AUTOMORPHISM GROUP SCHEMES

OF FINITE FIELD EXTENSIONS

Ke-Zheng Li

Graduate School of Academia Sinica/Max-Planck-Institut für Mathematik

0. Introduction.

Let S be a noetherian scheme and X → S be a flat projective morphism. The automorphism group scheme Aut(X/S) represents the following functor

𝔄𝔲𝔱_{X/S}: ((S-schemes)) → ((sets))

T ↦ {T -automorphisms of T ×_{S} X}

We study Aut(X/S) because of two reasons. First, Aut(X/S) is an important invariant of the S-scheme X, which often gives useful information about X; second, the structure of Aut(X/S) and its action on X are also interesting objects to study. The situation is usually non-trivial even when S = Spec(K) for a field K.

In this paper we mainly discuss the case when X → S is finite, especially when X = Spec(L), S = Spec(K) for a finite field extension L ⊃ K, in this case Aut(X/S) may play a role of a stronger tool than the Galois group Gal(L/K), and its structure is by no means trivial. This is also the necessary first step toward the calculation of Aut(X/S) in general case.

In §1 we give the definition and basic properties of automorphism group schemes, and then give some examples. This will show the non-triviality of the calculation of Aut(X/S), even when S = Spec(K) for a field K. In §2 we discuss the automorphism group schemes of finite field extensions. We first give a canonical decomposition for inseparable case and explain the calculation for the purely inseparable case. This almost reduces the general calculation to separable case. Finally we discuss the separable case and give more examples in §3.

Typeset by AMS-TEX

1. Automorphism group schemes.

Let X be a flat relatively projective scheme over a noetherian scheme S, then the following functor

𝔄𝔲𝔱_{X/S}: ((S-schemes)) → ((sets))

T ↦ {T -automorphisms of T ×_{S} X}

is represented by a locally quasi-projective S-scheme Aut(X/S), called the automorphism group scheme of X over S (if S = Spec(K) for a field K, then we also denote Aut(X/S) by Aut(X/K)). Explicitly, let Φ_{X/S} : Aut(X/S)×_{S}X → Aut(X/S)×_{S}X be the universal Aut(X/S)-automorphism, then an S-morphism f : T → Aut(X/S) corresponds to the base change of Φ_{X/S} via f.

The following are some basic properties of automorphism group schemes.

Lemma 1. Let S be a noetherian scheme and X be a flat relatively projective scheme over S.

i) Aut(X/S) commutes with base change, i.e. if T is a noetherian S-scheme we have Aut(X ×_{S} T/T) ≅ Aut(X/S) ×_{S} T.

ii) Aut(X/S) is an S-group scheme with a canonical action on X representing the following functor:

𝔄𝔠𝔱_{X/S}: ((S-group schemes)) → ((sets))
G ↦ {actions of G on X}

Explicitly, ρ_{X/S} = pr_{1} ∘ Φ_{X/S} : Aut(X/S) ×_{S} X → X is the canonical action, and each homomorphism h : G → Aut(X/S) corresponds to the action ρ_{X/S} ∘ (h ×_{S} id_{X}) : G ×_{S} X → X.

These are just abstract nonsense.

Lemma 2. Let S be a noetherian scheme and X → S be a flat finite morphism. Then Aut(X/S) is linear, i.e. locally over S, Aut(X/S) can be embedded into GL(n, S) as a closed subgroup scheme, where n is the local degree of X → S.

This is easy to check directly.

Remark 1. The assumption of relative projectivity is quite essential in the definition of Aut(X/S), in fact, if X → S is flat and quasi-projevtive but not projective, then 𝔄𝔲𝔱_{X/S} is usually not representable, not even when X = 𝔸_{S}^{1}. Note that the two concepts “representation” and “action” (of S-group schemes on X) are equivalent only when 𝔄𝔠𝔱_{X/S} is representable. When S = Spec(K) for a field K and X = 𝔸_{K}^{1}, 𝔄𝔠𝔱_{X/S} is representable only when ch(K) = 0.

Example 1. Let S = Spec(ℤ), X = ℙ_{ℤ}^{n}. Then for any S-scheme T, every T - automorphism of T ×_{S} X preserves O(1). Hence Aut(X/S) is isomorphic to a closed subgroup scheme of PGL(n + 1, S). On the other hand, PGL(n + 1, S) really acts on X. Hence Aut(X/S) ≅ PGL(n + 1, S).

Example 2. Let X be an abelian variety over an algebraically closed field k. Then by [L2, Lemma 2.3], one sees easily that Aut(X/k) ≅ X⋊Aut(X), where Aut(X) is the automorphism group of X endowed the discrete group scheme structure, and X acts on X by translation.

Example 3. Let C be a complete non-singular curve over a field K. Using the method of §2 and Lemma 3.1 below, one can calculate Aut(C/K) as follows: first calculate Aut(C ⊗_{K} K̄/K̄) (where K̄ is the algebraic closure of K), then by taking the quotient of an action of a finite group scheme one can get Aut(C/K). Therefore it reduces to consider the case when K is algebraically closed. If g(C) = 0 or 1, we can apply Example 1 or Example 2 to calculate Aut(C/K). If g(C) > 1, then Aut(C/K) is finite étale (cf. [L2]).

If X is a smooth projective variety of dimension > 1 over K, then Aut(X/K) need not to be reduced when ch(K) > 0 (see [L2, Example 2.8]).

Terminology. A morphism of schemes f : X → Y is called an epimorphism if f is set-theoretically onto and f^{∗} : O_{Y} → f_{∗}O_{X} is a monomorphism.

Let S be a noetherian scheme, X be an S-scheme of finite type and G be a group scheme of finite type over S. An action ρ : G ×_{S} X → X of G on X is called transitive if (ρ, pr_{2}) : G ×_{S} X → X ×_{S} X (intuitively (g, x) ↦ (gx, x)) is an epimorphism. If ρ has a geometric quotient X/G which is a locally closed subscheme of S, then ρ is transitive; on the other hand, if X → S is f.f. and ρ is transitive, then there is a geometric quotient X/G ≅ S.

Corollary 1. If X → S is finite étale of degree n, then Aut(X/S) is finite étale of degree n! over S, and the universal action ρ_{X/S} is transitive. In fact, (ρ, pr_{2}) : G ×_{S} X → X ×_{S} X is f.f. and there is a universal geometric quotient X/Aut(X/S) ≅ S.

Proof. Take an étale cover S′ → S such that X ×_{S} S′ is a disjoint union of n copies of S′. It is clear that (X ×_{S} S′/S′) is the discrete group scheme structure of the permutation group of the n copies of S′ (≅ 𝔖_{n}), in particular it acts transitively on X ×_{S} S′. Therefore Aut(X/S) → S is étale of degree n! because it is so after an étale base change, and ρ_{X/S} is transitive also because it is so after an étale base change. Q.E.D.

2. The case of finite field extensions.

In the following we study a special case: S = Spec(K), X = Spec(L), where L ⊃ K is a finite field extension. For simplicity we will denote G_{L/K} = Aut(X/S) in this case. We are interested with this case because of two reasons: First, the calculation of G_{L/K} can be a first step toward the general calculations of Aut(X/S), which is by no means trivial as we will see; Second, in any case L^{G_{L/K}} = K (see Proposition 1 below), while L^{Gal(L/K)} = K only when L ⊃ K is Galois, hence G_{L/K} is a useful tool which can take over the job of Gal(L/K) when L ⊃ K is not a Galois extension (especially when L ⊃ K is inseparable). For example, in many constructions one can make use of an arbitrary base field extension, and after the construction one can go back to the original base field whenever the construction commutes with the action of G_{L/K}.

Note that by definition there is a canonical isomorphism Gal(L/K) ≅ G_{L/K}(K) (the group of K-points of G_{L/K}). However, we will see G_{L/K} is larger than Gal(L/K) (i.e. it has points which are not K-points) unless L = K or L ⊃ K is a separable extension of degree 2.

For a group scheme G over K we will denote by m : G ×_{K} G → G the multiplication, ι : G → G the inverse and o : Spec(K) → G the unit section. When G is affine, we will only give m^{∗}, ι^{∗} and o^{∗} for the group scheme structure of G.

Example 1. Let K = ℚ, L = ℚ[x]/(x^{3} − 2) ≅ ℚ[\sqrt[3]{2} ]. Then G_{L/K} is an étale group scheme of degree 6 over ℚ (see Corollary 1.1). By direct calculation we can get G_{L/K} = Spec(R), where

R = ℚ[r, s, t]/(r − s^{2} − s + t^{3}/2 − 1, rs, rt, st) ≅ ℚ × ℚ[\sqrt{−3} ] × ℚ[\sqrt[3]{2} ] (1)

with group scheme structure given by

m^{∗}(r) = r ⊗ r + \frac{1}{3}(s ⊗ s + s^{2} ⊗ s^{2} + s^{3} ⊗ s^{3}) + \frac{1}{12}(2t^{2} ⊗ t + 2t ⊗ t^{2} + t^{3} ⊗ t^{3}), m^{∗}(s) = r ⊗s+s⊗r + \frac{1}{3}(2s⊗s−s^{2} ⊗s^{2} −s^{3} ⊗s^{3})+ \frac{1}{12}(4t^{2} ⊗t−2t⊗t^{2} −t^{3} ⊗t^{3}),

m^{∗}(t) = r ⊗ t + t ⊗ r + s^{2} ⊗ t + t ⊗ s,
ι^{∗}(r) = r, ι^{∗}(s) = s^{2}, ι^{∗}(t) = t, o^{∗}(r) = 1, o^{∗}(s) = o^{∗}(t) = 0

and the universal action ρ is given by

ρ^{∗}(x) = r ⊗ x + s ⊗ x + \frac{1}{2}t^{2} ⊗ x^{2}.

One observes that 2 and 3 appear in the denominators, this is because that L has bad reduction at 2 and 3 (but good reductions at the other places). One also see that L appears in the structure of G_{L/K}, and we will see this is a pretty general situation.

Example 2. Let K be a field of characteristic p > 0, and L = K[t^{\frac{1}{p}}] ≅ K[x]/(x^{p}−t) for some t ∈ K − K^{p}. By direct calculation we can get G_{L/K} = Spec(R), where

R = K[y_{0}, y_{1}, ..., y_{p−1}, y_{1}^{−1}]/(y_{0}^{p} + (y_{1}^{p} − 1)t + y_{2}^{p}t^{2} + ... + y_{p−1}^{p}t^{p−1})  (2)

and the universal action ρ is given by

ρ^{∗}(x) = y_{0} ⊗_{K} 1 + y_{1} ⊗_{K} x + ⋯ + y_{p−1} ⊗_{K} x^{p−1}

(for simplicity we omit the group scheme structure, e.g. when p = 2, the group scheme structure of G_{L/K} is given by m^{∗}(y_{0}) = y_{0} ⊗ 1 + y_{1} ⊗ y_{0}, m^{∗}(y_{1}) = y_{1} ⊗ y_{1}, ι^{∗}(y_{0}) = y_{0}y_{1}^{−1}, ι^{∗}(y_{1}) = y_{1}^{−1}, o^{∗}(y_{0}) = 0, o^{∗}(y_{1}) = 1). We see that dim(G_{L/K}) = p − 1 > 0 and G_{L/K} is non-commutative. Furthermore, G_{L/K} is integral and geometrically irreducible, but not normal, and not geometrically reduced.

For the general case we use the following decomposition method.

Proposition 1. Let L ⊃ K be a finite field extension. Denote by L′ the largest separable extension of K in L.

i) If ch(K) = p > 0, then the inclusion L′ ↪ L induces an epimorphism G_{L/K} → G_{L′/K}. Let H = ker(G_{L/K} → G_{L′/K}), then L^{H} = L′. Furthermore, for any finite extension Λ ⊃ L which is normal over K, H ⊗_{K} Λ ≅ ∏_{i=1}^{n} G_{L/L′} ⊗_{L′} Λ.

ii) G_{L/K} acts transitively on Spec(L), i.e. (ρ, pr_{2}) : G_{L/K} ×_{K} Spec(L) → Spec(L) ×_{K} Spec(L) is f.f., where ρ : G_{L/K} ×_{K} Spec(L) → Spec(L) is the universal action of G_{L/K} on Spec(L). Therefore we have Spec(L)/G_{L/K} ≅ Spec(K), or in other words L^{G_{L/K}} = K.

Proof. When L ⊃ K is separable, ii) holds by Corollary 1.1, so in the following we assume ch(K) = p > 0.

For any K-algebra A, any A-automorphism of A ⊗_{K} L induces an A-automorphism of A⊗_{K} L′, because L′ = K[L^{p^{m}}] for some m. Hence there is an induces homomorphism G_{L/K} → G_{L′/K}. If Λ ⊃ L is a finite extension which is normal over K, then L ⊗_{K} Λ ≅ ∏_{i=1}^{n} R, where R = L ⊗_{L′} Λ ≅ Λ[x_{1}, ..., x_{r}]/(x_{1}^{p^{m_{1}}} , ..., x_{r}^{p^{mr}}) for some r ≥ 0, m_{1}, ..., m_{r} > 0. By Lemma 1.1.i), G_{L/K} ⊗_{K} Λ ≅ Aut(Spec(L ⊗_{K} Λ)/Λ). Obviously any Λ-automorphism of Spec(L′ ⊗_{K} Λ) can be lifted to an Λautomorphism of Spec(L ⊗_{K} Λ), hence G_{L/K} → G_{L′/K} is an epimorphism (because it is an epimorphism after an f.f. base change). Furthermore, it is easy to see that H ⊗_{K} Λ ≅ ∏_{i=1}^{n} Aut(Spec(R)/Λ), and Spec(R)/Aut(Spec(R)/Λ) ≅ Spec(Λ). Thus ii) holds because Aut(Spec(R)/Λ) ≅ G_{L/L′} ⊗_{L′} Λ. Finally, it is it is easy to see that Aut(Spec(R)/Λ) acts transitively on Spec(R), hence Aut(Spec(L ⊗_{K} Λ)/Λ) acts transitively on Spec(L ⊗_{K} Λ), so G_{L/K} acts transitively on Spec(L). Q.E.D.

When ch(K) = p > 0, let H_{m} = ker(F_{G_{L/K}/K}^{m}), where m satisfies L^{p^{m}} ⊂ L′. Then H_{m} is a finite normal subgroup scheme of G_{L/K}, H_{m} ⊂ H, and L^{H_{m}} = L′. Therefore to get down from L to K we need only divide by finite group schemes. If L ⊃ K is normal, there exists an intermediate field L′′ of K ⊂ L such that L′′ ⊃ K is purely inseparable (or trivial) and L ⊃ L′′ is separable. In this case L ≅ L′ ⊗_{K} L′′, hence for any K-algebra A, an A-automorphism of A ⊗_{K} L′′ is equivalent to an A ⊗_{K} L′-automorphism of A ⊗_{K} L. Therefore G_{L′′/K} ⊗_{K} L′ ≅ G_{L/L′}. Imitating Example 2, the calculation of G_{L′′/K} is not hard, though the results are rather complicated. Furthermore, the semidirect product G = H_{m}⋊Gal(L/K) is a finite subgroup scheme of G_{L/K} such that L^{G} = K.

With the decomposition method in Proposition 1 and the above way to calculate the purely inseparable case, it remains to study the calculation of separable case for the general calculation of G_{L/K}.

3. Separable case.

In the following we consider the case when L ⊃ K is separable. The following lemma is a general method to study schemes using flat base change (see [Ra]).

Lemma 1. Let T → S be a finite f.f. morphism of locally noetherian schemes, and G ⊂ Aut(T/S) be a flat finite subgroup scheme acting freely and transitively on T. Then there is an equivalence of categories

((S-schemes))/∼ ↔ \begin{matrix}( \\ ( & T -schemes together with an action of G compatible with the action of G on T & ) \\ & & ) \end{matrix}/∼
Y ↦ Y ×_{S} T
X/G ←∣ X

(where the S-schemes are all assumed to be separated, and an action of G on X is assumed to have a geometric quotient X/G such that X → X/G is flat, these all hold if we only consider locally quasi-projective schemes).

Proof. By assumption we have T/G ≅ S and T ×_{S} T ≅ G ×_{S} T. We need to show that for any separated T -scheme X together with an action of G compatible with the action of G on T, we have X ≅ T ×_{S} (X/G).

Denolte by ρ the action of G on X, Y = X/G and π : X → T the projection. We claim that G acts freely on X. By the assumptions on G we have an isomorphism

θ : G ×_{S} X ≅ (G ×_{S} T) ×_{T} X →^{≃} (T ×_{S} T) ×_{T} X ≅ T ×_{S} X  (5)

By the compatibility of the actions of G on X and T, we have a commutative diagram

\begin{matrix}G ×_{S} X & ⟶^{(ρ,pr_{2})} & X ×_{S} X \\ ↓^{θ} & & ↓^{π×_{S}id_{X}} \\ T ×_{S} X & ⟶^{id} & T ×_{S} X \end{matrix}  (6)

Hence G ×_{S} X → X ×_{S} X is a closed immersion because it is a section of the separated morphism π ×_{S} id_{X}. Therefore X ×_{Y} X ≅ G ×_{S} X. We get

\begin{matrix}(T ×_{S} Y) ×_{Y} X ≅ T ×_{S} X ≅ (T ×_{S} T) ×_{T} X \\ ≅ & (G ×_{S} T) ×_{T} X ≅ G ×_{S} X ≅ X ×_{Y} X \end{matrix}  (7)

Since X → Y is f.f., we see X → T ×_{S} Y is an isomorphism. Q.E.D.

Corollary 1. Let Λ ⊃ K be a finite Galois extension. Then there is an equivalence of categories

\{K-schemes\}/ ∼ ↔ \{ \begin{matrix}Λ-schemes together with an action of Gal(Λ/K) \\ compatible with the action of Gal(Λ/K) on Λ \end{matrix} \}/∼

Y ↦ Y ⊗_{K} Λ

X/Gal(Λ/K) ←∣ X

(where the schemes are all assumed to be separated, and an action of Gal(Λ/K) on X is assumed to have a geometric quotient Y such that X → Y is flat). In particular, we have an equivalence of categories

\{ \begin{matrix}étale K-schemes of degree n \\ which are split over Λ \end{matrix} \}/∼ ↔ Hom(Gal(Λ/K), 𝔖_{n})/Aut(𝔖_{n})

where Gal(Λ/K) acts on Spec(Λ× ⋯^{n} ×Λ) by σ(a_{1}, ..., a_{n}) = (σa_{σ1}, ..., σa_{σn}) (∀σ ∈

Gal(Λ/K)) via the Λ-algebra structure Λ → Λ × ⋯ × Λ.

Let X = Spec(Λ× ⋯^{n} ×Λ) with a permutatione action ρ of H = Gal(Λ/K). Then ρ gives a homomorphism η : Gal(Λ/K) → Per(X) ≅ 𝔖_{n}. Let Y be the Kscheme corresponding to X in Lemma 2. Then we have a one-to-one correspondence

μ_{X} : {H-orbits of X} → {points of Y }

where for any x ∈ X, the H-orbit C_{x} of x satisfies μ_{X}(C_{x}) = Spec(K_{x}) for a field K_{x} ⊃ K, with #(C_{x}) = [K_{x} : K] (= the degree of μ_{X}(C_{x}) over K). Furthermore, if H_{x} ⊂ Gal(Λ/K) is the stabilizer of x under ρ, then K_{x} is isomorphic to the intermediate field of Λ ⊃ K corresponding to H_{x}. (Note that for different choices of representatives x of an orbit, the stabilizers H_{x} just form a conjugacy class of subgroups of Gal(Λ/K), which corresponds to just one K-isomorphism class of intermediate fields.) Thus |H_{x}| = [Λ : K_{x}].

In particular, if Y = Spec(L), where L is a separable field extension of K, we can take Λ to be a Galois closure of L ⊃ K (and X = Y ⊗_{K} Λ), in this case η is a monomorphism.

Let Λ ⊃ K be a finite Galois extension. Let Y = Spec(A) be an étale Kscheme of degree n which splits over Λ, and X = Y ⊗_{K} Λ. Denote G = Aut(Y/K), G′ = G ⊗_{K} Λ. Then by Lemma 1.1.i), G′ is isomorphic to Per(X) ≅ 𝔖_{n} with the discrete group scheme structure over Λ. By Corollary 1 we see that H acts on both X(Λ) and G′(Λ) ≅ 𝔖_{n}, and it is easy to see that σ(g(x)) = σ(g)(σ(x)) (∀σ ∈ H, g ∈ G′, x ∈ X), hence

σ(g) = η(σ) ∘ g ∘ η(σ)^{−1} ∀σ ∈ H, g ∈ 𝔖_{n}  (8)

Therefore we have a one-to-one correspondence

μ : {η(H)-conjugacy classes of 𝔖_{n}} → {points of G} where for any g ∈ 𝔖_{n}, the μ(H)-conjugacy class C_{g} of g satisfies μ(C_{g}) = Spec(K′) for a field K′ ⊃ K, with #(C_{g}) = [K′ : K] (= the degree of μ(C_{g}) over K). Furthermore, if H_{g} ⊂ H is the stabilizer of g under action (8), then K′ is isomorphic to the intermediate field of Λ ⊃ K corresponding to H_{g}. In particular |H_{g}| = [Λ : K′]. Summarizing we get

Proposition 1. Let Λ ⊃ K be a finite Galois extension, Y be an étale scheme of degree n over K which splits completely over Λ, X = Y ⊗_{K} Λ, G = Aut(Y/K), H = Gal(Λ/K), and η : H → 𝔖_{n} be the induced permutative representation of H on X ≅ Spec(Λ× ⋯^{n} ×Λ). Then i) Under the equivalence of categories in Corollary 1 (with n! instead of n), G

corresponds to the conjugation action of H on 𝔖_{n} given by (8). ii) There is a one-to-one correspondence

μ : {η(H)-conjugacy classes of 𝔖_{n}} → {points of G}

where for any g ∈ 𝔖_{n}, the η(H)-conjugacy class C_{g} of g satisfies μ(C_{g}) = Spec(K′) for the intermediate field K′ of Λ ⊃ K corresponding to the stabilizer H_{g} of g in H under action (7), and #(C_{g}) = [K′ : K], |H_{g}| = [Λ : K′]. Hence G ≅ Spec(∏_{i=1}^{r} K_{i}) for some intermediate fields K_{i} of Λ ⊃ K, and ∑_{i=1}^{r}[K_{i} : K] = n!.

In particular, if Y = Spec(L) for a separable extension L ⊃ K and Λ is a Galois closure of L ⊃ K, then H can be viewed as a subgroup of 𝔖_{n}, and H_{g} is just the centralizer of g in H.

It is natural to ask: For a given extension L ⊃ K, what intermediate fiels appear as some K_{i}? The following example answers this in the case when L ⊃ K is Galois.

Example 1 (The case L = Λ). We use the notation in Proposition 1. Let L ⊃ K be a Galois extension. Then X is in one-to-one correspondence with Gal(L/K), hence we can identify X as H with the action of H by left multiplication. We show that when n ≠ 2, any subgroup H′ ⊂ H is equal to H_{g} for some g ∈ G′.

We may assume H′ ≠ H. Let r = i_{H}(H′). Take u_{1} = e, u_{2}, ..., u_{r} such that H = H′u_{1} ∪ ⋯ ∪ H′u_{r}. If r > 2, let g ∈ 𝔖_{n} be defined by g(hu_{1}) = hu_{1}, g(hu_{i}) = hu_{i+1} (2 ≤ i < r), g(hu_{r}) = hu_{2} (∀h ∈ H′). Clearly g ∘ ϕ_{h} = ϕ_{h} ∘ g ∀h ∈ H′, where ϕ_{h} : H′ → H′ is the left multiplication by h. But for u ∈ H − H′ we have g ∘ ϕ_{u} ≠ ϕ_{u} ∘ g, because g(u · e) ≠ u · e. If r = 2, then |H′| > 1. Take a ≠ e ∈ H′ and define g ∈ 𝔖_{n} by g(hu_{i}) = hau_{i} (∀i > 1), g(hu_{1}) = hu_{1} ( ∀h ∈ H′). Again we have g ∘ ϕ_{h} = ϕ_{h} ∘ g (∀h ∈ H′) but g ∘ ϕ_{u} ≠ ϕ_{u} ∘ g (∀u ∈ H − H′), because g(u · e) ≠ u. In any case we have H_{g} = H′.

By Proposition 1, we see that every intermediate field of L ⊃ K is isomorphic to some K_{i} over K.

Example 2.1 shows that when L ⊃ K is not Galois, there may be a K_{i} which is not isomorphic to an intermediate field of L ⊃ K over K. On the other hand, if L ≠ Λ, there may be an intermediate field K′ of Λ ⊃ K which is not isomorphic to any K_{i} over K, for example when n > 4 and Gal(Λ/K) ≅ 𝔖_{n}, we can identify Gal(Λ/K) with Per(X) ≅ 𝔖_{n}, there is no element of 𝔖_{n} whose centralizer is A_{n}, hence there is no K_{i} such that [K_{i} : K] = 2. Furthermore, in general case there may not be a subgroup scheme of G (of degree n) which acts freely and transitively on Spec(L), for example if [L : K] = 5 and Gal(Λ/K) ≅ 𝔖_{5}, then since all of the elements of order 5 in 𝔖_{5} are in one and the same conjugacy class, they correspond to a K_{i} with [K_{i} : K] = 24, hence there is no subgroup scheme of G of degree 5.

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[Ra] M. Raynaud: Schémas en groupes de type (p, ..., p), Bull. Soc. math. France 102 (1974), 241-280 [Ro] M. Rosenlicht: Some basic theorems on algebraic groups, Amer. J. Math. 78

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[TO] J. Tate & F. Oort: Group schemes of prime order, Ann. Sc. Ecole Norm. Sup. 3 (1970), 1-21