ABSTRACT
Let G be a group and R an associative ring. A G−grading on R is a decomposition of R as an additive group into the direct sum of subgroups R = ⊕g∈GRg such that RgRh ⊆ Rgh for any g, h ∈ G. The subgroup Rg is called the homogeneous component of degree g and elements of Rg are called homogeneous of degree g. The subgroup Re, where e is the identity element of G, is called the identity component. The support of R is defined by SuppR = {g ∈ G : Rg 6= 0}. If H = SuppR is a subgroup of G, we say that the group H supports the G−grading on R. If R is an algebra over a field F , a G−grading on R is called fine if dimF (Rg) ≤ 1 for all g ∈ G. We refer the reader to [5] for more details on G-gradings on algebra over a field F . Our first object in this paper is to observe that the notion of a fine grading
on a central simple algebra is basically the same as the notion of a projective basis. Recall that a finite dimensional F-algebra R is said to have a projective
basis if it has a basis {a1, a2, . . . , an} of invertible elements, such that for every pair i, j there is an m such that aiaj = λijam for some λij ∈ F×. It is often convenient to view such an algebra R as a twisted group algebra. Let Γ be the subgroup of R× generated by a1, a2, . . . , an and F×, and let G = Γ/F×, then R is isomorphic to the twisted group algebra FαG where the cohomology class α ∈ H2(G,F×) is defined by the central extension
1 −→ F× −→ Γ −→ G −→ 1 1
(Recall that for each finite group G and two-cocycle f in Z2(G,F×) one can form the twisted group algebra F fG = ⊕σ∈GFxσ where xσxτ = f(σ, τ)xστ for all σ, τ ∈ G. The resulting F-algebra depends (up to F-isomorphism) only on the class α = [f ] in H2(G,F×), and so we denote the algebra by FαG.) The precise statement that relates projective bases to fine gradings for
F-central simple algebras is as follows:
Theorem 1. Let F be a field. Let R be a finite dimensional central simple algebra over a field F graded by a group G such that dimF (Rg) ≤ 1 for all g ∈ G. Then H = SuppR is a subgroup of G and R ∼= FαH for some α ∈ H2(H,F×). The converse of Theorem 1 is clear. If R is an F-algebra isomorphic to a
twisted group algebra FαH, α ∈ H2(H,F×), then H supports a fine grading on R over F . Theorem 1 is proved in Section 2. We will call a group H, such that FαH is F-central simple for some F
and α ∈ H2(H,F×), a group of central type (nonclassically). The class α, such that FαH is F-central simple, is said to be nondegenerate. Recall that classically a finite group Γ is of central type if it has an irreducible complex representation of the maximal possible degree, namely
√ [Γ : Z(Γ)]
(where Z(Γ) denotes the center of Γ). It is easy to see that H is of central type nonclassically if and only if there exists a classical central type group Γ such that H ∼= Γ/Z(Γ). Isaacs and Howlett proved, using the classification of finite simple groups, that such Γ, and hence H, is solvable ([6, Theorem 7.3]). In this paper, we refer to “central type” only in the nonclassical sense. Thus by Theorem 1 we obtain that the groups H that support fine grad-
ings of central simple algebras are precisely the groups of central type. Theorem 1 allows us to give a precise description of the groups H that
support fine gradings of F-central division algebras. Let Λp be the following list of p-groups:
(1) H is abelian of symmetric type, that is H ∼=∏(Zpni × Zpni ) (2) H ∼= H1 × H2 where H1 = Zpn o Zpn = 〈 pi, σ | σpn = pipn =
1 and σpiσ−1 = pips+1 〉 where 1 ≤ s < n and 1 6= s if p = 2 and H2 is an abelian group of symmetric type of exponent ≤ ps
(3) H ∼= H1 ×H2 where H1 = Z2n+1o (Z2n×Z2) =
〈 pi, σ, τ
∣∣∣∣ pi2n+1 = σ2n = τ2 = 1, στ = τσ,σpiσ−1 = pi3, τpiτ−1 = pi−1 〉
and H2 is an abelian group of symmetric type of exponent ≤ 2 Combining Theorem 1 with [1, Theorem 1] and [2, Corollary 3] we have
Theorem 2. Let H be a finite group that is the support of a fine grading on an F-central division algebra D. Then
(1) H is a nilpotent group. (2) If H ∼= P1 × . . . × Pr is the decomposition of H into the product
of its Sylow pi-subgroups then Pi belongs to Λpi. In particular, the commutator subgroup H ′ of H is cyclic.
