ON THE DESCENDING CENTRAL SEQUENCE OF ABSOLUTE GALOIS GROUPS

IDO EFRAT AND JÁN MINÁČ

Abstract. Let p be an odd prime number and F a field containing a primitive pth root of unity. We prove a new restriction on the group-theoretic structure of the absolute Galois group G_{F} of F. Namely, the third subgroup G_{F}^{(3)} in the descending p-central sequence of G_{F} is the intersection of all open normal subgroups N such that G_{F} /N is 1, ℤ/p^{2}, or the extra-special group M_{p^{3}} of order p^{3} and exponent p^{2}.

1. Introduction

Let q = p^{d} be a prime power and let G be a profinite group. The descending q-central sequence of G is defined inductively by

G^{(1)} = G, G^{(i+1)} = (G^{(i)})^{q}[G^{(i)}, G], i = 1, 2, ….

Thus G^{(i+1)} is the closed subgroup of G generated by all powers h^{q} and all commutators [h, g] = h^{−1}g^{−1}hg, where h ∈ G^{(i)} and g ∈ G.

Now suppose that q = p. Let F be a field containing a primitive pth root of unity ζ_{p}, and let G = G_{F} be its absolute Galois group. Let M_{p^{3}} be the unique nonabelian group of order p^{3} and exponent p^{2} (see §8).

Main Theorem. For p ≠ 2 and for G = G_{F} as above, G^{(3)} is the intersection of all open normal subgroups N of G such that G/N is isomorphic to one of 1, ℤ/p^{2}, and M_{p^{3}}.

Determining the profinite groups which are realizable as absolute Galois groups of fields is a major open problem in Galois theory. Our Main Theorem appears to be simple yet powerful restriction on the possible structure of such groups, and on their quotients G_{F} /G_{F}^{(3)}. These quotients an extremely important invariant of fields, carrying a substantial information about their arithmetical structure. For example, when p = 2 it encodes the orderings and the Witt ring of quadratic forms of F ([MSp90], [MSp96]) as well as some non-trivial valuations [MMS04]. Further, it encodes the entire mod 2 Galois cohomology ring of G_{F} [AKM99, Th. 3.14]. In a forthcoming joint work with S. Chebolu we show that G_{F} /G_{F}^{(3)} can be in fact thought of as a group-theoretic analog of Galois cohomology of F for any p, and use these results to provide new examples of profinite groups which are not realizable as absolute Galois groups of fields.

2000 Mathematics Subject Classification. Primary 12F10; Secondary 12G05, 12E30.

Key words and phrases. descending central sequence, absolute Galois group, Galois cohomology, embedding problem, W -group, Bockstein map.

Ján Mináč was supported in part by National Sciences and Engineering Council of Canada grant R0370A01.

The analog of our Main Theorem for p = 2 was discovered by Villegas in a different formalism [Vil88]. The second author and Spira reformulated and reproved it in [MSp96, Cor. 2.18] using the descending 2-central sequence of G_{F}. Namely, then G^{(3)} = G_{F}^{(3)} is the intersection of all open normal subgroups N of G such that G/N is isomorphic to 1, ℤ/2, ℤ/4, or to the dihedral group D_{4} = M_{8} of order 8.

A main difference between the case p > 2 and the case p = 2 is the existence in the former case of elements in H^{2}((ℤ/p)^{n}, ℤ/p) which are not expressible as sums of cup products of elements in H^{1}((ℤ/p)^{n}, ℤ/p). To handle this new kind of elements we study the Bockstein homomorphism β_{G}: H^{1}(G, ℤ/p) → H^{2}(G, ℤ/p) and its relation to Galois theory.

Our approach is purely cohomological. Thus we prove the Main Theorem more generally for profinite groups G which satisfy two simple conditions on their lower cohomology. These conditions are known to hold for G = G_{F}, with F as above, where they are consequences of the following two Galois-theoretic facts (see §3 for details and terminology):

(i) the Galois symbol K_{2}^{M}(F)/p → H^{2}(G, ℤ/p) is injective (it is actually bijective by the Merkurjev–Suslin theorem, which is a special case of the Rost–Voevodsky’s theorem); and

(ii) β_{G} is the cup product by the Kummer element (ζ_{p}) ∈ H^{1}(G, ℤ/p).

More generally, when q = p^{d} is an arbitrary prime power and F is a field containing a primitive qth root of unity, we characterize G_{F}^{(3)} as the intersection of all open normal subgroups N of G_{F} such that G_{F} /N belongs to a certain cohomologically defined class of finite groups (Theorem 5.2). This is based on the natural generalizations of (i) and (ii) above, as well as the following additional property of G_{F}:

(iii) the map H^{1}(G_{F} , ℤ/q) → H^{1}(G_{F} , ℤ/p^{i}), 1 ≤ i ≤ d, is surjective.

Our analysis applies also to p = 2. Thus we give a new cohomological proof of the above-mentioned result of [Vil88] and [MSp96], and generalize it to profinite groups G satisfying the appropriate conditions on their lower cohomology. We also show that the group ℤ/2 can be omitted from the list unless F is a Euclidean field (Corollary 11.4).

The paper is organized as follows: In §2 we collect various cohomological preliminaries, especially facts related to the Bockstein homomorphism β_{G} and its connections with roots of unity and cup products. In §3 we introduce the key notion of a profinite group of Galois relation type. It axiomatizes the cohomological properties of absolute Galois groups that we need for our proofs ((i)–(iii) above). In §4 we define an abelian group Ω(G) and a homomorphism Λ_{G}: Ω(G) → H^{2}(G, ℤ/q). These extend the cup product ∪: H^{1}(G, ℤ/q)^{⊗2} → H^{2}(G, ℤ/q), but take into account also the Galois-theoretic role of β_{G}. Our axioms on G imply that Ker(Λ_{G}) is generated by elements of simple type (Definition 4.2 and Proposition 4.3). These simple type elements are in turn related to cohomologically defined open subgroups N of G of index dividing q^{3}, which we call “distinguished subgroups”. In §5 we translate the above result about Ker(Λ_{G}) to the language of distinguished subgroups, and prove the crucial Theorem 5.2: for G of Galois relation type, G^{(3)} is the intersection of all distinguished subgroups of G.

In §§6–10 we build a “dictionary” between the images under Λ_{G} of simple type elements of Ω(G) and some special group extensions. The solutions of the resulting embedding problems correspond to distinguished subgroups of G. This is then used in §11 to prove the Main Theorem and the analogous results for p = 2 in the general setting of profinite groups of Galois relation type.

In §12 we study G/G^{(3)} for G of Galois relation type. As a corollary we recover some known “automatic realization” results in Galois theory. Our approach seems to provide a good explanation why these curious automatic realization results are true. Finally, in §13 we give examples showing that all the finite groups in our lists are indeed necessary.

We thank P. Deligne, L. Moret-Baily, A. Shalev, and T. Weigel for their interest in this work and their comments related to talks given at the Israel Mathematical Union 2008 conference and the 2008 conference on Profinite Groups in ESI, Vienna. We also thank Y. Tschinkel for his kind encouragement.

2. cohomological preliminaries

Let p be a prime number, let q = p^{d} be a power of p, and let G be a profinite group. We write H^{i}(G) for the profinite cohomology group H^{i}(G, ℤ/q), where G acts trivially on ℤ/q. Thus H^{1}(G) = Hom(G, ℤ/q) consists of all continuous group homomorphisms G → ℤ/q. We consider H^{∗}(G) = ⨁_{i=0}^{∞} H^{i}(G) as a graded anti-commutative ring with respect to the cup product ∪.

A) Normal Subgroups. Let N be a normal closed subgroup of G. Then G acts canonically on H^{i}(N). Denote the group of all G-invariant elements of H^{i}(N) by H^{i}(N)^{G}. For i = 1 this action is given by φ ↦ φ^{g}, where φ^{g}(n) = φ(g^{−1}ng) for g ∈ G and n ∈ N. Thus H^{1}(N)^{G} consists of all homomorphisms φ: N → ℤ/q which are trivial on N^{q}[N, G].

The next lemma provides a fundamental connection between the descending q-central sequence of G and cohomology.

Lemma 2.1. For a normal closed subgroup N of G one has

⋂{Ker(φ) | φ ∈ H^{1}(N)^{G}} = N^{q}[N, G].

Proof. Consider the natural projection π: N → N̄ = N/N^{q}[N, G]. The abelian torsion group N̄ has Pontryagin dual H^{1}(N̄). By the Pontryagin duality [NSW00, Th. 1.1.8], ⋂_{φ̄∈H^{1}(N̄)} Ker(φ̄) = {0}, whence ⋂_{φ̄∈H^{1}(N̄)} π^{−1}(Ker(φ̄)) = N^{q}[N, G]. Further, if φ̄ ∈ H^{1}(N̄) and φ = inf_{N}(φ̄), then Ker(φ) = π^{−1}(Ker(φ̄)). Finally, by the previous remarks, inf_{N}: H^{1}(N̄) → H^{1}(N)^{G} is an isomorphism. The assertion follows. □

Corollary 2.2. There is a natural non-degenerate pairing

N/N^{q}[N, G] × H^{1}(N)^{G} → ℤ/q.

Corollary 2.3. G^{(i)}/G^{(i+1)} is dual to H^{1}(G^{(i)})^{G} for i ≥ 1.

B) Spectral sequences. Let N be a closed normal subgroup of G. Recall that the Hochschild–Serre spectral sequence

E_{2}^{ij} = H^{i}(G/N, H^{j}(N)) ⇒ H^{i+j}(G)

induces the 5-term exact sequence (2.1)

0 → H^{1}(G/N) ⟶^{inf_{G}} H^{1}(G) ⟶^{res_{N}} H^{1}(N)^{G} ⟶^{trg_{G/N}} H^{2}(G/N) ⟶^{inf_{G}} H^{2}(G).

Here trg_{G/N} is the differential d_{2}^{0,1} of the spectral sequence [NSW00, §2.1]. If N′ is another closed normal subgroup of G and N′ ≤ N, then the projection G/N′ → G/N and the restriction map res_{N′}: H^{j}(N) → H^{j}(N′) induce a spectral sequence morphism from H^{i}(G/N, H^{j}(N)) ⇒ H^{i+j}(G) to H^{i}(G/N′, H^{j}(N′)) ⇒ H^{i+j}(G) [NSW00, pp. 78–79]. In particular, there is a commutative diagram

(2.2)  \xymatrix{H^{1}(N)^{G} \ar[r]^{^{trg^{G/N}}} \ar[d]_{_{res_{N′}}} & H^{2}(G/N) \ar[d]^{_{inf_{G/N′}}} \\ H^{1}(N′)^{G} \ar[r]^{_{trg_{G/N′}}} & H^{2}(G/N′).}

C) Connecting homomorphisms. Let n, m be positive integers. The exact sequences

0 → ℤ/n → ℤ/mn → ℤ/m → 0

0 → \frac{1}{n}ℤ/ℤ ↪ \frac{1}{mn}ℤ/ℤ ⟶^{n} \frac{1}{m}ℤ/ℤ → 0

of trivial G-modules give rise to connecting homomorphisms

\begin{aligned}β_{G,m,n} & : H^{1}(G, ℤ/m) → H^{2}(G, ℤ/n) \\ β′_{G,m,n} & : H^{1}(G, \frac{1}{m}ℤ/ℤ) → H^{2}(G, \frac{1}{n}ℤ/ℤ), \end{aligned}

respectively. When m = n = q is our fixed p-power, we abbreviate

β_{G} = β_{G,q,q}

and call it the Bockstein homomorphism of G. Note that it is functorial in G. We now relate β_{G,m,n} to some other connecting homomorphisms and cup products.

Lemma 2.4. Suppose q = 2. For ψ ∈ H^{1}(G) one has β_{G}(ψ) = ψ ∪ ψ.

Proof. This is straightforward when G ≅ ℤ/2. In the general case, it follows by inflating from G/ Ker(ψ) to G. □

Next let ϵ: H^{1}(G, ℚ/ℤ) → H^{2}(G, ℤ) be the connecting map arising from the short exact sequence of trivial G-modules

0 → ℤ ↪ ℚ → ℚ/ℤ → 0.

Since ℚ is cohomologically trivial, ϵ is in fact an isomorphism. Let

j_{m}: \frac{1}{m}ℤ/ℤ ⟶^{∼} ℤ/m, π_{n}: ℤ → ℤ/n

be the natural maps. A routine computation gives:

Lemma 2.5. β_{G,m,n} ∘ j_{m}^{∗} = π_{n}^{∗} ∘ ϵ on H^{1}(G, \frac{1}{m}ℤ/ℤ).

Now let F be a field. Set (ℚ/ℤ)′ = ⨁_{l≠char F} (ℚ/ℤ)_{l}, where for l prime (ℚ/ℤ)_{l} is the l-primary component of ℚ/ℤ. Assume that char F |̸ n, m. For an integer r consider the r-th Tate twists (\frac{1}{n}ℤ/ℤ)(r), (ℤ/n)(r), and (ℚ/ℤ)′(r) [NSW00, Def. 7.3.6]. Let ι_{n}: (\frac{1}{n}ℤ/ℤ)(r) ⟶^{∼} (ℤ/n)(r) be the isomorphism of multiplication by n. Thus μ_{n} = (ℤ/n)(1) is the G_{F} -module of nth roots of unity, and ι_{n} = j_{n} when r = 0. The exact sequence

(2.3)  0 → (\frac{1}{n}ℤ/ℤ)(r) ↪ (ℚ/ℤ)′(r) ⟶^{n} (ℚ/ℤ)′(r) → 0

gives rise to a connecting homomorphism

δ^{i,r}: H^{i}(G_{F} , (ℚ/ℤ)′(r)) → H^{i+1}(G_{F} , (\frac{1}{n}ℤ/ℤ)(r)).

Lemma 2.6. There is an equality of maps

\begin{aligned}δ^{i,r} ∪ id = id ∪δ^{j,s}: H^{i}(G_{F} , (ℚ/ℤ)′(r)) & × H^{j}(G_{F} , (ℚ/ℤ)′(s)) \\ & → H^{i+j+1}(G_{F} , (\frac{1}{n}ℤ/ℤ)(r + s)). \end{aligned}

Proof. Tensorizing (2.3) with (ℚ/ℤ)′(s) gives the same sequence but with (r + s)-twists, which is also exact. Therefore the composed map

\begin{matrix}H^{i}(G_{F} , (ℚ/ℤ)′(r)) × H^{j}(G_{F} , & (ℚ/ℤ)′(s)) ⟶^{∪} H^{i+j}(G_{F} , (ℚ/ℤ)′(r + s)) \\ ⟶^{δ^{i+j,r+s}} H^{i+j+1}(G_{F} , (\frac{1}{n}ℤ/ℤ)(r + s)) \end{matrix}

breaks as δ^{i,r} ∪ id [GS06, Prop. 3.4.8]. Similarly, it breaks also as id ∪δ^{j,s}, and the equality follows. □

For F and n as above, consider the Kummer homomorphism

κ_{n}: F^{×} = H^{0}(G_{F} , F_{sep}^{×}) → H^{1}(G_{F} , μ_{n}).

Lemma 2.7. (a) β′_{G_{F} ,m,n} is the restriction of δ^{1,0} to H^{1}(G_{F} , \frac{1}{m}ℤ/ℤ). (b) ι_{n}^{∗} ∘ δ^{0,1} = κ_{n} ∘ ι_{m}^{∗} on H^{0}(G_{F} , (\frac{1}{m}ℤ/ℤ)(1)).

Proof. For every r there is a commutative diagram with exact rows

[figure: 0; (\frac{1}{n}ℤ/ℤ)(r); (\frac{1}{mn}ℤ/ℤ)(r); n; (\frac{1}{m}ℤ/ℤ)(r); 0; 0; (\frac{1}{n}ℤ/ℤ)(r); (ℚ/ℤ)′(r); n; (ℚ/ℤ)′(r); 0.]

It gives rise to a commutative square of connecting homomorphisms

(2.4)  [figure: H^{i}(G_{F} , (\frac{1}{m}ℤ/ℤ)(r)); δ; H^{i+1}(G_{F} , (\frac{1}{n}ℤ/ℤ)(r)); H^{i}(G_{F} , (ℚ/ℤ)′(r)); δ^{i,r}; H^{i+1}(G_{F} , (\frac{1}{n}ℤ/ℤ)(r)).]

For i = 1 and r = 0 we have δ = β′_{G_{F} ,m,n}, and the left vertical map in (2.4) is an embedding. This proves (a).

Next take in (2.4) i = 0 and r = 1 and consider the resulting connecting map δ. From the commutative diagram with exact rows

[figure: 0; (\frac{1}{n}ℤ/ℤ)(1); ι_{n}; ≀; (\frac{1}{mn}ℤ/ℤ)(1); n; ι_{mn}; ≀; (\frac{1}{m}ℤ/ℤ)(1); ι_{m}; ≀; 0; 1; μ_{n}; μ_{mn}; n; μ_{m}; 1; 1; μ_{n}; F_{sep}^{×}; n; F_{sep}^{×}; 1,]

we get that ι_{n}^{∗} ∘ δ = κ_{n} ∘ ι_{m}^{∗}. Combined with (2.4), this gives (b). □

Corollary 2.8. Let d = gcd(m, n).

(a) There is an equality of maps

\begin{matrix}ι_{d}^{∗}∘ & (β′_{G_{F} ,m,n} ∪ id) = ι_{m}^{∗} ∪ (κ_{n} ∘ ι_{m}^{∗}): \\ H^{1}(G_{F} , \frac{1}{m}ℤ/ℤ) × H^{0}(G_{F} , (\frac{1}{m}ℤ/ℤ)(1)) → H^{2}(G_{F} , μ_{d}). \end{matrix}

(b) There is an equality of maps

β_{G_{F} ,m,n} ∪ id = id ∪κ_{n}: H^{1}(G_{F} , ℤ/m) × H^{0}(G_{F} , μ_{m}) → H^{2}(G_{F} , μ_{d}).

Proof. As (\frac{1}{m}ℤ/ℤ) ⊗ (\frac{1}{n}ℤ/ℤ) ≅ \frac{1}{d}ℤ/ℤ, Lemma 2.6 gives

ι_{d}^{∗} ∘ (δ^{1,0} ∪ id) = ι_{d}^{∗} ∘ (id ∪δ^{0,1}) = ι_{m}^{∗} ∪ (ι_{n}^{∗} ∘ δ^{0,1})

on H^{1}(G_{F} , \frac{1}{m}ℤ/ℤ)×H^{0}(G_{F} , (\frac{1}{m}ℤ/ℤ)(1)). By Lemma 2.7, this restricts to (a). (b) follows from (a). □

See [Led05, p. 91], [GS06, Lemma 7.5.10], and [Koc02, Th. 8.13] for related results.

D) Cohomology of finite abelian p-groups. For a profinite group G, let H_{dec}^{i}(G) be the decomposable part of H^{i}(G), i.e., its subgroup generated by cup products of elements of H^{1}(G). In this subsection we show that when G = (ℤ/q)^{n}, the group H^{2}(G) is generated by H_{dec}^{i}(G) and the image of β_{G}. In fact, for every finite abelian p-group G of exponent divisible by q = p^{d}, the structure of H^{∗}(G) as a graded ring was computed by Chapman (for p ≠ 2) and by Townsley-Kulich (for p = 2), in terms of generators and relations ([Cha82], [TK88]). Since the identification of the Bockstein elements as generators is somewhat implicit in [Cha82] and [TK88], we outline an alternative proof of the required result. It is based on the following decomposition of H^{2}(G) to its symmetric and skew-symmetric parts, as studied by Tignol and Amitsur ([TA85], [Tig86]); see also Massy [Mas87].

Let G be a finite abelian group and A be a finite trivial G-module. Call a map a: G × G → A skew-symmetric if it is ℤ-bilinear and a(σ, σ) = 0 for all σ ∈ G. Then a(σ, τ) = −a(τ, σ) for σ, τ ∈ G. The set Skew(G, A) of all such maps forms an abelian group under addition.

For a 2-cocycle f ∈ Z^{2}(G, A) define a_{f} ∈ Skew(G, A) by a_{f}(σ, τ) = f(σ, τ) − f(τ, σ). We call f symmetric if a_{f} = 0. Since the action of G on A is trivial, 2-coboundries are symmetric. Let H^{2}(G, A)_{sym} be the subgroup of H^{2}(G, A) consisting of all cohomology classes of symmetric 2-cocyles. The map f ↦ a_{f} induces a homomorphism Ψ with a split exact sequence [TA85, Prop. 1.3]

(2.5)  0 → H^{2}(G, A)_{sym} → H^{2}(G, A) ⟶^{Ψ} Skew(G, A) → 0.

For ϵ as above, ϵ ∪ id gives by [Tig86, Prop. 1.5] an isomorphism

(2.6)  H^{1}(G, ℚ/ℤ) ⊗_{ℤ} A ⟶^{∼} H^{2}(G, A)_{sym},

Now let A = ℤ/q.

Proposition 2.9. For G = (ℤ/q)^{n} one has Ψ(H_{dec}^{2}(G)) = Skew(G, ℤ/q).

Proof. Write G = ⟨σ_{1}⟩×⋯×⟨σ_{n}⟩ with σ_{i} of order q. Take χ_{1}, …, χ_{n} ∈ H^{1}(G) such that χ_{i}(σ_{i}) = 1 for all i, and χ_{i}(σ_{k}) = 0 for i ≠ k. For distinct i, j the cohomology class χ_{i} ∪ χ_{j} is represented by the 2-cocyle (σ, τ) ↦ χ_{i}(σ)χ_{j}(τ). Hence

(Ψ(χ_{i} ∪ χ_{j}))(σ_{k}, σ_{l}) = χ_{i}(σ_{k})χ_{j}(σ_{l}) − χ_{i}(σ_{l})χ_{j}(σ_{k})

is 1 if (i, j) = (k, l), is −1 if (i, j) = (l, k), and is 0 otherwise.

Now given a ∈ Skew(G, ℤ/q), take φ = ∑_{i<j} a(σ_{i}, σ_{j}) · χ_{i} ∪ χ_{j}. For k < l we get (Ψ(φ))(σ_{k}, σ_{l}) = a(σ_{k}, σ_{l}). But maps in Skew(G, ℤ/q) are determined by their values on (σ_{k}, σ_{l}), k < l. Hence Ψ(φ) = a. □

Proposition 2.10. Let G be a finite abelian p-group. Then β_{G} maps H^{1}(G) isomorphically onto H^{2}(G)_{sym}.

Proof. As (ℤ/q) ⊗_{ℤ} (ℤ/q) ≅ ℤ/q, the isomorphism (2.6) coincides with

(π_{q}^{∗} ∘ ϵ) ∪ id: H^{1}(G, ℚ/ℤ) ⊗ H^{0}(G) → H_{sym}^{2}(G),

where π_{q}: ℤ → ℤ/q is the natural map. Moreover, H^{0}(G) = ℤ/q, so

H^{1}(G, \frac{1}{q}ℤ/ℤ) ⊗ H^{0}(G) = H^{1}(G, ℚ/ℤ) ⊗ H^{0}(G).

By Lemma 2.5, (2.6) is therefore also given by

(β_{G} ∘ j_{q}^{∗}) ∪ id: H^{1}(G, \frac{1}{q}ℤ/ℤ) ⊗ H^{0}(G) → H_{sym}^{2}(G),

and the latter isomorphism may be identified with β_{G}. □

Corollary 2.11. Let G = (ℤ/q)^{n}.

(a) H^{2}(G) is generated by H_{dec}^{2}(G) and by the image of β_{G}.

(b) When q = 2 one has H^{2}(G) = H_{dec}^{2}(G).

Proof. (a) follows from Proposition 2.9, Proposition 2.10, and the exact sequence (2.5). (b) follows from (a) and Lemma 2.4. □

3. Groups of Galois relation type

Let G be again a profinite group. The cup product ∪: H^{1}(G) × H^{1}(G) → H^{2}(G) uniquely extends to a homomorphism

(3.1)  ∪: H^{1}(G) ⊗_{ℤ} H^{1}(G) → H^{2}(G), α ↦ ∪α.

Definition 3.1. We say that G has Galois relation type if:

(i) the kernel of the homomorphism (3.1) is generated by elements of the form ψ ⊗ ψ′, where ψ, ψ′ ∈ H^{1}(G);

(ii) there exists ξ ∈ H^{1}(G) such that for every ψ ∈ H^{1}(G) one has

ψ ∪ ξ + β_{G}(ψ) = 0; and (iii) the natural map H^{1}(G) = H^{1}(G, ℤ/q) → H^{1}(G, ℤ/p^{i}) is sur-

jective for 1 ≤ i ≤ d (where q = p^{d}).

As a main example, consider a field F of characteristic ≠ p and containing a (fixed) primitive qth root of unity ζ_{q}. Let G_{F} be the absolute Galois group of F. Let K_{i}^{M}(F) be the ith Milnor K-group of F, and consider the Galois symbol K_{i}^{M}(F)/q → H^{i}(G_{F} , ℤ/q). It is an isomorphism for i = 1, 2, by the Kummer theory and the Merkurjev– Suslin theorem ([MeSu82], [GS06, Th. 8.6.5]), respectively. Moreover, it induces a commutative square

(3.2)  \xymatrix{(F^{×}/(F^{×})^{q}) ⊗_{ℤ} (F^{×}/(F^{×})^{q}) \ar[r]^{∼} \ar[d] & H^{1}(G_{F}) ⊗_{ℤ} H^{1}(G_{F}) \ar[d]^{∪} \\ K_{2}^{M}(F)/q \ar[r]^{∼} & H^{2}(G_{F}).}

Here the left vertical map is given by

∑_{i=1}^{n}(a_{i}(F^{×})^{q} ⊗ b_{i}(F^{×})^{q}) ↦ ∑_{i=1}^{n}{a_{i}, b_{i}} + qK_{2}^{M}(F)

and is surjective. Its kernel is the Steinberg group, generated by all a(F^{×})^{q} ⊗ b(F^{×})^{q} with 1 ∈ a(F^{×})^{q} + b(F^{×})^{q} [Efr06, §24.1]. We obtain:

Proposition 3.2. G = G_{F} has Galois relation type.

Proof. By definition, the Steinberg group is generated by elements a(F^{×})^{q} ⊗ b(F^{×})^{q} which are mapped to 0 in K_{2}^{M}(F)/q. Now use the surjectivity (resp., injectivity) of the upper (resp., lower) horizontal map in (3.2) to deduce (i).

By Corollary 2.8(b), for ψ ∈ H^{1}(G_{F}) one has β_{G_{F}} (ψ)∪ζ_{q} = ψ∪κ_{q}(ζ_{q}), where on the left hand side we consider ζ_{q} as an element of H^{0}(G_{F} , μ_{q}). Identifying μ_{q} with ℤ/q via ζ_{q}^{i} ↦ ī, we get β_{G_{F}} (ψ) = ψ ∪ κ_{q}(ζ_{q}) in H^{2}(G_{F} , ℤ/q) = H^{2}(G_{F} , μ_{q}). Thus (ii) holds by taking ξ = −κ_{q}(ζ_{q}).

Finally, let 1 ≤ i ≤ d. By Kummer’s theory, the natural epimorphism F^{×}/(F^{×})^{q} → F^{×}/(F^{×})^{p^{i}} yields an epimorphism H^{1}(G_{F} , ℤ/q) → H^{1}(G_{F} , ℤ/p^{i}), proving (iii). □

Remark 3.3. Using also the surjectivity of the Galois symbol in dimension 2, one can strengthen Proposition 3.2 to Galois groups G = Gal(E/F), where E/F is a Galois extension, F contains a primitive qth root of unity, and E has no proper p-extensions. Indeed, then H^{1}(G_{E}) = 0. In (3.2) all maps are surjective. Applying it for E, we obtain that H^{2}(G_{E}) = 0 as well. It therefore follows from the Hochschild–Serre spectral sequence that inf_{G_{F}} : H^{i}(G) → H^{i}(G_{F}) is an isomorphism for i = 1, 2 [NSW00, Prop. 2.1.3]. Since the cup product and the Bockstein homomorphisms commute with inflation, conditions (i)–(iii) for G_{F} now transform into the analogous conditions for G.

4. Cohomology elements of simple type

For a profinite group G we define an abelian group Ω(G) by

Ω(G) = \begin{cases}H^{1}(G) ⊗_{ℤ} H^{1}(G), & if q = 2, \\ (H^{1}(G) ⊗_{ℤ} H^{1}(G)) ⊕ H^{1}(G), & if q ≠ 2. \end{cases}

Define a homomorphism Λ_{G}: Ω(G) → H^{2}(G) as follows:

\begin{matrix}Λ_{G}(α) = ∪α, & if q = 2, \\ Λ_{G}(α_{1}, α_{2}) = ∪α_{1} + β_{G}(α_{2}), & if q ≠ 2. \end{matrix}

The map G ↦ Ω(G) is functorial. Given an epimorphism G_{1} → G_{2} of profinite groups, the inflation map inf_{G_{1}} : H^{1}(G_{2}) → H^{1}(G_{1}) induces a homomorphism inf_{G_{1}} : Ω(G_{2}) → Ω(G_{1}) with a commutative square:

(4.1)  \xymatrix{Ω(G_{2}) \ar[r]^{^{inf^{G_{1}}}} \ar[d]_{Λ_{G_{2}}} & Ω(G_{1}) \ar[d]^{Λ_{G_{1}}} \\ H^{2}(G_{2}) \ar[r]^{_{inf_{G_{1}}}} & H^{2}(G_{1}).}

Lemma 4.1. Assume that G has Galois relation type and let G̃ = G/G^{(2)}. Then ΛG̃ is surjective.

Proof. For 1 ≤ i ≤ d, the natural map H^{1}(G) → H^{1}(G, ℤ/p^{i}) is just the natural map Hom(G̃, ℤ/q) → Hom(G̃, ℤ/p^{i}), so by Definition 3.1(iii), it is surjective. Since additionally G̃ is abelian of exponent dividing q, it is therefore an inverse limit of finite groups G̃_{j} of the form (ℤ/q)^{n_{j}}. By Corollary 2.11, each H^{2}(G̃_{j}) is generated by the images of ∪ and βG̃_{j} (and of ∪ only, if q = 2). Hence each ΛG̃_{j} is surjective. Conclude that ΛG̃ = \underrightarrow{lim} ΛG̃_{j} is surjective. □

Definition 4.2. We say that α ∈ Ω(G) has simple type if either: (i) q = 2 and α = ψ ⊗ ψ′ for some ψ, ψ′ ∈ H^{1}(G); or (ii) q ≠ 2 and α = (ψ ⊗ ψ′, ψ) for some ψ, ψ′ ∈ H^{1}(G).

We call M = Ker(ψ)∩Ker(ψ′) is a kernel of α (it may depend on ψ, ψ′). Observe that M is a normal open subgroup of G and that (ψ, ψ′) induce an embedding of G/M in (ℤ/q)^{2}. Hence G^{(2)} ≤ M. Note that inflation homomorphisms map simple type elements to simple type elements.

Proposition 4.3. Assume that G has Galois relation type. Then the group Ker(Λ_{G}) is generated by elements of simple type.

Proof. For q = 2, this is just Definition 3.1(i).

So suppose that q ≠ 2 and let α ∈ Ker(Λ_{G}). There exists ψ_{0} ∈ H^{1}(G) with α − (0, ψ_{0}) ∈ (H^{1}(G) ⊗ H^{1}(G)) ⊕ {0}. Take ξ ∈ H^{1}(G) as in Definition 3.1(ii). Thus ψ_{0} ∪ ξ + β_{G}(ψ_{0}) = 0, i.e., Λ_{G}(ψ_{0} ⊗ ξ, ψ_{0}) = 0. Let α′ = α − (ψ_{0} ⊗ ξ, ψ_{0}). Then α′ ∈ (H^{1}(G) ⊗ H^{1}(G)) ⊕ {0} and Λ_{G}(α′) = Λ_{G}(α) = 0. By Definition 3.1(i), there exist ψ_{i}, ψ′_{i} ∈ H^{1}(G), i = 1, …, n, with α′ = ∑_{i=1}^{n}(ψ_{i} ⊗ ψ′_{i}, 0) and ψ_{i} ∪ ψ′_{i} = 0 for all i. For each i we have Λ_{G}(ψ_{i} ⊗ ξ, ψ_{i}) = 0. Then

(4.2) α = (ψ_{0} ⊗ ξ, ψ_{0}) + ∑_{i=1}^{n}(ψ_{i} ⊗ (ψ′_{i} + ξ), ψ_{i}) − ∑_{i=1}^{n}(ψ_{i} ⊗ ξ, ψ_{i}).

Here all summands are simple type elements in Ker(Λ_{G}). □

Lemma 4.4. Let α ∈ Ker(Λ_{G}) have simple type and kernel M. Then there exist φ ∈ H^{1}(M)^{G} and ᾱ ∈ Ω(G/M) of simple type and with trivial kernel, such that inf_{G}(ᾱ) = α and Λ_{G/M}(ᾱ) = trg_{G/M}(φ).

Proof. Take ψ, ψ′ as in Definition 4.2 with M = Ker(ψ) ∩ Ker(ψ′). There exist ψ̄, ψ̄′ ∈ H^{1}(G/M) such that inf_{G}(ψ̄) = ψ and inf_{G}(ψ̄′) = ψ′. We define ᾱ ∈ Ω(G/M) to be ψ̄⊗ ψ̄′, if q = 2, and (ψ̄⊗ ψ̄′, ψ̄), if q ≠ 2. Thus ᾱ has simple type and trivial kernel, and inf_{G}(ᾱ) = α. By (4.1) and (2.1), there is a commutative diagram with an exact row

\xymatrix{ & Ω(G/M) \ar[r]^{^{inf^{G}}} \ar[d]_{Λ_{G/M}} & Ω(G) \ar[d]^{Λ_{G}} \\ H^{1}(M)^{G} \ar[r]^{_{trg_{G/M}}} & H^{2}(G/M) \ar[r]^{_{inf_{G}}} & H^{2}(G).}

It yields φ ∈ H^{1}(M)^{G} as required. □

Definition 4.5. Call a subgroup N of G distinguished if there is an open subgroup M of G and elements φ ∈ H^{1}(M)^{G} and ᾱ ∈ Ω(G/M), with ᾱ of simple type and with trivial kernel, such that

Λ_{G/M}(ᾱ) = trg_{G/M}(φ), N = Ker(φ).

In this case we say that M, φ, ᾱ are data for N.

Remark 4.6. Since ᾱ ∈ Ω(G/M) has trivial kernel, G/M embeds in (ℤ/q)^{2}. Hence (G : N) = (G : M)(M : N)|q^{3} and G^{(2)} ≤ M. Also, the exponent of G/N divides q^{2}.

Example 4.7. For every ψ ∈ H^{1}(G), the subgroup M = Ker(ψ) of G is distinguished. Indeed, take ψ̄ ∈ H^{1}(G/M) with inf_{G}(ψ̄) = ψ and set ᾱ = 0 ∈ Ω(G/M). Trivially, ᾱ = ψ̄ ⊗ 0 if q = 2, and ᾱ = (0 ⊗ ψ̄, 0) if q ≠ 2. Thus ᾱ has simple type and trivial kernel. For φ = 0 ∈ H^{1}(M)^{G} we have trg_{G/M}(φ) = Λ_{G/M}(ᾱ) = 0 and M = Ker(φ).

5. G^{(3)} as an intersection

Let G be again a profinite group, and let Δ_{G} be the intersection of all distinguished subgroups of G.

Proposition 5.1. G^{(3)} ≤ Δ_{G} ≤ G^{(2)}.

Proof. Let N be a distinguished subgroup of G. Thus there exists an open normal subgroup M of G and φ ∈ H^{1}(M)^{G} such that Ker(φ) = N and G^{(2)} ≤ M. Hence Lemma 2.1 gives

G^{(3)} = (G^{(2)})^{q}[G^{(2)}, G] ≤ M^{q}[M, G] ≤ Ker(φ) = N.

Consequently, G^{(3)} ≤ Δ_{G}.

By Lemma 2.1 again, ⋂_{ψ∈H^{1}(G)} Ker(ψ) = G^{(2)}. Since each Ker(ψ) is distinguished (Example 4.7), we get that Δ_{G} ≤ G^{(2)}. □

Theorem 5.2. If G has Galois relation type, then G^{(3)} = Δ_{G}.

Proof. By Proposition 5.1, G^{(3)} ≤ Δ_{G}.

For the converse inclusion, let G̃ = G/G^{(2)}. It follows from Lemma 2.1 (with N = G) that the map res_{G^{(2)}} : H^{1}(G) → H^{1}(G^{(2)}) is trivial. Hence, by (2.1), inf_{G}: H^{1}(G̃) → H^{1}(G) is an isomorphism. Consequently, inf_{G}: Ω(G̃) → Ω(G) is also an isomorphism.

Now let φ ∈ H^{1}(G^{(2)})^{G}. By Lemma 4.1, ΛG̃ is surjective, so there exists α̃ ∈ Ω(G̃) with trgG̃(φ) = ΛG̃(α̃). By (4.1) and (2.1),

Λ_{G}(inf_{G}(α̃)) = inf_{G}(ΛG̃(α̃)) = inf_{G}(trgG̃(φ)) = 0.

By Proposition 4.3 we may therefore write inf_{G}(α̃) = ∑_{i=1}^{n} α_{i}, where α_{1}, …, α_{n} ∈ Ker(Λ_{G}) have simple type.

For each 0 ≤ i ≤ n let M_{i} be a kernel for α_{i}. Recall that G^{(2)} ≤ M_{i}, so (4.1) again gives a commutative diagram

(5.1)  \xymatrix{Ω(G/M_{i}) \ar[r]^{^{infG̃}} \ar[d]_{Λ_{G/M_{i}}} & Ω(G̃) \ar[r]^{^{inf^{G}}} \ar[d]^{ΛG̃} & Ω(G) \ar[d]^{Λ_{G}} \\ H^{2}(G/M_{i}) \ar[r]^{_{infG̃}} & H^{2}(G̃) \ar[r]^{_{inf_{G}}} & H^{2}(G).}

Lemma 4.4 gives rise to ᾱ_{i} ∈ Ω(G/M_{i}) of simple type and with trivial kernel and to φ_{i} ∈ H^{1}(M_{i})^{G} such that inf_{G}(ᾱ_{i}) = α_{i} and Λ_{G/M_{i}}(ᾱ_{i}) = trg_{G/M_{i}}(φ_{i}). In particular, Ker(φ_{i}) is distinguished. For each i let α̃_{i} = infG̃(ᾱ_{i}). It also has simple type, and one has α_{i} = inf_{G}(α̃_{i}). By (5.1), inf_{G}(ΛG̃(α̃_{i})) = 0. Moreover,

inf_{G}(α̃) = ∑_{i=1}^{n} α_{i} = ∑_{i=1}^{n} inf_{G}(α̃_{i}).

But inf_{G}: Ω(G̃) → Ω(G) is an isomorphism, so α̃ = ∑_{i=1}^{n} α̃_{i}.

Next (2.1) and (2.2) give a commutative diagram with an exact row:

\xymatrix{ & H^{1}(M_{i})^{G} \ar[r]^{^{trg^{G/Mi}}} \ar[d]_{_{res_{G(2)}}} & H^{2}(G/M_{i}) \ar[d]^{_{infG̃}} \\ 0 \ar[r] & H^{1}(G^{(2)})^{G} \ar[r]^{_{trgG̃}} & H^{2}(G̃).}

Using this and (5.1) we compute:

trgG̃(φ) = ΛG̃(α̃) = ∑_{i=1}^{n} ΛG̃(α̃_{i}) = ∑_{i=1}^{n} ΛG̃(infG̃(ᾱ_{i})) = ∑_{i=1}^{n} infG̃(Λ_{G/M_{i}}(ᾱ_{i}))

= ∑_{i=1}^{n}(infG̃ ∘ trg_{G/M_{i}})(φ_{i}) = ∑_{i=1}^{n}(trgG̃ ∘ res_{G^{(2)}})(φ_{i}).

Since trgG̃ is injective, φ = ∑_{i=1}^{n} res_{G^{(2)}}(φ_{i}), so by Proposition 5.1,

Ker(φ) ≥ ⋂_{i=1}^{n} Ker(res_{G^{(2)}}(φ_{i})) = G^{(2)} ∩ ⋂_{i=1}^{n} Ker(φ_{i}) ≥ G^{(2)} ∩ Δ_{G} = Δ_{G}.

Since φ ∈ H^{1}(G^{(2)})^{G} was arbitrary, we deduce from Lemma 2.1 that

G^{(3)} = (G^{(2)})^{q}[G^{(2)}, G] = ⋂_{φ∈H^{1}(G^{(2)})^{G}} Ker(φ) ≥ Δ_{G}.  □

Corollary 5.3. Let G be a profinite group of Galois relation type. Then G^{(3)} is an intersection of normal open subgroups N of G with G/N of order dividing q^{3} and exponent dividing q^{2}.

6. Extensions

Let Ḡ be a finite group and A a finite trivial Ḡ-module. We consider central extensions

ω : 0 → A ⟶^{f} B ⟶^{g} Ḡ → 1.

For a group isomorphism θ: Ḡ′ → Ḡ define an extension

ω^{θ} : 0 → A ⟶^{f} B ⟶^{θ^{−1}∘g} Ḡ′ → 1.

When there is a commutative diagram of central extensions

[figure: ω :; 0; A; f; B; g; h; ≀; Ḡ; 1; ω′ :; 0; A′; f; B′; g′; Ḡ; 1,]

with h an isomorphism, ω and ω′ are called equivalent. Let Ext(Ḡ, A) be the set of all equivalence classes [ω] of extensions ω as above.

The Baer sum [CE56, Ch. XIV, §1] of central extensions

ω_{i} : 0 → A ⟶^{f_{i}} B_{i} ⟶^{g_{i}} Ḡ → 1, i = 1, 2,

is the central extension

0 → A ⟶^{\overline{(f_{1},1)}=\overline{(1,f_{2})}} B ⟶^{g_{1}=g_{2}} Ḡ → 1.

where for the fibred product B_{1} ×Ḡ B_{2} we set

B = (B_{1} ×Ḡ B_{2})/{(f_{1}(a), f_{2}(a)^{−1}) | a ∈ A}.

This induces an abelian group structure on Ext(Ḡ, A), which is functorial in Ḡ (contravariantly) and in A (covariantly).

There is a canonical isomorphism Ext(Ḡ, A) ≅ H^{2}(Ḡ, A) which is functorial in both Ḡ and A [NSW00, Th. 1.2.5]. Specifically, the cohomology class of an inhomogeneous normalized 2-cocycle α: Ḡ^{2} → A corresponds to the class of [ω], where B = A×Ḡ as sets, and the group law is given for a, b ∈ A and σ, τ ∈ Ḡ by

(6.1)  (a, σ) ∗ (b, τ) = (a + b + α(σ, τ), στ).

Conversely, given ω as above, choose a set-theoretic section s: Ḡ → B of g with s(1) = 1. The map α: Ḡ × Ḡ → A, given by α(σ̄_{1}, σ̄_{2}) = s(σ̄_{1})s(σ̄_{2})s(σ̄_{1}σ̄_{2})^{−1}, is an inhomogenous normalized 2-cocyle whose cohomology class corresponds to [ω].

Remark 6.1 ([GS06, Remark 3.3.11], [Led05, p. 33]). Let Ḡ → G̃ be an epimorphism and let A be a G̃-module, whence a Ḡ-module in the natural way. Then infḠ: H^{2}(G̃, A) → H^{2}(Ḡ, A) corresponds to the map infḠ: Ext(G̃, A) → Ext(Ḡ, A) sending the class of

ω̃ : 0 → A ⟶^{f} B ⟶^{g} G̃ → 1

to the class of

infḠ(ω̃) : 0 → A ⟶^{(f,1)} B ×G̃ Ḡ ⟶^{(b,σ̄)↦σ̄} Ḡ → 1.

In particular we have:

Lemma 6.2. Suppose that Ḡ = G̃ × G̃′ and let ω̃ be as above. Then infḠ(ω̃) is equivalent to

0 → A ⟶^{(f,1)} B × G̃′ ⟶^{g×id} G̃ × G̃′ → 1.

Proof. Use the commutative triangle

\xymatrix{B ×G̃ Ḡ \ar[rr]^{(b,(σ̃,σ̃′))↦(b,σ̃′)} \ar[dr]_{(b,(σ̃,σ̃′))↦(σ̃,σ̃′)} &  & B × G̃′ \ar[dl]^{g×id} \\  & Ḡ}

where the horizontal map is an isomorphism. □

7. Embedding problems

Let G be a profinite group. The following proposition is due to Hoechsmann [Hoe68, 2.1]:

Proposition 7.1. Let M be an open normal subgroup of G. Consider the embedding problem

(7.1)  \xymatrix{ &  &  &  & G \ar[dl]_{Φ} \ar[d] \\ ω : & 0 \ar[r] & A \ar[r] & B \ar[r] & G/M \ar[r] & 1}

where A is a finite G/M-module, and let α ∈ H^{2}(G/M, A) be the cohomology class corresponding to [ω]. Then the restriction map Φ ↦ φ = Φ|_{M} is a bijection between

(a) the continuous homomorphisms Φ: G → B making (7.1) commutative; and

(b) elements φ of H^{1}(M, A)^{G} with trg_{G/M}(φ) = α. In particular, there exists Φ as in (a) if and only if inf_{G}(α) = 0.

Here the last sentence follows from the bijection using (2.1).

Now suppose that A = ℤ/q with the trivial G-action, where q = p^{d}.

Proposition 7.2. Let M be an open normal subgroup of G and let ᾱ ∈ Ω(G/M) have simple type and trivial kernel. The following conditions on an open subgroup N of M are equivalent:

(a) N is a distinguished subgroup of G with data M, ᾱ;

(b) N is normal in G and there is a commutative diagram

\xymatrix{ &  &  &  & G/N \ar[d] \ar[dl]_{h} \\ ω : & 0 \ar[r] & ℤ/q \ar[r] & B \ar[r] & G/M \ar[r] & 1,}

where ω is an extension corresponding to Λ_{G/M}(ᾱ), the vertical map is the natural projection, and h is a monomorphism.

Proof. (a)⇒(b): By assumption, there exists φ ∈ H^{1}(M)^{G} such that Λ_{G/M}(ᾱ) = trg_{G/M}(φ) ∈ H^{2}(G/M) and N = Ker(φ). In particular, N is normal in G. Choose a central extension ω as above corresponding to Λ_{G/M}(ᾱ). Proposition 7.1 yields a continuous homomorphism Φ: G → B such that (7.1) commutes and φ = Φ|_{M}. Then N = Ker(φ) = M ∩ Ker(Φ). Consequently, Φ induces a homomorphism h: G/N → B whose restriction to M/N is injective. It follows that h is also injective.

(b)⇒(a): Lift h to a homomorphism Φ: G → B with kernel N. Then (7.1) commutes. Then φ = Φ|_{M} ∈ H^{1}(M)^{G}. By Proposition 7.1, Λ_{G/M}(ᾱ) = trg_{G/M}(φ) and N = M ∩ Ker(Φ) = Ker(φ), giving (a). □

8. Special extensions

Proposition 7.2 allows an explicit determination of the distinguished subgroups N of a profinite group G by means of the quotients G/N. We now carry this computation for q = p prime, based on an analysis of several central extensions of small p-groups. We first recall the structure of the nonabelian groups of order p^{3}. When p = 2 these are:

• the dihedral group of order 8,

D_{4} = ⟨r, s | r^{4} = s^{2} = (rs)^{2} = 1⟩;

• the quaternionic group

Q_{8} = ⟨r, s | r^{4} = 1, [r, s] = r^{2} = s^{2}⟩.

For p odd, there are two isomorphism types of groups of order p^{3}: • the Heisenberg group of order p^{3} and exponent p,

H_{p^{3}} = ⟨r, s, t | r^{p} = s^{p} = t^{p} = 1, [r, t] = [s, t] = 1, [r, s] = t⟩; • the extra-special group of order p^{3} and exponent p^{2},

M_{p^{3}} = ⟨r, s | r^{p^{2}} = s^{p} = 1, [r, s] = r^{p}⟩.

8.1. Remarks. (a) When p = 2, M_{8} = D_{4}. However, we will keep in this case the traditional notation D_{4}, and write M_{p^{3}} only when p ≠ 2.

(b) Let G be one of the groups D_{4}, Q_{8}, when p = 2, or H_{p^{3}}, M_{p^{3}}, when p ≠ 2. Then the unique normal subgroup of G of order p is its center, which coincides with the Frattini subgroup G^{(2)} [MN77, §3.1]. Therefore G^{(3)} = (G^{(2)})^{p}[G^{(2)}, G] = 1.

(c) In M_{p^{3}} (for p ≠ 2) one has r^{j}s^{i} = s^{i}r^{(1+ip)j} for all i, j ≥ 0. In particular, [s, r^{p}] = 1. Further, by induction, (s^{i}r^{j})^{k} = s^{ki}r^{(1+(k−1)ip/2)kj} for k ≥ 0. It follows that (s^{i}r^{j})^{p} = 1 if and only if p|j.

We define epimorphisms from these groups onto (ℤ/p)^{2} as follows:

θ: D_{4} → (ℤ/2)^{2}, r ↦ (¯1, ¯1), s ↦ (¯0, ¯1);
λ: H_{p^{3}} → (ℤ/p)^{2}, r ↦ (¯1, ¯0), s ↦ (¯0, ¯1), t ↦ (¯0, ¯0);
λ′: M_{p^{3}} → (ℤ/p)^{2}, r ↦ (¯1, ¯0), s ↦ (¯0, ¯1).

Remark 8.2. For later use we note that no proper subgroup of D_{4} (resp., M_{p^{3}}) is mapped surjectively by θ (resp., λ′).

The following central extensions will be needed in the sequel:

ω_{0} : 0 → ℤ/p ⟶^{id} ℤ/p → 0 → 0;

ω_{1} : 0 → ℤ/p ⟶^{ī↦(ī,¯0)} (ℤ/p)^{2} ⟶^{(ī,j̄)↦j̄} ℤ/p → 0;

ω_{2} : 0 → ℤ/p ⟶^{ī↦\overline{pi}} ℤ/p^{2} ⟶^{ī↦ī} ℤ/p → 0;

ω_{3} : 0 → ℤ/2 ⟶^{ī↦r^{2i}} D_{4} ⟶^{θ} (ℤ/2)^{2} → 0;

ω_{4} :  0 → ℤ/p ⟶^{ī↦t^{i}} H_{p^{3}} ⟶^{λ} (ℤ/p)^{2} → 0  (p ≠ 2);

ω_{5} : 0 → ℤ/p ⟶^{ī↦r^{pi}} M_{p^{3}} ⟶^{λ′} (ℤ/p)^{2} → 0 (p ≠ 2);

ω_{6} : 0 → ℤ/p ⟶^{ī↦(\overline{pi},0)} (ℤ/p^{2}) ⊕ (ℤ/p) ⟶^{(ī,j̄)↦(ī,j̄)} (ℤ/p)^{2} → 0.

Thus [ω_{0}], [ω_{1}] are the trivial classes of Ext(0, ℤ/p), Ext(ℤ/p, ℤ/p), respectively, and [ω_{1}] is the inflation of [ω_{0}]. Likewise

(8.1)  inf_{(ℤ/p)^{2}}([ω_{2}]) = [ω_{6}]

relative to the projection pr_{1} : (ℤ/p)^{2} → ℤ/p on the first coordinate.

Lemma 8.3. For p ≠ 2, the Baer sum of [ω_{4}] and [ω_{6}] is [ω_{5}].

Proof. Let

B̃ = ⟨r̃, s̃, t̃ | r̃^{p^{2}} = s̃^{p} = t̃^{p} = 1, [r̃, t̃] = [s̃, t̃] = 1, [r̃, s̃] = t̃⟩.

There is a commutative square

[figure: B̃; f; j; H_{p^{3}}; λ; (ℤ/p^{2}) ⊕ (ℤ/p); (ī,j̄)↦(ī,j̄); (ℤ/p)^{2} ,]

where j maps r̃, s̃, t̃ to r, s, t, respectively, and f maps r̃, s̃, t̃ to (¯1, ¯0), (¯0, ¯1), (¯0, ¯0), respectively. Moreover, this square is cartesian, i.e.,

(j, f): B̃ → H_{p^{3}} ×_{(ℤ/p)^{2}} ((ℤ/p^{2}) ⊕ (ℤ/p))

is an isomorphism. The Baer sum is therefore the equivalence class of

0 → ℤ/p ⟶^{ī↦t̃^{i}} B = B̃/⟨t̃^{i} · r̃^{−pi})| ī ∈ ℤ/p⟩ ⟶^{λ∘j} (ℤ/p)^{2} → 0.

Hence B is obtained from B̃ by adding the relation t̃ = r̃^{p}. Using Remark 8.1(c) we deduce that

B ≅ ⟨r, s | r^{p^{2}} = s^{p} = 1, [s, r^{p}] = 1, [r, s] = r^{p}⟩ = M_{p^{3}},

and the Baer sum is [ω_{5}]. □

9. Extensions and simple type elements

We assume again that q = p is prime. Let Ḡ be a finite group. In this section we compute the extensions corresponding to ΛḠ(ᾱ) for ᾱ ∈ Ω(Ḡ) of simple type and trivial kernel. Some of these facts are quite well-known in a Galois setting, as is systematically described in Ledet’s book [Led05] (see also [Frö85, 7.7], [MN77]), but we derive them in a more abstract group-theoretic setting.

A) Cup products. Let ψ̄, ψ̄′ ∈ H^{1}(Ḡ). We compute the extensions corresponding to ψ̄ ∪ ψ̄′ ∈ H^{2}(Ḡ) in various situations. We use the notation ωψ̄ as in the beginning of §6. For the uniformity of the presentation we use this notation also when Ḡ ≅ ℤ/2.

Proposition 9.1. Suppose that Ker(ψ̄) ∩ Ker(ψ̄′) = 1.

(a) If ψ̄ = ψ̄′ = 0, then ψ̄ ∪ ψ̄′ corresponds to ω_{0}.

(b) If ψ̄ ≠ 0, ψ̄′ = 0 (resp., ψ̄ = 0, ψ̄′ ≠ 0), then ψ̄∪ ψ̄′ corresponds

to ωψ̄ ; 1 (resp., ωψ̄′ ; 1).

(c) If p = 2 and ψ̄ = ψ̄′ ≠ 0, then ψ̄ ∪ ψ̄′ corresponds to ωψ̄ ; 2.

(d) If p ≠ 2 and ψ̄, ψ̄′ ≠ 0 are 𝔽_{p}-linearly dependent, then ψ̄ ∪ ψ̄′ corresponds to ωψ̄ ; 1.

(e) If p = 2 and ψ̄, ψ̄′ are 𝔽_{p}-linearly independent, then ψ̄ ∪ ψ̄′ corresponds to ω_{3}^{(ψ̄,ψ̄′)}.

(f) If p ≠ 2 and ψ̄, ψ̄′ are 𝔽_{p}-linearly independent, then ψ̄ ∪ ψ̄′ corresponds to ω_{4}^{(ψ̄,ψ̄′)}.

Proof. Consider the central extension

ω : 0 → ℤ/p ⟶^{f} B ⟶^{g} Ḡ → 1

corresponding to ψ̄ ∪ ψ̄′. An inhomogeneous normalized 2-cocyle Ḡ × Ḡ → ℤ/p representing ψ̄∪ψ̄′ is given by (σ, τ) ↦ ψ̄(σ)·ψ̄′(τ). Therefore B = (ℤ/p) × Ḡ, with the group law

(9.1)  (a, σ) ∗ (b, τ) = (a + b + ψ̄(σ)ψ̄′(τ), στ)

for a, b ∈ ℤ/p and σ, τ ∈ Ḡ (see (6.1)). The trivial element of B is (0, 1), and one has f(a) = (a, 1) and g(a, σ) = σ for a ∈ ℤ/p and σ ∈ Ḡ. By induction,

(a, σ)^{i} = (ia + \frac{i(i−1)}{2} ψ̄(σ)ψ̄′(σ), σ^{i}), i = 0, 1, 2, ….

We examine the various possibilities.

(a) Immediate.

(b) Here ψ̄ (resp., ψ̄′) is an isomorphism Ḡ → ℤ/p and B is just the direct product (ℤ/p) × Ḡ. The assertion follows.

(c) The assumptions imply that ψ̄ = ψ̄′: Ḡ → ℤ/2 is an isomorphism. Let σ_{0} be the generator of Ḡ. Then (0, σ_{0})^{2} = (1, 1) and (0, σ_{0})^{4} = (0, 1) in B. Hence B ≅ ℤ/4 and ω is equivalent to ωψ̄ ; 2.

(d) Here ψ̄: Ḡ → ℤ/p is an isomorphism. Since p ≠ 2 and ∪ is alternate, ψ̄ ∪ ψ̄′ = 0. Hence ω, and therefore also ωψ̄ split, so B ≅ (ℤ/p)^{2}. Moreover, pick b ∈ B such that (ψ̄ ∘ g)(b) = ¯1. Then the map B → (ℤ/p)^{2}, f(¯1) ↦ (¯1, ¯0), b ↦ (¯0, ¯1), is an isomorphism making the following diagram commutative:

[figure: ω :; 0; ℤ/p; f; B = (ℤ/p) × Ḡ; g; ≀; Ḡ; ≀; ψ̄; 1; ω_{1} :; 0; ℤ/p; ī↦(ī,¯0); (ℤ/p)^{2}; (ī,j̄)↦j̄; ℤ/p; 0.]

Thus ω is equivalent to ωψ̄ ; 1.

(e), (f) Here (ψ̄, ψ̄′): Ḡ → (ℤ/p)^{2} is an isomorphism. Take σ_{1}, σ_{2} ∈ Ḡ with

ψ̄(σ_{1}) = 1, ψ̄(σ_{2}) = 0, ψ̄′(σ_{1}) = 0, ψ̄′(σ_{2}) = 1.

When p = 2, we set r̃ = (1, σ_{1}σ_{2}), s̃ = (0, σ_{2}) and compute in B:

r̃^{2} = (1, 1), r̃^{4} = (0, 1), s̃^{2} = (0, 1), r̃s̃ = (0, σ_{1}), (r̃s̃)^{2} = (0, 1).

We get an isomorphism B ≅ D_{4}, r̃ ↦ r, s̃ ↦ s, and a diagram

[figure: ω :; 0; ℤ/2; f; B = (ℤ/2) × Ḡ; g; ≀; Ḡ; (ψ̄,ψ̄′); ≀; 1; ω_{3} :; 1; ℤ/2; ī↦r^{2i}; D_{4}; θ; (ℤ/2)^{2}; 0]

which is commutative with exact rows. Hence ω is equivalent to ω_{3}^{(ψ̄,ψ̄′)}.

For p odd, B has exponent p. Set r̃ = (0, σ_{1}), s̃ = (0, σ_{2}), t̃ = (1, 1). Then

r̃t̃= t̃r̃ = (1, σ_{1}), s̃t̃= t̃s̃ = (1, σ_{2}), r̃s̃ = t̃s̃r̃ = (1, σ_{1}σ_{2}).

This gives an isomorphism B ≅ H_{p^{3}}, r̃ ↦ r, s̃ ↦ s, t̃ ↦ t, and a commutative diagram

[figure: ω :; 0; ℤ/p; f; B = (ℤ/p) × Ḡ; g; ≀; Ḡ; (ψ̄,ψ̄′); ≀; 1; ω_{4} :; 0; ℤ/p; ī↦t^{i}; H_{p^{3}}; λ; (ℤ/p)^{2}; 0.]

Therefore ω is equivalent in this case to ω_{4}^{(ψ̄,ψ̄′)}. □

B) Bockstein elements.

Proposition 9.2. If 0 ≠ ψ̄ ∈ H^{1}(Ḡ) and Ḡ ≅ ℤ/p, then βḠ(ψ̄) corresponds to ωψ̄ ; 2.

Proof. As a connecting homomorphism in a cohomology exact sequence, βḠ: H^{1}(Ḡ) → H^{2}(Ḡ) is defined as follows [NSW00, Ch. I, §3]: let pr: ℤ/p^{2} → ℤ/p be the natural projection. Given a nonzero ψ̄ ∈ H^{1}(Ḡ), we consider it as an inhomogeneous 1-cocycle, and lift it to a map ψ̂: Ḡ → ℤ/p^{2} with ψ̄ = pr ∘ψ̂. Then the map

χ: Ḡ × Ḡ → ℤ/p, χ(σ_{1}, σ_{2}) = ψ̂(σ_{1}) + ψ̂(σ_{2}) − ψ̂(σ_{1}σ_{2})

is a normalized 2-cocycle with cohomology class βḠ(ψ̄).

On the other hand, ψ̄: Ḡ → ℤ/p is an isomorphism, so ψ̂ is a section of the epimorphism ψ̄^{−1} ∘ pr: ℤ/p^{2} → Ḡ. By the remarks in §6, the cohomology class βḠ(ψ̄) of χ therefore corresponds to the extension

ωψ̄ ; 2 :  0 → ℤ/p → ℤ/p^{2} ⟶^{ψ̄^{−1}∘pr} Ḡ → 1.  □

Corollary 9.3. Suppose that p ≠ 2 and let ψ̄, ψ̄′ ∈ H^{1}(Ḡ) be 𝔽_{p}- linearly dependent, ψ̄ ≠ 0. Then ΛḠ(ψ̄ ⊗ ψ̄′, ψ̄) corresponds to ωψ̄ ; 2.

Proof. Since ∪ is alternate, ψ̄ ∪ ψ̄′ = 0. Hence ΛḠ(ψ̄ ⊗ ψ̄′, ψ̄) = βḠ(ψ̄). Now use Proposition 9.2. □

When ψ̄, ψ̄′ are 𝔽_{p}-linearly independent the computation is more involved. It is sufficient for us to consider only the case p ≠ 2.

Proposition 9.4. Suppose that p ≠ 2. Let ψ̄, ψ̄′ ∈ H^{1}(Ḡ) be 𝔽_{p}- linearly independent with Ker(ψ̄) ∩ Ker(ψ̄′) = 1. Then ΛḠ(ψ̄ ⊗ ψ̄′, ψ̄) corresponds to ω_{5}^{(ψ̄,ψ̄′)}.

Proof. We may decompose Ḡ = G̃ × G̃′, where G̃, G̃′ ≅ ℤ/p and there exists ψ̃ ∈ H^{1}(G̃) with infḠ(ψ̃) = ψ̄. By Proposition 9.2, βG̃(ψ̃) corresponds to ωψ̃ ; 2. Hence βḠ(ψ̄) = infḠ(βG̃(ψ̃)) corresponds to infḠ(ωψ̃ ; 2). By Lemma 6.2, this extension is

0 → ℤ/p ⟶^{ī↦(\overline{pi},1)} (ℤ/p^{2}) × G̃′ ⟶^{(ψ̃^{−1}∘pr,id)} G̃ × G̃′ → 1,

where pr: ℤ/p^{2} → ℤ/p is again the natural projection. But the latter extension is equivalent to

0 → ℤ/p ⟶^{ī↦(\overline{pi},0)} (ℤ/p^{2}) × (ℤ/p) ⟶^{(ψ̄^{−1}∘pr,(ψ̄′)^{−1})} Ḡ = G̃ × G̃′ → 1,

which is ω_{6}^{(ψ̄,ψ̄′)}.

Now by Proposition 9.1(f), ψ̄∪ψ̄′ corresponds to ω_{4}^{(ψ̄,ψ̄′)}. It therefore follows from Lemma 8.3 that ψ̄∪ψ̄′ +βḠ(ψ̄) corresponds to ω_{5}^{(ψ̄,ψ̄′)}. □

C) Summary. Putting together the results of the previous two subsections we obtain:

Proposition 9.5. Let ᾱ ∈ Ω(Ḡ) have simple type and trivial kernel. Then ΛḠ(ᾱ) ∈ H^{2}(Ḡ) corresponds to one of the following extensions:

(i) when p = 2: ω_{0}, ωψ̄ ; 1, ωψ̄ ; 2, ω_{3}^{(ψ̄,ψ̄′)};

(ii) when p ≠ 2: ω_{0}, ωψ̄ ; 1, ωψ̄ ; 2, ω_{5}^{(ψ̄,ψ̄′)},

where ψ̄, ψ̄′ are taken as above.

Proof. When p = 2 we have ᾱ = ψ̄ ⊗ ψ̄′, with ψ̄, ψ̄′ ∈ H^{1}(Ḡ) and Ker(ψ̄) ∩ Ker(ψ̄′) = 1. Furthermore, ΛḠ(ᾱ) = ψ̄ ∪ ψ̄′. Now apply Proposition 9.1.

When p ≠ 2 we write ᾱ = (ψ̄⊗ψ̄′, ψ̄) where again ψ̄, ψ̄′ ∈ H^{1}(Ḡ) and Ker(ψ̄) ∩ Ker(ψ̄′) = 1. Here ΛḠ(ᾱ) = ψ̄ ∪ ψ̄′ + βḠ(ψ̄). If ψ̄ = 0, then this corresponds to either ω_{0} or ωψ̄ ; 1, by Proposition 9.1(a)(b). If ψ̄ ≠ 0 and ψ̄, ψ̄′ are 𝔽_{p}-linearly dependent, then ΛḠ(ᾱ) = βḠ(ψ̄) corresponds to ωψ̄ ; 2, by Proposition 9.2. Finally, if ψ̄, ψ̄′ are 𝔽_{p}-linearly independent, then by Proposition 9.4, ΛḠ(ᾱ) corresponds to ω_{5}^{(ψ̄,ψ̄′)}. □

One has the following converse result:

Proposition 9.6. When p = 2 let i ∈ {0, 1, 2, 3} and when p ≠ 2 let i ∈ {0, 1, 2, 5}. Let (ℤ/p)^{s} be the right group in ω_{i} (so s = 0, 1, 1, 2, 2 for i = 0, 1, 2, 3, 5, respectively). Let θ: Ḡ ⟶^{∼} (ℤ/p)^{s} be an isomorphism. There exists ᾱ ∈ Ω(Ḡ) of simple type and with trivial kernel such that ΛḠ(ᾱ) ∈ H^{2}(Ḡ) corresponds to ω_{i}^{θ}.

Proof. We may assume that Ḡ = (ℤ/p)^{s} and θ = id. Let pr_{j}: (ℤ/p)^{2} → ℤ/p be the projection on the jth coordinate, j = 1, 2.

When p = 2 we take ᾱ = ψ̄ ⊗ ψ̄′, where

(ψ̄, ψ̄′) = (0, 0), (id_{ℤ/2}, 0), (id_{ℤ/2}, id_{ℤ/2}), (pr_{1}, pr_{2}),

to obtain using Proposition 9.1(a)(b)(c)(e) ω_{0}, ω_{1}, ω_{2}, ω_{3}, respectively.

When p ≠ 2 we take ᾱ = (ψ̄⊗ψ̄′, ψ̄), where (ψ̄, ψ̄′) = (0, 0), (0, id_{ℤ/p}), to obtain using Proposition 9.1(a)(b) the extensions ω_{0}, ω_{1}, respectively. Also, take ᾱ = (ψ̄ ⊗ ψ̄′, ψ̄), where ψ̄ = ψ̄′ = id_{ℤ/p} to obtain using Corollary 9.3 the extension ω_{2}. Finally, ᾱ = (ψ̄ ⊗ ψ̄′, ψ̄), where ψ̄ = pr_{1}, ψ̄′ = pr_{2}, gives using Proposition 9.4 the extension ω_{5}. □

10. Lifting of homomorphisms

We now apply the computations of the previous section to solve some specific embedding problems.

Lemma 10.1. Let G be a profinite group and ψ: G → ℤ/p an epimorphism. Then β_{G}(ψ) = 0 if and only if ψ factors via the natural map ℤ/p^{2} → ℤ/p.

Proof. Let Ḡ = G/ Ker(ψ) ≅ ℤ/p and let π: G → Ḡ be the natural map. There exists ψ̄ ∈ H^{1}(Ḡ) with ψ = ψ̄ ∘ π and inf_{G}(ψ̄) = ψ. Then inf_{G}(βḠ(ψ̄)) = β_{G}(ψ). By Proposition 9.2, βḠ(ψ̄) corresponds to ωψ̄ ; 2. It follows from the last sentence of Proposition 7.1 that β_{G}(ψ) = 0 if and only if the following embedding problem is solvable

\xymatrix{ & G \ar[dl]_{Φ} \ar[d]^{ψ} \\ ℤ/p^{2} \ar[r] & ℤ/p \ar[r] & 0.}

Note that if the homomorphism Φ exists, then it must be surjective. □

In the next proposition let r, s be the generators of M_{p^{3}} as in §8.

Proposition 10.2. Let p ≠ 2 and let G be a profinite group of Galois relation type. Every epimorphism ψ: G → ℤ/p breaks via one of the epimorphisms:

(i) the natural map ℤ/p^{2} → ℤ/p; (ii) the map λ′′: M_{p^{3}} → ℤ/p, defined by r ↦ ¯1, s ↦ ¯0.

Proof. If β_{G}(ψ) = 0, then by Lemma 10.1, ψ breaks via the map (i).

Next assume that β_{G}(ψ) ≠ 0. Since G has Galois relation type, there exists ξ ∈ H^{1}(G) with ψ ∪ ξ + β_{G}(ψ) = 0. In particular, ψ ∪ ξ ≠ 0. Since p ≠ 2 and the cup product is alternate, ψ and ξ are 𝔽_{p}-linearly independent. Now let Ḡ = G/(Ker(ψ) ∩ Ker(ξ)) ≅ (ℤ/p)^{2}, and let π: G → Ḡ be the canonical map. Take ψ̄, ξ̄ ∈ H^{1}(Ḡ) with ψ = ψ̄ ∘ π, ξ = ξ̄∘ π, inf_{G}(ψ̄) = ψ, and inf_{G}(ξ̄) = ξ. Then

inf_{G}(ΛḠ(ψ̄ ⊗ ξ̄, ψ̄)) = Λ_{G}(ψ ⊗ ξ, ψ) = 0.

By Proposition 9.4, ΛḠ(ψ̄⊗ξ̄, ψ̄) corresponds to ω_{5}^{(ψ̄,ξ̄)}. It follows again from Proposition 7.1 that the embedding problem

\xymatrix{ & G \ar[dl]_{Φ} \ar[d]^{(ψ,ξ)} \\ M_{p^{3}} \ar[r]^{λ′} & (ℤ/p)^{2} \ar[r] & 0}

is solvable. By Remark 8.2, no proper subgroup of M_{p^{3}} is mapped surjectively by λ′. Therefore Φ is surjective. As before, let pr_{1} : (ℤ/p)^{2} → ℤ/p be the projection on the first coordinate. We deduce that ψ breaks via the epimorphism pr_{1} ∘λ′, which is just λ′′. □

Next let r, s be the generators of D_{4} as in §8. We write G(p) for the maximal pro-p quotient of the profinite group G. One has the following analog of Proposition 10.2 for p = 2.

Proposition 10.3. Let p = 2 and let G be a profinite group of Galois relation type and such that G(2) ≅ ℤ̸/2. Every epimorphism ψ: G → ℤ/2 factors via one of the epimorphisms:

(i) the natural map ℤ/4 → ℤ/2; (ii) the map θ′: D_{4} → ℤ/2 , defined by r ↦ ¯1, s ↦ ¯0; (iii) the map θ′′: D_{4} → ℤ/2, defined by r ↦ ¯0, s ↦ ¯1.

Proof. Let ξ be as in Definition 3.1(ii). By the assumptions, G(2) ≠ 1, ℤ/2. Hence, if G(2) is pro-cyclic, then ψ factors via the map (i). We may therefore assume that G(2) is not pro-cyclic.

If β_{G}(ψ) = 0, then by Lemma 10.1, ψ factors via the map (i).

Next we assume that ψ, ξ + ψ are 𝔽_{2}-linearly independent. Let Ḡ = G/(ker(ψ) ∩ Ker(ξ)) and let π: G → G/Ḡ be the natural map. There exist ψ̄, ξ̄ ∈ H^{1}(Ḡ) with ψ = ψ̄ ∘ π, ξ = ξ̄ ∘ π, inf_{G}(ψ̄) = ψ and inf_{G}(ξ̄) = ξ. Note that Ker(ψ̄) ∩ Ker(ξ̄+ ψ̄) = Ker(ψ̄) ∩ ker(ξ̄) = 1. By Proposition 9.1(e), ψ̄ ∪ (ξ̄+ ψ̄) corresponds to the extension ω_{3}^{(ψ̄,ξ̄+ψ̄)}. By the choice of ξ and Lemma 2.4,

inf_{G}(ψ̄ ∪ (ξ̄+ ψ̄)) = ψ ∪ ξ + ψ ∪ ψ = 2β_{G}(ψ) = 0.

Proposition 7.1 therefore implies that the embedding problem

\xymatrix{ & G \ar@{->>}[dl]_{Φ} \ar[d]^{(ψ,ξ+ψ)} \\ D_{4} \ar[r]^{θ} & (ℤ/2)^{2} \ar[r] & 0}

is solvable. Since no proper subgroup of D_{4} is mapped by θ surjectively onto (ℤ/2)^{2} (Remark 8.2), Φ is surjective. We deduce that ψ factors via the epimorphism pr_{1} ∘θ, which is just θ′.

Finally assume that β_{G}(ψ) ≠ 0 and ψ, ξ + ψ are 𝔽_{2}-linearly dependent. As β_{G}(ψ) = ψ ∪ ξ, necessarily ξ ≠ 0. But ψ ≠ 0, so ψ = ξ.

Now G(2) is not pro-cyclic, so there exists ψ′ ∈ H^{1}(G) such that ψ, ψ′ are 𝔽_{2}-linearly independent. Then ψ′, ξ + ψ′ are also 𝔽_{2}-linearly independent. By the argument above, the embedding problem

\xymatrix{ & G \ar@{->>}[dl]_{Φ} \ar@{->>}[d]^{(ψ′,ξ+ψ′)} \\ D_{4} \ar[r]^{θ} & (ℤ/2)^{2} \ar[r] & 0}

is solvable. Composing with the map σ: (ℤ/2)^{2} → ℤ/2, (ī, j̄) ↦ \overline{i + j}, we obtain that ψ = ξ factors via σ ∘ θ, which is just θ′′. □

11. The main results

Let G be a again a profinite group and q = p a prime number.

Theorem 11.1. The following conditions on a normal open subgroup N of G are equivalent:

(a) N is distinguished;

(b) (i) When p = 2, G/N is isomorphic to one of the groups

1, ℤ/2, (ℤ/2)^{2}, ℤ/4, D_{4};

(ii) When p ≠ 2, G/N is isomorphic to one of the groups

1, ℤ/p, (ℤ/p)^{2}, ℤ/p^{2}, M_{p^{3}}.

Proof. (a)⇒(b): Let N be distinguished and M, ᾱ, φ data for N. Thus N = Ker(φ). Set Ḡ = G/M and consider a central extension

ω : 0 → ℤ/p → B → Ḡ → 1

corresponding to ΛḠ(ᾱ). By Proposition 7.2, G/N embeds in B.

If p = 2, then by Proposition 9.5, ω is equivalent to an extension of one of the forms ω_{0}, ωψ̄ ; 1, ωψ̄ ; 2, ω_{3}^{(ψ̄,ψ̄′)}. Then G/N embeds in one of the groups ℤ/2, (ℤ/2)^{2}, ℤ/4, D_{4}, and is therefore as in (i).

If p ≠ 2, then by Proposition 9.5, ω is equivalent to an extension of one of the forms ω_{0}, ωψ̄ ; 1, ωψ̄ ; 2, ω_{5}^{(ψ̄,ψ̄′)}. Then G/N embeds in one of the groups ℤ/p, (ℤ/p)^{2}, ℤ/p^{2}, M_{p^{3}}, and is therefore as in (ii).

(b)⇒(a): By Example 4.7, G itself is distinguished. We may therefore assume that G/N is nontrivial. Hence it is isomorphic to the middle group B of ω_{i} where i ∈ {0, 1, 2, 3}, if p = 2, and i ∈ {0, 1, 2, 5}, if p ≠ 2. Therefore there is an open normal subgroup M of G such that N ≤ M and the following diagram commutes:

[figure: ω :; 0; ℤ/p; G/N; ≀; G/M; ≀; θ; 1; ω_{i} :; 0; ℤ/p; B; (ℤ/p)^{s}; 0,]

where θ is an isomorphism. Then ω, ω_{i}^{θ} are equivalent. By Proposition 9.6, ω_{i}^{θ} corresponds to Λ_{G/M}(ᾱ) ∈ H^{2}(G/M) for some ᾱ ∈ Ω(G/M) of simple type and with trivial kernel, and therefore so does ω. Conclude from Proposition 7.2 that N is distinguished. □

We deduce the following stronger form of the Main Theorem:

Corollary 11.2. Suppose that p ≠ 2 and let G be a profinite group of Galois relation type. Then G^{(3)} is the intersection of all normal open subgroups N of G with G/N isomorphic to one of 1, ℤ/p^{2}, M_{p^{3}}.

Proof. By Theorems 5.2 and 11.1, G^{(3)} is the intersection of all normal open subgroups N of G with G/N isomorphic to one of 1, ℤ/p, ℤ/p^{2}, M_{p^{3}}. By Proposition 10.2, ℤ/p can be omitted from this list. □

For p = 2 Theorem 5.2 and Theorem 11.1 give:

Corollary 11.3. Let p = 2 and let G be of Galois relation type. Then G^{(3)} is the intersection of all normal open subgroups N of G such that G/N is isomorphic to one of the groups 1, ℤ/2, ℤ/4, D_{4}.

By Proposition 3.2, this extends [MSp96, Cor. 2.18], which proves it for G = G_{F}, F a field. Combined with Proposition 10.3 it gives:

Corollary 11.4. Let p = 2 and let G be a profinite group of Galois relation type such that G(2) ≅ ℤ̸/2. Then G^{(3)} is the intersection of all normal open subgroups N of G such that G/N is isomorphic to one of the groups 1, ℤ/4, D_{4}.

11.5. Remarks. (a) The converse of Corollary 11.4 also holds: if G(2) ≅ ℤ/2, then G^{(3)} is not an intersection as above.

(b) Let F be a field of characteristic ≠ 2 and let G = G_{F}. Then G(2) ≅ ℤ/2 if and only if F is a Euclidean field, i.e., the set (F^{×})^{2} of all nonzero squares in F is an ordering on F ([Bec74], [Efr06, §19.2]). Therefore, by (a), the Euclidean fields are those fields for which the group ℤ/2 cannot be omitted from the list in Corollary 11.3.

12. The structure of G/G^{(3)}

When p = 2 and G = G_{F} for a field F, the quotient G/G^{(3)} is the W -group of F, studied in [MSp90], [MSp96], and [MMS04]. It encodes much of the “real” arithmetic structure of F. We now give some restrictions on the group structure of G/G^{(3)} also for p odd.

Proposition 12.1. Let G be a profinite group of Galois relation type with G/G^{(3)} nonabelian.

(a) If p ≠ 2, then M_{p^{3}} is a quotient of G/G^{(3)}.

(b) If p = 2, then D_{8} is a quotient of G/G^{(3)}.

Proof. By our assumption, G^{(3)} cannot be an intersection of open normal subgroups N of G with G/N abelian. When p ≠ 2 (resp., p = 2) Corollary 11.2 (resp., Corollary 11.3) yields an open normal subgroup N of G with G/N ≅ M_{p^{3}} (resp., G/N ≅ D_{4}). The natural epimorphism h: G → Ḡ = G/N maps G^{(3)} to Ḡ^{(3)}, which is trivial by Remark 8.1(b). Hence h induces an epimorphism h̄: G/G^{(3)} → Ḡ. □

We recover the following known “automatic realizations”:

Corollary 12.2. Suppose that F is a field of characteristic ≠ p and containing a root of unity of order p.

(a) ([Bra89]) If p ≠ 2 and H_{p^{3}} is realizable as a Galois group over F, then M_{p^{3}} is also realizable as a Galois group over F.

(b) ([MS91, Prop. 2.1]) If p = 2 and Q_{8} is realizable as a Galois group over F, then D_{4} is also realizable over F.

Proof. When p ≠ 2 (resp., p = 2), take Ḡ = H_{p^{3}} (resp., Ḡ = Q_{8}). Thus Ḡ is a quotient of G = G_{F}, and as Ḡ^{(3)} = 1 (by Remark 8.1(b)), also of G/G^{(3)}. Hence G/G^{(3)} is nonabelian. Now apply Proposition 12.1. □

The next fact was earlier proved in [BLMS07, Th. A.3] when G = G_{F} for a field F containing a primitive pth root of unity.

Proposition 12.3. Let p ≠ 2 and let G be a profinite group of Galois relation type. Every element of G/G^{(3)} of order p belongs to G^{(2)}/G^{(3)}.

Proof. It suffices to show that the elements of order p in G/G^{(3)} are in the kernel of every epimorphism ψ̄: G/G^{(3)} → ℤ/p. Now ψ̄ lifts to a unique epimorphism ψ: G → ℤ/p. By Proposition 10.2, ψ breaks via an epimorphism π: Ḡ → ℤ/p, where either Ḡ = ℤ/p^{2} and π is the natural projection, or Ḡ = M_{p^{3}} and π = λ′′ (where λ′′ maps the generators r, s of M_{p^{3}} to ¯1, ¯0, respectively). In both cases, Ḡ^{(3)} = 1, by Remark 8.1(b) again. Therefore there is a commutative triangle

\xymatrix{ & G/G^{(3)} \ar@{->>}[d]^{ψ̄} \ar@{->>}[dl] \\ Ḡ \ar@{->>}[r]^{π} & ℤ/p.}

In both cases π is trivial on elements of Ḡ of order p (for Ḡ = M_{p^{3}} this follows from Remark 8.1(c)). The claim follows. □

Remark 12.4. Proposition 12.3 is no longer true when p = 2. For instance, G = ℤ/2(≅ G_{ℝ}) has Galois relation type, yet G/G^{(3)} = ℤ/2 and G^{(2)}/G^{(3)} = 1. More generally, take G = G_{F} for a field F of characteristic ≠ 2. Then G/G^{(3)} contains an involution which is not in G^{(2)}/G^{(3)} if and only if F is formally real [MSp90, Th. 2.7].

Example 12.5. Suppose that p ≠ 2 and that G has Galois relation type. By Proposition 12.3, G/G^{(3)} cannot be isomorphic to (ℤ/p)^{I}, with I ≠ ∅, to H_{p^{3}}, nor to M_{p^{3}} (see Remark 8.1(c)).

Remark 12.6. By the celebrated Artin–Schreier theorem, an absolute Galois group of a field is either 1, ℤ/2, or is infinite. Our results provide a new cohomological proof of this fact in characteristic 0, as follows.

Assume that F is a field of characteristic 0 with G = G_{F} finite. If G ≅ ℤ/p with p ≠ 2, then F contains a primitive pth root of unity. By Proposition 3.2, G has Galois relation type, contrary to what we have seen in Example 12.5. This shows that G is a finite 2-group.

Next suppose that G contains an element of order 4. We may then assume that G ≅ ℤ/4. Let K be its unique subgroup of order 2 and write H^{1}(K) = {0, ψ}. The map res_{K}: H^{1}(G) → H^{1}(K) is trivial. Hence the Kummer element κ_{2}(−1) ∈ H^{1}(K) (which comes from H^{1}(G)) is zero. By Corollary 2.8(b), β_{K}(ψ) = ψ ∪ κ_{2}(−1) = 0. On the other hand, there are no epimorphisms K → ℤ/4, contrary to Lemma 10.1.

Hence, G consists of involutions. By Proposition 10.3, G ≅ 1, ℤ/2.

13. Examples

We first give examples showing that non of the groups listed in Corollaries 11.3 and 11.2 can be omitted from these lists.

Example 13.1. Taking G = G_{ℂ} = 1 we see that the trivial group cannot be removed from the above lists.

Example 13.2. For p = 2 and G = G_{ℝ} = ℤ/2 one has G^{(3)} = 1. This shows that ℤ/2 cannot be removed from the list in Corollary 11.3.

Example 13.3. Let F be a finite field and let G = G_{F} = ℤ̂. Then G/G^{(3)} ≅ ℤ/p^{2}. Therefore ℤ/p^{2} cannot be removed from the lists.

Example 13.4. Take p = 2 and F = ℝ((t)). Then G = G_{F} = ⟨τ, ϵ | ϵ^{2} = (τϵ)^{2} = 1⟩ [Efr06, §22.1]. There is an epimorphism G ↦ D_{4}, τ ↦ r, ϵ ↦ s (with notation as in §8). Hence G/G^{(3)} is non-abelian. Now let N_{0} be the intersection of all closed normal subgroups N of G such that G/N is isomorphic to one of 1, ℤ/2,and ℤ/4. Then G/N_{0} is abelian (in fact, it is isomorphic to (ℤ/2)^{2}). Consequently, N_{0} ≠ G^{(3)}. Therefore D_{4} cannot be removed from the list in Corollary 11.3.

Example 13.5. Let p ≠ 2. Dirichlet’s theorem on primes in arithmetical progressions yields n ≥ 0 with l = p(pn + 1) + 1 prime. Let ζ_{p^{2}} be in the algebraic closure of 𝔽_{l}. Then 𝔽_{l} contains the pth roots of unity, but does not contain a p^{2}th root of unity ζ_{p^{2}}. Therefore the maximal pro-p Galois group G_{𝔽_{l}}(p) has a generator σ̄ such that σ̄(ζ_{p^{2}}) = ζ_{p^{2}}^{1+p}. Lift σ̄ to some σ ∈ G = G_{ℚ_{l}}(p). Also let τ be a generator of the inertia group of G. Then G is generated by τ and σ, subject to the defining relation στσ^{−1} = τ^{1+p} [Efr06, Example 22.1.6].

Now let N_{0} be the intersection of all closed normal subgroups N of G with G/N isomorphic to 1 or ℤ/p^{2}. Then G/N_{0} is abelian. Since there is an epimorphism G → M_{p^{3}}, τ ↦ r, σ ↦ s (notation as in §8), G/G^{(3)} is non-abelian, so N_{0} ≠ G^{(3)} (in fact, G/N_{0} ≅ (ℤ/p^{2}) × (ℤ/p) while G/G^{(3)} ≅ (ℤ/p^{2}) ⋊ (ℤ/p^{2}) = ⟨τ̃⟩ ⋊ ⟨σ̃⟩, with action σ̃τ̃σ̃^{−1} = τ̃^{1+p}). Thus M_{p^{3}} cannot be removed from the list in Corollary 11.2.

Our final two examples show that in Corollaries 11.3 and 11.2 one cannot omit the assumption that G has Galois relation type.

Example 13.6. Let p = 2 and let G = Q_{8}. Then G has no normal subgroups N with G/N ≅ ℤ/4 or G/N ≅ D_{4}, and has three distinct normal subgroups N with G/N ≅ ℤ/2, all containing the center Z(G).

Thus the intersection of all normal subgroups N of G as in Corollary 11.3 is Z(G)(≅ ℤ/2). On the other hand, G^{(3)} = 1 (Remark 8.1(b)).

Example 13.7. Let p ≠ 2 and let G = ℤ/p or G = H_{p^{3}}. Then G has no quotients isomorphic to ℤ/p^{2} or M_{p^{3}}. Thus the intersection in Corollary 11.2 is G itself. But by Remark 8.1(b), G^{(3)} = 1.

In this respect, the Main Theorem is a genuine structural result about absolute Galois groups.

Remark 13.8. In view of Corollaries 11.3 and 11.2, the previous two examples show that Q_{8} (when p = 2) and ℤ/p, H_{p^{3}} (when p ≠ 2) do not have Galois relation type. This can be seen directly as follows.

For G = Q_{8} and p = 2, one has a graded ring isomorphism

H^{∗}(Q_{8}) ≅ 𝔽_{2}[x, y, z]/(x^{2} + xy + y^{2}, x^{2}y + xy^{2}),

where x, y, z have degrees 1, 1, 4, respectively (see [Ade97, p. 811, Example], [AM04, Ch. IV, Lemma 2.10]). In this ring, no product of nonzero elements of degree 1 vanishes, yet x^{2} + xy + y^{2} = 0. Hence condition (i) of Definition 3.1 is not satisfied for G = Q_{8}.

For G = ℤ/p and p ≠ 2 one has H^{∗}(G) ≅ 𝔽_{p}[x, y]/(x^{2}), where x, y have degrees 1, 2, respectively, and (with the obvious abuse of notation) β_{G}(x) = y [Eve91, §3.2]. Here ∪: H^{1}(G) × H^{1}(G) → H^{2}(G) is the zero map, but β_{G}(x) ≠ 0. Hence (ii) of Definition 3.1 is not satisfied.

For G = H_{p^{3}} and p ≠ 2, the structure of H^{∗}(G) is considerably more complicated, and was computed by Leary [Lea92, Th. 6 and Th. 7]. Here as well, ∪: H^{1}(G) × H^{1}(G) → H^{2}(G) is the zero map, but β_{G} is nontrivial. Therefore condition (ii) of Definition 3.1 is not satisfied.

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Mathematics Department, Ben-Gurion University of the Negev, P.O. Box 653, Be’er-Sheva 84105, Israel

E-mail address: efrat@math.bgu.ac.il

Mathematics Department, University of Western Ontario, London, Ontario, Canada N6A 5B7

E-mail address: minac@uwo.ca