II, J. Reine Angew. Math. 448,(1994),97-190.

[2] Külshammer, B.; Robinson, G.R., Characters of Relatively Projective Modules II, J. London Math. Soc.,(2),36,(1987), 59-67.

[3] Robinson, G.R., Dade’s Projective Conjecture for p-Solvable Groups, J. Algebra, 229,(2000),234-248.

[4] Robinson, G.R., Cancellation Theorems related to Conjectures of Alperin and Dade, preprint.