A group-theoretic consequence of the Donald–Flanigan conjecture

Murray Gerstenhaber and David J. Green

16 December 1993

Abstract

For a finite group G and a prime p dividing the order of G, Donald and Flanigan conjecture that the group algebra \overline{𝔽}_{p}G can be deformed into a semisimple (hence rigid) algebra. We demonstrate that this implies that for some element g of G, the centralizer C_{G}(g) of g in G has a normal subgroup of index p. The method is to observe that the Donald–Flanigan deformation must be a jump. This implies that there is a non-trivial class in H^{1}(𝔽_{p}G, 𝔽_{p}G); therefore this Hochschild cohomlogy group must be nontrivial. Using a standard result linking Hochschild and group cohomology one sees that some H^{1}(C_{G}(g), 𝔽_{p}) must be non-zero. The result follows immediately.

1 Introduction

Let G be a finite group, p be a prime dividing its order |G|, and k be an algebraically closed field of characteristic p. The Donald–Flanigan conjecture [2] asserts that the group algebra kG, which is not semisimple, can nevertheless be deformed to a semisimple algebra. It is, in a sense, a p-modular version of Maschke’s theorem. M. Schaps [16] refined the conjecture to assert that the deformation could moreover be so chosen that the matric blocks of the resulting algebra had the same dimensions as those of ℂG, the complex group algebra. If true, the conjecture may have important consequences for modular representation theory. We show here that it has as a corollary a purely group theoretic property with a simple statement: there must exist in G an element g whose centralizer C_{G}(g) has a normal subgroup of index p. (Equivalently, the index of the derived group, C_{G}(g)′, is divisible by p.) We have recently learned that, with knowledge of this implication, Fleischmann, Janiszczak and Lempken [6] have in fact proven a somewhat stronger assertion with the virtue that its proof can be reduced to the case where the group is simple: If G is a finite group of order divisible by a prime p then G contains a p-singular element g whose p-part is not contained in the commutator subgroup of C_{G}(g). Their proof consists of an efficient examination of all simple groups and they cite the result in fact as new evidence for the Donald–Flanigan conjecture.

We will call a semisimple deformation of kG a “DF deformation”. Such a deformation need not be unique; there may be others in addition to a “Schaps” deformation with the correct components.

It was originally an observation of Knörr that the element g whose existence our Corollary asserts may in fact be assumed to be p-singular, i.e., to have order divisible by p. On the other hand it need not be a p-element (one whose order is a power of p). For the sporadic simple Janko group J_{4} and p = 3, no 3element serves. The Rudvalis, Thompson and Tits groups Ru, Th and ^{2}F_{4}(2)′ also produce counterexamples, for the primes 3, 5 and 3 respectively. In all four cases the Sylow p-subgroups are extraspecial of order p^{3} and exponent p. Curiously, in M_{23} with p = 2 there is a suitable g which is a product of three disjoint 7-cycles and hence has order prime to 2.

The present conjecture asserts precisely, as we shall see, that the Hochschild cohomology group H^{n}(𝔽_{p}G, 𝔽_{p}G) of 𝔽_{p}G with coefficients in itself as a bimodule is non-trivial for n = 1. Of course, the Donald–Flanigan conjecture requires that it be non-trivial for n = 2, else there could be no deformation. But it is implicit that a DF deformation be a jump deformation, and this implies, as we shall see, that also H^{1} ≠ 0. The concept of a jump deformation is fundamental to both parts of what we show. We recapitulate that theory, but first review briefly the present state of the DF and Schaps conjectures.

2 History

The remarkable Donald–Flanigan conjecture was proven by them only for abelian groups. The first subsequent progress was not until 1988 when Schaps [18] solved the problem for groups G with cyclic Sylow p-subgroups. The theorem was proven by the first author and Schaps [13] for G with an abelian normal Sylow p-subgroup, by Erdmann and Schaps [5] for characteristic p = 2 when the Sylow 2-subgroups of G are dihedral, and most recently by the first author and Schaps [14] for the symmetric groups. It is natural to extend the problem by considering not only deformations of the entire group algebra kG but of blocks of that algebra, i.e., its indecomposable direct summands B. To each is associated a defect group δ(B), but a deformation of the group algebra of δ(B) does not automatically give a deformation of B nor conversely. Nevertheless, to date the few cases where B has been proven to have a semisimple deformation are precisely the cases where kδ(B) has been proven to have one. This is partly a consequence of the observation that for these B the endomorphism ring of an indecomposable projective of B is isomorphic to the group algebra of δ(B), so deformation of B gives a deformation of kδ(B), cf. [4], [3] and [5]. They have shown for arbitrary p that every tame block with dihedral defect group has a semisimple deformation.

Another direction was initiated by Michler [15] who asked when a DF deformation of a block could be lifted to a deformation which is generically a maximal order descending modulo p to the block. He proved this for a block with cyclic defect group. Independently, Schaps [17] proved a liftability result for p–p′ metacyclic groups. A recent approach to the full problem, due to Schaps, is to consider liftable deformations of Coxeter groups as deformations of Hecke algebras. At present it appears that all groups of order less than 32 are known to have DF deformations except the extraspecial 3-group of order 3^{3} and exponent 3. This group, which occurs in three of the examples in §1, may be presented as a central extension of C_{3} × C_{3} by C_{3}, where if c generates the central C_{3} and a, b descend to generators of the quotient C_{3} × C_{3} we have aba^{−1}b^{−1} = c.

3 Jump deformations

Henceforth “δ” will denote the Hochschild coboundary operator. For an exposition of the algebraic deformation theory, cf. [12]; the original papers are [7, 8, 9, 10, 11]. (The observation that there can be no jump deformations when the first cohomology vanishes is already contained in [12] but for completeness we recapitulate the main ideas here.)

Let A be an algebra over a ring k (here not necessarily a field) with multiplication α: A × A → A. A (formal) deformation of A is a power series α_{t} = α + tα_{1} + t^{2}α_{2} + ⋯ where all α_{i} are k-bilinear maps A × A → A (extended to be k[[t]]-bilinear) and which is formally associative, i.e., where α_{t}(α_{t}(a, b), c) = α_{t}(a, α_{t}(b, c)), all a, b, c ∈ A. This is equivalent to having, for each n, the identity

∑_{i+j=n; i,j>0} {α_{i}(α_{j}(a, b), c) − α_{i}(a, α_{j}(b, c))} = δα_{n}(a, b, c),

where δ is the Hochschild coboundary operator. In particular, δα_{1} = 0, so α_{1} ∈ Z^{2} = Z^{2}(A, A), the group of Hochschild 2-cocycles. If α_{1} = ⋯ = α_{n−1} = 0 and α_{n} ≠ 0 then δα_{n} = 0 and the deformation “begins” with α_{n}. The deformation α_{t} is equivalent to a second, α′_{t} = α + tα′_{1} + t^{2}α′_{2} + ⋯ if there exists Φ_{t} = 1(= id_{A}) + tϕ_{1} + t^{2}ϕ_{2} + ⋯ where all ϕ_{i} are k-linear maps A → A such that α′_{t}(a, b) = Φ_{t}^{−1}α_{t}(Φ_{t}a, Φ_{t}b) = Φ_{t}^{−1}α_{t}(Φ_{t} ⊗ Φ_{t})(a ⊗ b). This implies, in particular, that α′_{1} = α_{1} + δϕ_{1}. Thus, up to equivalence of deformations, only the class [α_{1}] ∈ H^{2}(A, A) is significant, and by successive equivalences we may assume that α_{t} begins with an α_{n} not a coboundary. Denote the deformed algebra (whose underlying module is A[[t]]) with multiplication α_{t} by A_{t}. Write k[[t]] = k_{t}. We call α_{t} a jump deformation if α_{(1+u)t}, considered as a deformation of α_{t} over k_{t}[[u]], is equivalent to α_{t}. Intuitively, a deformation “infinitely close” to α_{t} is equivalent to it. There must then exist Ψ_{u} = 1 + uψ_{1} + u^{2}ψ_{2} + ⋯, where the ψ_{i} are themselves now k_{t}-linear maps from A_{t} to A_{t}, such that α_{(1+u)t} = Ψ_{u}^{−1}α_{t}(Ψ_{u} ⊗ Ψ_{u}). Collecting powers of u, the left hand side has the form α_{t} + uα_{t,1} + u^{2}α_{t,2} + ⋯, where α_{t,1} = tα_{1} + 2t^{2}α_{2} + 3t^{3}α_{3} + · · ·. But note that we must have α_{t,1} = δ_{t}ψ_{1}, where δ_{t} is the Hochschild coboundary operator of the deformed algebra A_{t}. To see what this implies we must consider both obstructions to derivations and the behaviour of cohomology under deformation.

4 Derivations

If ϕ is a derivation of a ℚ-algebra A then exp tϕ = 1+tϕ+(t^{2}/2!)ϕ^{2}+⋯ is a formal automorphism, but in characteristic p > 0 we must ask when, given ϕ, there is an automorphism Φ_{t} of the form 1 + tϕ + t^{2}ϕ_{2} + t^{3}ϕ_{3} + · · ·. The corresponding identities are then

∑_{i+j=n; i,j>0} ϕ_{i} ⌣ ϕ_{j} = −δϕ_{n}.

Taking, as we may, ϕ_{i} = ϕ^{i}/i! for i < p, the obstruction to finding ϕ_{p} is the cohomology class of −(ϕ_{1}⌣ϕ_{p−1}+ϕ_{2}⌣ϕ_{p−2}+⋯+ϕ_{p−1}⌣ϕ_{1}) (which is guaranteed to be a cocycle) in H^{2}(A, A). If this vanishes, then the next obstruction, which is again in H^{2}, is at the p^{2}-place; if this is passed then the following is at the p^{3}-place, and so forth. If we have an expression of the form 1+t^{m}ϕ_{m}+t^{m+1}ϕ_{m+1}+⋯+t^{n}ϕ_{n} which is an automorphism of A[t]/t^{n+1}, then ϕ_{m} is necessarily a derivation, and if mp^{r} ≤ n it necessarily passes at least r obstructions. While in accordance with what one expects, proofs of some of the foregoing statements are not completely obvious: cf. [10].

5 Effect of deformations on cohomology

If A_{t} is a deformation of A, then the deformed coboundary operator δ_{t} of A_{t} can be written as a power series δ +tδ_{1} +t^{2}δ_{2} +⋯ where each δ_{i} maps the n-cochains C^{n} = C^{n}(A, A) linearly into the (n + 1)-cochains. (In fact using the graded Lie multiplication on ⨁ C^{n} introduced in [7], if F ∈ C^{n} then δF = −[α, F] and δ_{i}F is just −[α_{i}, F].) Suppose now for simplicity that k is a field. An n-cocycle z is liftable to A_{t} if there is a series z_{t} = z + tz_{1} + t^{2}z_{2} + ⋯ with z_{i} ∈ C^{n} such that δ_{t}z_{t} = 0; denote the space of these by LZ^{n}. It is liftable modulo t^{r} if δ_{t}z_{t} ≡ 0 (mod t^{r}); denote the module of these by L_{r−1}Z^{n}. Then Z^{n} = L_{0}Z^{n} ⊃ L_{1}Z^{n} ⊃ ⋯ ⊃ LZ^{n} ⊃ B^{n}, the coboundaries; the last inclusion follows because every δy is liftable to δ_{t}y. Now δ_{t}z_{t} = 0 is equivalent to having δz_{r} = −(δ_{1}z_{r−1} + δ_{2}z_{r−2} + ⋯ + δ_{r}z). Denote the right hand side by y, the obstruction cocycle. The obstruction to extending a lifting z +tz_{1} +⋯+t^{r−1}z^{r−1} modulo t^{r} to one modulo t^{r+1} is the class [y] ∈ H^{n+1}. Observe that if z + tz_{1} + ⋯ + t^{r−1}z_{r−1} is a lifting modulo t^{r}, then δ_{t}(z + tz_{1} + ⋯ + t^{r−1}z_{r−1}) has the form t^{r}(δ_{1}z_{r−1} + δ_{2}z_{r−2} + ⋯ + δ_{r}z) + higher terms, so if y is the obstruction cocycle to extending some lifting mod t^{r} to one mod t^{r+1} then t^{r}y can be lifted to a δ_{t}-coboundary (and in particular, y itself is liftable). Such y (now in Z^{n+1}) will be called r-jump cocycles. The space of them is denoted J_{r}Z^{n+1}, and the union by JZ^{n+1}, so we have LZ^{n+1} ⊃ JZ^{n+1} ⊃ ⋯ ⊃ J_{r}Z^{n+1} ⊃ J_{r−1}Z^{n+1} ⊃ ⋯ ⊃ J_{0}Z^{n+1} = B^{n+1}. Now if we extend coefficients from k[[t]] to its quotient field k((t)), then any y ∈ J_{r}Z^{n+1} can already be lifted to a coboundary. It is easy to see that every element of Z^{n}(A_{t}, A_{t}) is the lifting of an element of Z^{n}, so over the field k((t)), we have H_{t}^{n} = H^{n}(A_{t}, A_{t}) = LZ^{n}/JZ^{n}, the liftable cocycles modulo the jump cocycles. (An evident strengthening which we shall not need, that L_{r−1}Z^{n}/L_{r}Z^{n} ≅ J_{r}Z^{n+1}/J_{r+1}Z^{n+1}, leads to the invariance of the Euler– Poincaré characteristic under deformation whenever it is defined.) All we shall need of this is the

Lemma 1 If [z] ∈ H^{n}(A, A) is a jump class then it is the obstruction to lifting a cocycle in Z^{n−1}(A, A). In particular if H^{n−1}(A, A) = 0 then there can be no jump classes.

6 Implications of a jump

Suppose that α_{t} = α+t^{r}α_{r}+t^{r+1}α_{r+1}+⋯ is a jump deformation of A with [α_{r}] ≠ 0. From §3, this implies that rt^{r}α_{r} + (r + 1)t^{r+1}α_{r+1} + ⋯ becomes a coboundary in A_{t}, so rα_{r} is a jump cocycle, and hence so is α_{r} unless we are in characteristic p and p | r. In fact, it is then possible that α_{t,1} = 0, which will be the case if α_{t} is actually a power series in t^{p}. In that case α_{(1+u)t} = α_{t} + u^{p}(t^{p}α_{p} + 2t^{2p}α_{2p} + ⋯) + ⋯, and Ψ_{u} might actually have begun with u^{s}ψ_{s} for some s > 1. But then Ψ_{u}^{−1}α_{t}(Ψ_{u}⊗Ψ_{u}) will begin α_{t}+u^{s}δ_{t}ψ_{s}+u^{s+1}(⋯), and this must agree with α_{(1+u)t}. Suppose the r above is mp^{ℓ} with p | m̸ so α_{(1+u)t} = α_{t}+u^{p^{ℓ}}(mt^{mp^{ℓ}}α_{mp^{ℓ}} +⋯)+· · ·. Then surely s ≤ p^{ℓ}, inequality being possible if δ_{t}ψ_{s} = 0, and if s = p^{ℓ} then again α_{mp^{ℓ}} is a jump cocycle. If s < p^{ℓ} then ψ_{s} is a derivation of A_{t} which we may write as a power series in t whose first non-zero term must then be a derivation of A. Were it inner, i.e., a coboundary, then it could be lifted to a coboundary of A_{t}, and there can be no obstruction to extending a coboundary to an automorphism— and in fact to an inner automorphism. But that would have no effect on the equivalence. Thus, either the first non-zero term of α_{t} is a jump cocycle or we have found an outer derivation of A. But in either case, we have

Lemma 2 If α_{t} is a jump deformation of A then H^{1}(A, A) ≠ 0.

7 The corollary to the DF conjecture

Suppose again that G is a finite group, p a prime dividing |G| and k an algebraically closed field of characteristic p. A DF deformation of kG, if one exists, is necessarily a jump deformation; the deformed algebra, being semisimple, has no cohomology in positive dimensions and therefore no further deformations. It follows now that H^{1}(kG, kG) ≠ 0, and therefore that H^{1}(𝔽_{p}G, 𝔽_{p}G) ≠ 0, since the former is obtained from the latter just by extending coefficients.

We have now seen that if the DF conjecture is true then the Hochschild cohomology group H^{1}(𝔽_{p}G, 𝔽_{p}G) must be non-trivial. With the following standard theorem one can translate this requirement into group cohomology with trivial coefficients.

Theorem 3 ([1] 2.11.2) There is an isomorphism of additive groups

H^{n}(𝔽_{p}G, 𝔽_{p}G) ≅ ⨁ H^{n}(C_{G}(g), 𝔽_{p}),

where the sum is over a set of representatives g of conjugacy classes in G .

Therefore there must exist a g in G for which H^{1}(C_{G}(g), 𝔽_{p}) is non-trivial. But this is just the set of group homomorphisms from C_{G}(g) to the additive group of 𝔽_{p}, and there is a non-trivial homomorphism if and only if C_{G}(g) has a normal subgroup of index p. Therefore we have

Theorem 4 If the DF conjecture holds for a group G then there is a g ∈ G whose centralizer has a normal subgroup of index p.

Note that our consequence of the DF conjecture is in a sense dual to Cayley’s theorem that G contains an element of order p.

Acknowledgements We would like to express our thanks to Professor Michler, who organised the meeting in Essen at which this research was done. The first author is partially supported by the N.S.A., and his visit to Essen was supported by the Institut für Experimentelle Mathematik. The second author is grateful for the support of the Volkswagen–Stiftung.

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University of Pennsylvania, Philadelphia PA 19104–6395

mgersten@pennsas.upenn.edu Institut für Experimentelle Mathematik, Ellernstrasse 29, D–45326 Essen, Germany.

david@exp-math.uni-essen.de