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Norm formulas for finite groups and induction from elementary abelian subgroups

It is known that the norm map N_G for a finite group G acting on a ring R is surjective if and only if for every elementary abelian subgroup E of G the norm map N_E for E is surjective. Equivalently, there exists an element x_G in R with N_G(x_G) = 1 if and only for every elementary abelian subgroup E there exists an element x_E in R such that N_E(x_E) = 1. When the ring R is noncommutative, it is an open problem to find an explicit formula for x_G in terms of the elements x_E. In this paper we present a method to solve this problem for an arbitrary group G and an arbitrary group action on a ring.Using this method, we obtain a complete solution of the problem for the quaternion and the dihedral 2-groups,and for a group of order 27. We also show how to reduce the problem to the class of (almost) extraspecial p-groups.

Let G be a finite group acting by ring automorphisms on an arbitrary (not necessarily commutative) ring R with unit. For any subgroup U of G the norm map N U : R → R U is defined for all x ∈ R by Here g(x) denotes the value in R of the action of g on x and R U the subring of U -invariant elements in R.
The question of the surjectivity of the map N G onto R G has well-known interpretations in topics such as Galois theory, algebraic number fields and, most importantly for this paper, integral group representations.
In [2, Theorem 1] the first-named author and Ginosar proved that N G is surjective onto R G if and only if N E is surjective onto R E for every elementary abelian subgroup E of G. This generalizes Chouinard's theorem [7] that asserts that a Z[G]-module is projective if and only if for every elementary abelian subgroup E of G it is projective as a Z[E]-module.
The map N U being R U -linear, it is surjective onto R U if and only if the unit 1 of R belongs to the image of N U . We can therefore rephrase Aljadeff and Ginosar's result as follows: there is an element x G ∈ R such that N G (x G ) = 1 if and only if there is an element x E ∈ R such that N E (x E ) = 1 for every elementary abelian subgroup E of G. Using this statement, Shelah observed (see [2,Proposition 6]) that there exist formulas expressing x G polynomially in terms of the elements x E and the elements of G.
The aim of this paper is to find such a formula for any given finite group G. We would like to point out that such a formula should be defined over Z and be independent of the ring on which the group acts. In this way it has a universal character though it may not be unique. With such a formula it is possible to construct a projective G-basis for any finitely generated projective Z[G]-module M out of given projective E-bases of M , one for each elementary abelian subgroup E of G.
We assume that G is not an elementary abelian group (otherwise the problem is trivial). The smallest group that is not elementary abelian is the cyclic group C 4 of order 4. This case was solved by Péter P. Pálfy who found the formula where σ denotes a generator of C 4 . The meaning of this formula is the following: if C 4 acts on some ring R and there is an element x E ∈ R such that x E + σ 2 (x E ) = 1 (i.e., N E (x E ) = 1 for the (unique) elementary abelian subgroup E of G), then for x G ∈ R given by (0.1) we have This was the first formula of this kind to be found. The next step is due to the authors: they showed in [3] how to obtain formulas for all abelian groups. Such formulas can be complicated even in simple cases. For instance, if G = C 9 is a cyclic group of order 9 (with generator σ) and x E ∈ R is an element of norm one for the subgroup of order 3, then is a formula (with 22 monomials) for an element of norm one for G. It should also be noted that the first-named author obtained formulas for arbitrary groups acting on commutative rings (see [1]).
We are thus left with the case of nonabelian groups acting on noncommutative rings (noncommutative rings are important for us because we want to be able to apply the formulas to rings of endomorphisms). In this paper we present a general method that allows to solve the problem for arbitrary groups and arbitrary ring actions. The idea is to translate the task of finding formulas for norm one elements x G for a group G acting on a ring R into a more accessible system of equations in R, where the variables are symbols b(σ), one for each generator σ in a presentation of G. The equations are obtained from the relations in the presentation. Such a system has solutions, and we will show how each solution yields a formula for x G . We illustrate this method by solving this system of equations for the quaternion and dihedral families of 2-groups, and a group of order 27. This provides the first examples of norm one formulas for nonabelian groups acting on noncommutative rings.
We also show that the problem for a general group G can be reduced to a smaller class of groups, namely the class of extraspecial and almost extraspecial p-groups that are subquotients of G. For instance, in order to solve the problem for the quaternion and dihedral groups mentioned above, it is sufficient to solve it for the quaternion group Q 8 and the dihedral group D 8 of order 8.
The paper is organized as follows. In Section 1 we explain precisely what we mean by a formula for a group G and we introduce a ring that is universal for the situation under consideration.
In Section 2 we present three reductions, first a straightforward one to pgroups, then a reduction to extraspecial and almost extraspecial p-groups. Finally we show how to solve the problem for the product of two p-groups once we have solutions for each of them.
After some cohomological preliminaries in Section 3 we explain our method to solve the problem for an arbitrary p-group in Section 4. This involves solving the above-mentioned system of equations and a further problem that we solve in Section 5. More precisely, assuming we are given an element x ∈ R such that N U (x) = 1, we will show in Section 5 how to express any 1-cocycle β : U → R explicitly as a 1-coboundary, i.e., how to find an explicit formula for an element w ∈ R such that β(g) = g(w) − w for all g ∈ U . Once we have solved the abovementioned system of equations and we have a formula for w, we obtain a complete explicit solution to the problem of finding a formula for a norm one element for G.
In Sections 6-8 we apply our method to two important families of nonabelian groups, namely the quaternion groups Q 2 n and the dihedral groups D 2 n , and to a group of order 27.
All groups considered in this paper are finite, and all rings have units. We denote a cyclic group of order n by C n .

Formulas for a group
For any group G we denote E G the set of elementary abelian subgroups of G. Recall that a group E is elementary abelian if it is isomorphic to C r p for some prime number p and some integer r ≥ 1. Clearly, Definition 1.1.-A formula for a finite group G is a polynomial Φ G in noncommuting variables g(x E ), where g ∈ G and E ∈ E G , and with coefficients in Z, satisfying the following condition: whenever G acts by automorphisms on a ring R and (x E ) E∈E G is a family of elements of R such that N E (x E ) = 1 for all E ∈ E G , then the element x G ∈ R obtained by replacing in Φ G each variable g(x E ) by the value of the action of g on the element x E ∈ R satisfies N G (x G ) = 1.
In order to clarify Definition 1.1, we consider the free noncommutative ring generated by symbols of the form g(X E ), where g runs over all elements of G and E runs over all elements of E G . To simplify notation, we set e(X E ) = X E when e is the neutral element of G. The group G acts by ring automorphisms on R free (G) as follows: if h ∈ G and g(X E ) is a generator of R free (G), then for g, h ∈ G, and E ∈ E G . Let I be the two-sided ideal of R free (G) generated by all elements of the form for any g ∈ G and any E ∈ E G . The ideal is preserved by the G-action on R free (G). Let R univ (G) be the quotient ring R univ (G) = R free (G)/I with the induced G-action. By definition of R univ (G), for any E ∈ E G we have Proof.-First observe that Φ G is a polynomial with integer coefficients in noncommutative variables g(X E ) indexed by G × E G . Suppose that G acts on a ring R and that ( this is the element of R obtained by replacing each variable g(X E ) in Φ G by the value of the action of g on x E ∈ R. We have The proof above also shows that R univ (G) is the universal G-ring with a family of elements (X E ) E∈E G such that N E (X E ) = 1.
As we have already pointed out in the introduction, there is a formula for every finite group G. Let us give a quick proof of this fact using the ring R univ (G): indeed by (1.1), we have N E (X E ) = 1 for every elementary abelian subgroup E of G. Therefore by [2,Theorem 1] there exists Φ G ∈ R univ (G) such that N G (Φ G ) = 1. By Proposition 1.2 this is a formula for G. Since finding such a formula for G amounts to constructing a norm-one element in R univ (G), we see that the problem is a pure group-theoretical question.
The right-hand sides of (0.1) and (0.2) provide formulas for the cyclic groups C 4 and C 9 , respectively. If E is an elementary abelian group, then Φ E = x E is clearly a formula for E.
In our search for formulas for a group G, the following rephrasing of [2, Theorem 1] will be useful. To state it, we need the following notation: E max G denotes the set of maximal elements of E G with respect to inclusion, and E 0 G a subset of E max G such that any element of E max G is conjugated in G to exactly one element of E 0 G . Proposition 1.3.-Let G be a finite group acting on a ring R by ring automorphisms. We assume that G is not elementary abelian. Then the following statements are equivalent: . . , g r } be a set of representatives for the right cosets of U in G. Define It is obvious. (Note that the assumption that G is not elementary abelian is needed for the implication 2) ⇒ 3).) 3) ⇒ 1): This is nontrivial; it follows from [2, Theorem 1]. 5) ⇒ 4): Let g ∈ G and E ∈ E G . For x E ∈ R such that N E (x E ) = 1, define : Any E ∈ E G is a subgroup of an element of E max G . Then proceed as for 1) ⇒ 2).
It can be seen from (1.2) and (1.3) that the number of variables in a formula Φ G for G can be reduced; we can restrict ourselves to the variables g(x E ), where g ∈ G and where E ∈ E max G or E ∈ E 0 G .

Three reductions
In this section we reduce in three steps the problem of finding a formula for G to the problem of finding formulas for smaller groups of a special type.

First reduction
We start by reducing the problem to p-groups, where p is a prime number. Given a group G, let n = p a 1 1 · · · p a r r be the factorization of the order n of G in prime factors, where p 1 , . . . , p r are distinct prime numbers, r ≥ 2, and the exponents a 1 , . . . , a r are positive integers. Choose integers d 1 , . . . , d r such that For every i = 1, . . . , r, let S i be a Sylow p i -subgroup (of order p a i i ) of G. The following result implies that, in order to find a formula for a group G, it is sufficient to find a formula for a Sylow p-subgroup of G for each prime number p dividing the order of G.
then Φ G is a formula for G.
Proof.-Suppose we are given a ring R on which G acts and elements x E ∈ R such that N E (x E ) = 1, one for each elementary abelian subgroup E of G. Replacing each variable x E (E ∈ E S i ⊂ E G ) in the polynomial Φ S i by the element x E ∈ R, we obtain an element x i ∈ R such that N S i (x i ) = 1 for each i = 1, . . . , r. Let us check that N G (x G ) = 1 for x G = d 1 x 1 + · · · + d r x r . Indeed, Consequently, N G (x G ) = d 1 n/p a 1 1 + · · · + d r n/p a r r = 1. Let us illustrate Proposition 2.1 in the case of the symmetric group S 3 . Let s be a transposition and t be a cyclic permutation in S 3 . The subgroups E s and E t generated respectively by s and t are elementary abelian. Then by Proposition 2.1,

Second reduction
We next reduce the problem from arbitrary p-groups to p-groups that are extraspecial or almost extraspecial. Recall that a p-group G is extraspecial (respectively almost extraspecial) if G fits into a central extension of the type where r ≥ 1, and the center of G is isomorphic to C p (respectively to C p 2 ). For a complete description of (almost) extraspecial groups, see for instance [5] (see also [8,Chapter 5]). Note that with this definition no abelian group is extraspecial and that the only abelian almost extraspecial group is C p 2 . For any p-group G we denote F G the set of isomorphism classes of groups U satisfying the following conditions : (i) U is a subquotient (i.e., a homomorphic image of a subgroup) of G and (ii) U is extraspecial or almost extraspecial. We have F H ⊂ F G whenever H is a subquotient of G.
The following result states that, in order to find a formula for a finite pgroup G, it is sufficient to have formulas for all groups in F G .
Theorem 2.2.-For any finite p-group G there is an algorithm whose output is a formula Φ G for G and whose inputs are formulas Φ H for all H ∈ F G .
Before we prove the theorem, we establish two intermediate results. Proof.-If G is abelian, then G is of order ≥ p 3 . Take an element g of order p in G. If G/ g is not elementary abelian, we are done. If G/ g is elementary abelian, say G/ g ∼ = C r p for some r that is necessarily at least 2, then by the classification of finite abelian p-groups we have G ∼ = C p 2 × C r−1 p . Let σ be an element of order p in C r−1 p (it exists since r − 1 ≥ 1) and h ∈ G be the element mapped to (0, σ) ∈ C p 2 × C r−1 p . Then G/ h contains an element of order p 2 , hence is not elementary abelian. Now assume that G is not abelian. Let g be an element of order p in the center Z(G) of G. If G/ g is not elementary abelian, we are done. Therefore we may assume that G/ g is elementary abelian. Observe that under this condition the commutator subgroup G ′ of G is the subgroup g generated by g. Since G is neither abelian, nor extraspecial, nor almost extraspecial, its center Z(G) is not isomorphic to C p or to C p 2 . Moreover, Z(G) is not cyclic of order ≥ p 3 since G/ g is elementary abelian. Therefore there is an element h ∈ Z(G) of order p such that h = g = G ′ , and so G/ h is not abelian, hence not elementary abelian.
Let G be a p-group that is neither elementary abelian, nor extraspecial, nor almost extraspecial. By Lemma 2.3 there is a subgroup U of order p in the center of G such that G/U is not elementary abelian. We fix such a subgroup U .
Let π : G → G/U be the natural projection. Let Φ G/U be a formula for G/U , and for eachĒ ∈ E G/U let Φ π −1 (Ē) be a formula for the proper subgroup π −1 (Ē) of G (it is a proper subgroup because G/U is not elementary abelian). Set (2.1) Equality (2.1) defines a noncommutative polynomial with integer coefficients in the variables g(x E ), where g ∈ G and E ∈ E G . This is a consequence of the following observations on the right-hand side of (2.1). Firstly, N U (Φ π −1 (Ē) ) has the following meaning: we replace each monomial ) whose meaning has just been explained. In this way, each variableḡ(xĒ), wherē is multiplied on the right by the variable x U , which makes sense since U ∈ E G . Proposition 2.4.-With the previous notation, Φ G is a formula for G.
Proof.-Suppose G acts on a ring R and we have elements x E ∈ R such that N E (x E ) = 1, one for each E ∈ E G . In particular, we have an element x U ∈ R such that N U (x U ) = 1.
For eachĒ ∈ E G/U , let x π −1 (Ē) be the element of R obtained from the formula Φ π −1 (Ē) by replacing each variable h(x E ), where h ∈ π −1 (Ē) and E ∈ E π −1 (Ē) (⊂ E G ), by the value of the action of h on the element x E ∈ R. By definition of a formula, we have The element N U (x π −1 (Ē) ) clearly belongs to the subring R U . Let The element z G/U belongs to R U because the inputs in its definition are in this subring. To conclude, let Proof of Theorem 2.2.-We proceed by induction on the order of G. Suppose G is of order p 3 . If G is elementary abelian or extraspecial, we are done (note that G cannot be almost extraspecial). Otherwise, G ∼ = C p 3 or G ∼ = C p 2 × C p , and in both cases F G = {C p 2 }. Fix a central subgroup U of order p such that G/U is not elementary abelian. Then G/U , being of order p 2 , is isomorphic to C p 2 , which is almost extraspecial. The group G/U has a unique elementary abelian subgroupĒ of order p whose lifting π −1 (Ē) to G, being of order p 2 , is either isomorphic to C 2 p (elementary abelian) or C p 2 (belonging to F G ). In both cases, by Proposition 2.4, (2.1) yields a formula for Φ G in terms of Φ C p 2 .
Let G be a group of order p n with n ≥ 4. Suppose we have proved the theorem for all groups of order at most p n−1 . As above, we may assume that G is neither elementary abelian, nor extraspecial, nor almost extraspecial. We again fix a central subgroup U of order p such that G/U is not elementary abelian. By the induction hypothesis, there is an algorithm whose output is a formula Φ G/U for G/U and whose inputs are formulas for all groups in F G/U . The lifting π −1 (Ē) to G of eachĒ ∈ E G/U is a proper subgroup of G since G/U is not elementary abelian. By the induction hypothesis again, there is an algorithm whose output is a formula Φ π −1 (Ē) for π −1 (Ē) and whose inputs are formulas for all groups in F π −1 (Ē) . Therefore by Proposition 2.4, (2.1) yields an algorithm whose output is a formula Φ G for G and whose inputs are formulas for all groups in the sets F π −1 (Ē) or in F G/U . We conclude by observing that

Third reduction
We now consider the case when G is the product of two groups.
Theorem 2.5.-Let G = G 1 ×G 2 be a product of two p-groups G 1 and G 2 . There is an algorithm whose output is a formula Φ G for G and whose inputs are formulas for all groups in Before we give the proof of this theorem, we establish a result similar to Proposition 2.4.
Consider the case when G = H × C, where H is a p-group and C is an elementary abelian p-group of rank one, i. e., C ∼ = C p . Set The right-hand side of (2.2) has the following meaning. Firstly, becomes a polynomial in the variables g(x E ′ ×C ), where g ∈ G and E ′ × C ∈ E G . Therefore, the right-hand side of (2.2) is a noncommutative polynomial with integer coefficients and in the right set of variables.
Proposition 2.6.-With the previous notation, Φ G is a formula for G.
Proof.-Suppose G acts on a ring R and we have elements Moreover, since the inputs belong to R C , so does z. Now let x G = zx C . Then, using the R C -linearity of N C , we obtain Proof of Theorem 2.5.-(a) Assume first that G 2 is elementary abelian. Let us prove by induction on the order of G 2 that there is an algorithm whose output is a formula Φ G for G and whose inputs are formulas for all groups in F G 1 . Since F G 2 = ∅, it will prove Theorem 2.5 in this case.
If G 2 is of order p, we appeal to Proposition 2.6: Formula (2.2), in which we have replaced H by G 1 and C by G 2 , yields an algorithm whose output is a formula Φ G for the group G and whose input is a formula for G 1 . Therefore, by Theorem 2.2 there is an algorithm whose output is a formula for G and whose inputs are formulas for all groups in F G 1 .
If G 2 is of order > p, we write G 2 = G ′ 2 × C, where C is cyclic of order p (the subgroup G ′ 2 is elementary abelian). Reasoning as above, we obtain an algorithm whose output is a formula for G = G 1 × G ′ 2 × C and whose input is a formula for G 1 ×G ′ 2 . By induction there is an algorithm whose output is a formula for G 1 × G ′ 2 and whose inputs are formulas for all groups in F G 1 . Therefore there is an algorithm whose output is a formula for G and whose inputs are formulas for all groups in F G 1 .
(b) Let G 2 be an arbitrary p-group of order p n with n ≥ 2 and suppose we have proved Theorem 2.5 for all groups G = G 1 × G ′ 2 such that the order of G ′ 2 is at most p n−1 . By Part (a) we may assume that G 2 is not elementary abelian.
Let U be a central subgroup of G 2 of order p and be the natural projection. By Proposition 2.4, Formula (2.1) yields an algorithm whose output is a formula for G and whose inputs are formulas for G/U and for π −1 (Ē), whereĒ runs over E G/U . By induction there is an algorithm whose output is a formula for G/U = G 1 × G 2 /U and whose inputs are formulas for all groups 2) shows how to obtain a formula for π −1 (Ē) from a formula for E 1 × N . By Part (a) there is an algorithm whose output is a formula for E 1 × N and whose inputs are formulas for all groups in F N . Summing up and observing that F G 2 /U ⊂ F G 2 and F N ⊂ F G 2 , we conclude that there is algorithm whose output is a formula for G and whose inputs are formulas for all By Theorem 2.2 it suffices to have a formula for C p 2 in order to obtain formulas for all abelian groups. A formula for C p 2 was obtained in [3, Corollary 1]. We recall it here for the sake of completeness: let σ be a generator of C p 2 and E be the unique abelian elementary subgroup of C p 2 . Then (c) Theorem 2.5 may help get a better reduction than Theorem 2.2 for nonabelian groups. For instance, take the product G = Q 8 × Q 8 of two copies of the quaternion group of order 8. By [8,Chapter 5] the central product G 1 of Q 8 with itself, which is a quotient of G, is an extraspecial group; it belongs to F G . Theorem 2.2 therefore suggests that a formula for the group G 1 (of order 32) is needed to obtain a formula for G. Nevertheless, by Theorem 2.5 only formulas for the groups in F Q 8 = {C 4 , Q 8 } are needed.

Cohomological preliminaries
In this section we present two results needed in the sequel. The first one is an important consequence of the existence of a norm one element. Proof.-We proceed by induction on the order of G. Assume first that G is a cyclic group of order p with generator σ. Recall that the cohomology groups of a cyclic group are given for all j ≥ 1 by The even cohomology groups vanish since The vanishing of the odd cohomology groups follows from [3, Lemma 1]. Now let G be of order p n with n ≥ 2 and assume that the lemma holds for all p-groups of order < p n . Take a normal subgroup U of G of index p (such a subgroup always exists). Using Formula (1.2), out of the element x ∈ R satisfying N G (x) = 1 we derive an element x U ∈ R such that N U (x U ) = 1. The existence of x U implies by induction that H i (U, R) = 0 for all i > 0.
It then follows from the Lyndon-Hochschild-Serre spectral sequence that the inflation maps Then by the first part of the proof the cohomology groups H i (G/U, R U ) vanish for all i > 0, and so do the groups H i (G, R).
In the next sections we shall represent elements of the cohomology group H 1 (G, M ), where G is a group and M is a left G-module, by 1-cocycles of G with values in M . Recall from [6, Chapter X, § 4] that such a 1-cocycle (also called a crossed homomorphism) is a map β : G → M satisfying for all ∈ G. It is easy to check that a 1-coboundary is a 1-cocycle. 1-cocycles and 1-coboundaries are elements of the standard cochain complex whose cohomology is H * (G, M ). If δ is the differential in the standard cochain complex, then (3.2) can be rewritten as β = δ(m).

A method for finding formulas for p-groups
We now present a method for finding a formula for a given p-group G. It consists in taking a presentation of G and deriving from it a system of equations whose indeterminates are elements b(σ) ∈ R, one for each generator σ in the presentation. There is an equation for each relation in the presentation. Group cohomology guarantees that this system of equations has a solution. Once we have a solution, we again use homological algebra to obtain an explicit formula for G.
In view of Propositions 1.3 and 2.1 we may assume that G is a finite p-group (not elementary abelian) and that we have formulas for all proper subgroups of G. Let G act on a ring R. We assume the existence of x H ∈ R such that N H (x H ) = 1 for each proper subgroup H of G. Our aim is to give an explicit formula for x G ∈ R with N G (x G ) = 1 in terms of the elements x H and of the elements of G.
Fix a normal subgroup U of index p in G and choose an element σ ∈ G whose classσ generates the cyclic group G/U . Set x = x U ∈ R (this is one of the elements x H whose existence was assumed above); we have N U (x) = 1. A similar proof holds for y = xa.
To solve the problem for G, it is therefore sufficient to find an element a ∈ R U such that N G/U (a) = 1. In the rest of the section we show how to find such an element.
We start as in [3, Section 2] by considering the group B = Hom Z (Z[G], R) of Z-linear maps from the group ring Z[G] to R. The group G acts on the left on B by (gϕ)(s) = ϕ(sg) for all g, s ∈ G and ϕ ∈ B. The ring R is a G-submodule of B, where an element r ∈ R is identified with the element ϕ r ∈ B given by ϕ r (g) = g(r) for all g ∈ G. Let C = B/R be the quotient G-module. We denote q : B → C the canonical surjection.
Consider the following commutative square: The vertical maps δ in (4.1) are the connecting homomorphisms arising from the short exact sequences By Proposition 3.1 the group H 1 (U, R) vanishes because of the existence of the element x ∈ R satisfying N U (x) = 1. The maps δ are isomorphisms because B is a co-induced G-module and B U is a co-induced G/U -module, hence The horizontal maps in (4.1) are inflation maps. By the Lyndon-Hochschild-Serre spectral sequence the vanishing of H i (U, R) for i > 0 (see Proposition 3.1) implies that the lower inflation map Inf : H 2 (G/U, R U ) → H 2 (G, R) is an isomorphism. Therefore, the upper inflation map is an isomorphism as well. In view of this, of Proposition 1.3 and of Proposition 3.1, all groups in the square (4.1) vanish.
Let qα : G → C be the composition of α with q : B → C. By Lemma 4.2, qα 0 : G/U → C U represents an element in H 1 (G/U, C U ). It is easy to check that its image in H 1 (G, C) under the upper inflation map in the square (4.1) is represented by the map qα : G → C. Since qα 0 is a 1-cocycle, so is qα.
Our idea is to correct α : G → B as follows.
is a 1-cocycle with values in B.
We now claim that it suffices to perform the following three tasks in order to find a formula for G.
Task 1: Write the set of equations satisfied by the values b(g) ∈ R of the map b, obtained by expressing that α − b : G → B is a 1-cocycle. We can reduce the number of unknowns by choosing a presentation σ 1 , . . . , σ r | R 1 , . . . , R s of G.
Since the generators are of finite order, we may assume that each relation R i is a word in σ 1 , . . . , σ r (i.e., the inverses of σ 1 , . . . , σ r do not appear in R i ). Set By setting β(R i ) = 0 for all i = 1, . . . , s, we obtain a system (Σ) of s equations whose unknowns are b(σ 1 ), . . . , b(σ r ). It is an easy exercise to show that the values b(σ 1 ), . . . , b(σ r ) ∈ R determine uniquely a map b : G → R such that α − b is a 1-cocycle.
Task 2: By Lemma 4.3 the system of equations (Σ) derived in Task 1 has a solution b : G → R. Task 2 consists in writing down such a solution polynomially in terms of the given data.
Task 3: By Proposition 3.1 the existence of a norm one element x ∈ R for U implies the vanishing of H 1 (U, R). Hence, for any 1-cocycle β : U → R there is an element w ∈ R such that β(g) = (g −1)w for all g ∈ U . Give an explicit expression of such an element w as a noncommutative polynomial with integer coefficients in the variables u(x) and u(β(v)), where u, v ∈ U .
Once Tasks 1-3 are completed, we solve the problem for G as follows. Let b : G → R be a solution of the system (Σ) (in particular, we have an element b(σ) ∈ R). Then α − b is a 1-cocycle with values in B. Since B is cohomologically trivial, there is ψ : G → R such that α − b = δ(ψ). In particular, since α(g) = ϕ for g = σ, we obtain for all g ∈ U . Equations (4.6) imply that the restriction of b to U is a 1-cocycle with values in R. After performing Task 3, we have an element w ∈ R such that b(g) = (g − 1)w (4.7) for all g ∈ U . Relations (4.6-4.7) together imply (g − 1)(ψ + w) = 0 for all g ∈ U , which means that ψ + w is U -invariant. Proof.-(a) Relation (4.5) allows us to express a under the form To check the U -invariance of a, it is enough to check the U -invariance of since ϕ is U -invariant. For u ∈ U let u ′ ∈ U be such that uσ = σu ′ (recall that U is a normal subgroup of G). Therefore, in view of the U -invariance of ψ + w, for u ∈ U we obtain (b) Since σ p belongs to U , the U -invariance of ψ + w implies By Proposition 4.1, the element y = ax ∈ R (or y = xa) then yields an explicit norm one element for G with the appropriate form. This solves the problem for G.
Remark 4.5. Before we close this section, let us evaluate the level of difficulty of Tasks 1-3. We explained above how to perform Task 1; this is easy. For Task 3 we have a general method to solve it; it will be detailed in the next section.
At the moment we do not have a general method to solve Task 2, which consists in solving the system of equations (Σ) defined above. The solutions given in Sections 6-8 have been found in an ad hoc way; we nevertheless observe that they are of a very simple form. If we could prove in full generality that the solutions of (Σ) are of the form where H runs over all maximal proper subgroups of G, A H ∈ Z[G], and x H ∈ R satisfies N H (x H ) = 1, then solving (Σ) could be reduced to solving a system (Σ ′ ) of linear equations over the ring of integers Z, whose number of unknowns and of equations can be bounded in terms of G. More precisely, if r is the number of generators and s is the number of relations in the chosen presentation of the group G, and if m is the number of maximal proper subgroups of G, then the number of unknowns in (Σ ′ ) is rm|G| and the number of equations is s|G|. Note that the number of maximal proper subgroups of G is m = (p N − 1)(p − 1), where p N is the order of the quotient of G by its Frattini subgroup.

Writing a 1-cocycle as an explicit 1-coboundary
We consider a finite p-group U acting on a ring R. Assume that we have an element x ∈ R such that N U (x) = 1. The cohomology group H 1 (U, R) vanishes by Proposition 3.1. Therefore, given a 1-cocycle β : U → R, there exists w ∈ R such that β(g) = (g − 1)w for all g ∈ U . Our aim in this section is to explain how to obtain a formula for w in terms of x ∈ R, the elements of U , and the values of β (this is Task 3 of the previous section).
Let us start with the case when U = C p is a cyclic group of order p. We denote σ a generator of U . If β : U → R is a 1-cocycle of U with values in R, then by Lemma 3.2 (c) N U (β(σ)) = (1 + σ + · · · + σ p−1 )β(σ) = 0. (1 + σ + · · · + σ k−1 ) xσ −k β(σ) ∈ R. (5.1) The right-hand side of (5.1) is a noncommutative polynomial with integer coefficients in the variables u(x) and u(β(σ)), where u ∈ U . By Lemma 3.2 (b) we obtain for g = σ i , where 1 ≤ i ≤ p − 1, With Formula (5.1) we have thus expressed any 1-cocycle as a 1-coboundary in the case when U is a cyclic group of order p. Formula (5.1) is the prototype of formulas we wish to obtain for w in the general case.
To deal with a general finite p-group U , we proceed by induction on the order of U . Fix a normal subgroup U ′ of U of index p and choose σ ∈ U such that its classσ in U/U ′ generates U/U ′ . Following (1.2), set Then N U ′ (x ′ ) = 1. We assume that we know how to express any 1-cocycle γ : U ′ → R as the coboundary of an element of R expressed as a noncommutative polynomial with integer coefficients in u ′ (x ′ ) and In order to pass from U ′ to U we make use of a well-known construction due to Wall [9]. Let (C ′ * , d ′ ) be the standard resolution of Z by free left Z[U ′ ]-modules. In particular, For each q ≥ 0 we define a chain complex C * ,q of free left Z[U ]-modules by setting We define a differential d 0 : for all q ≥ 0. The chain complex C * ,q is a free resolution of such that (i) when p = 0, then d 1 : is given by Define a nonnegatively graded Z[U ]-module C W * for all r ≥ 0 by Observe that . The maps d W = k≥0 d k define a degree −1 differential on C W * and turn it into a resolution of Z by free left Z[U ]-modules.
Let us apply the functor Hom Z[U] (−, R) to the resolution (C W * , d W ). Define (the last isomorphism follows by adjunction). In particular, for all q ≥ 0. The differential d 0 turns into a degree +1 differential δ 0 : C p,q → C p+1,q . The maps d k (k ≥ 1) turn into maps δ k : C p,q → C p−k+1,q+k , which by (5.5) satisfy Then (C * W , δ W ) is a cochain complex whose cohomology groups are the groups H * (U, R).
The element x ′′ = N U ′ (x) belongs to R U ′ and we have The third relation in (5.7), together with (5.3) and (5.6), implies Since σ p belongs to the subgroup U ′ , the element σ p − 1 acts as 0 on R U ′ , and σ generates a cyclic group of order p in the automorphism group of the ring R U ′ . The element s ′ ∈ R U ′ is of norm zero for this cyclic group. Using Formula (5.1), we obtain an element w 2 ∈ R U ′ such that s ′ = (σ − 1)w 2 , explicitly in terms of s ′ , of σ, and of the element x ′′ appearing in (5.9). We claim that w = w 1 + w 2 ∈ R satisfies Equations (5.8). Indeed, On the other hand, δ 0 (w 2 ) = 0 since w 2 belongs to R U ′ . Therefore, This proves our claim and shows how to construct w for the cochain complex (C * W , δ W ). In order to express a 1-cocycle β : U → R in the standard cochain complex as a 1-coboundary, we use the comparison lemma between the resolution (C W * , d W ) and the standard resolution (C * , d) of Z by free left Z[U ]-modules (see, e.g., Proposition 1.2 in [6, Chapter V]).
is the identity map and is the Z[U ]-linear map whose restriction to the first summand Z[U × U ′ ] is induced by the natural inclusion of U ′ into U , and the restriction to the second summand Z[U ] is defined for all g ∈ U by θ 1 (g) = (g, σ) ∈ C 1 .
Proof.-The existence of θ * follows from the comparison lemma. Since C W 0 = C 0 = Z[U ], we can take θ 0 to be the identity map. It now suffices to check that for g ∈ U and h ∈ U ′ . On the summand Z[U ], by (5.4) we have When we apply the functor Hom Z[U] (−, R) to θ * : (C W * , d W ) → (C * , d), we obtain a cochain map inducing an isomorphism in cohomology. The cochain complex (C * (U, R), δ) is the standard cochain complex computing H * (U, R). Now, let β : U → R be a standard 1-cocycle. This is an element of C 1 (U, R) such that δ(β) = 0. Consider its image θ 1 (β) ∈ C 1 W . It is a 1-cocycle in (C * W , δ W ). By our investigation above we know how to construct w ∈ R such that θ 1 (β) = δ W (w). We claim the following.
To conclude it suffices to check that θ 1 is injective. Using the string of natural isomorphisms = Hom(U ′ , R) ⊕ R and Lemma 5.1, we easily see that the image θ 1 (β) of any standard 1-cocycle β ∈ Hom(U, R) is given by where β ′ is the restriction of β to U ′ and β(σ) is its value on σ. If θ 1 (β) = 0, then the restriction of β to U ′ is zero and β(σ) = 0. It follows from Lemma 3.2 (b) that β vanishes on all powers of σ. The cocycle condition (3.1) then implies that β vanishes on all elements of U . This proves the injectivity of θ 1 .
Summing up, we thus have obtained an inductive way (starting from cyclic groups) to express any 1-cocycle of a finite p-group (with values in a ring R) as the coboundary of an element w ∈ R, polynomially in terms of x, the values of the 1-cocycle, and the elements of the group. This is a vast generalization of [3, Lemma 1].
The smallest nonabelian p-groups are the two nonabelian groups of order 8, namely the quaternion group Q 8 , which has a unique elementary abelian subgroup of order 2, and the dihedral group D 8 , which has two nonconjugate maximal elementary abelian subgroups of order 4. Both Q 8 and D 8 are extraspecial groups.
In this section we apply the method of Section 4 in order to solve the problem for Q 8 and more generally for the generalized quaternion groups Q 2 n+2 (n ≥ 1).
The group G = Q 2 n+2 of order 2 n+2 (with n ≥ 1) has a presentation with two generators σ, τ and the relations Any element of the group can be written as σ i τ j , where i = 0, 1, . . . , 2 n+1 − 1 and j = 0, 1. We take U to be the cyclic group generated by σ. The quotient group G/U is cyclic of order 2 and generated by the class of τ . The group Q 2 n+2 has a unique elementary abelian subgroup, which is the group of order 2 generated by the central element τ 2 = σ 2 n .
We follow the method presented in Section 4. Let us first perform Task 1.
-The values b(σ) and b(τ ) ∈ R satisfy the system of three equations Proof.-Since the restriction of b to U is a 1-cocycle, the first equation follows from Lemma 3.2 (c). Applying Lemma 3.2 (e) to β = b − α, we obtain We derive the second equation of the lemma by recalling that ϕ is U -invariant. In order to prove the third equation, we use the third relation in (6.1). Since b is a 1-cocycle when restricted to U , we have b(σ 2 n ) = (1 + σ + · · · + σ 2 n −1 )b(σ) (6.2) by Lemma 3.2 (b). On the other hand, we have 3) The third equation then follows from τ 2 = σ 2 n and (6.2-6.3).
of R are solutions of the system of equations of Lemma 6.1.
Proof.-For the first equation we have We check the second equation: For the third equation we have We now complete Task 3, which is to find an explicit w ∈ R such that b(g) = (g − 1)w for g ∈ U . By Lemmas 6.1-6.2 we have N U (b(σ)) = 0 for b(σ) = (1 − στ )(x). Since U is cyclic, we may apply [3, Lemma 1]. We then obtain b(σ) = (σ − 1)w, where Observe that w is a noncommutative polynomial with 2 n+1 (2 n+1 − 1) monomials of degree ≤ 2 in terms of x.
For the special case when n = 1 and G = Q 8 is the quaternion group of order 8, we obtain the following element y ∈ R satisfying N Q 8 (y) = 1: The right-hand side of (6.6) contains 26 monomials of degree ≤ 3 in terms of x. If we wish to express y in terms of a norm one element x E for the elementary abelian subgroup E of Q 8 generated by σ 2 , it suffices by (0.1) to replace x in (6.6) by the polynomial We thus obtain a formula for Q 8 with 666 (= 2 · 3 2 + 24 · 3 3 ) monomials of degree ≤ 9. Remark 6.3. Observe that the group Q 8 is extraspecial, and if G = Q 2 n+2 (n ≥ 2), then F G = {C 4 , Q 8 , D 8 }.

The dihedral 2-groups
The dihedral group G = D 2 n+1 of order 2 n+1 (where n ≥ 2) has a presentation with two generators σ, τ and the relations τ σ = σ −1 τ and τ 2 = σ 2 n = 1. (7.1) Any element of the group can be written uniquely as σ i τ j , where i = 0, 1, . . . , 2 n −1 and j = 0, 1. Let U be the normal subgroup generated by σ 2 and τ . The quotient group G/U is the cyclic group of order 2 generated by the class of σ. Note that U is a dihedral group of order 2 n if n ≥ 3 and an elementary abelian group if n = 2. It contains the elementary abelian subgroup U 1 generated by u and τ , where u = σ 2 n−1 is the unique non-trivial central element of D 2 n+1 . Let x be an element of R such that N U (x) = 1. We denote H the cyclic group generated by σ (of order 2 n ). We follow the method presented in Section 4. Let us first perform Task 1.
The element w can be expressed (as a noncommutative polynomial with integer coefficients) in terms of the norm one element x, the elements of U , and the values b(σ), b(τ ) given in Lemma 7.2. Observe that here we need both x and x 2 ∈ R, which is not surprising since G has two nonconjugate maximal elementary abelian subgroups. As a consequence of Proposition 4.4, the element is U -invariant and N G/U (a) = 1. Hence, N G (y) = 1 for y = ax by Proposition 4.1.
If G = D 8 , then U = U 1 is elementary abelian of order 4, and we can use Example 5.3. Setting σ 1 = τ , σ 2 = σ 2 , r = b(τ ), and s = b(σ 2 ) in Formula (5.10), we obtain an explicit w with 48 monomials of degree at most 3. Hence for G = D 8 we have a norm one element y with 98 monomials of degree ≤ 4. (b) By Sections 2, 6, 7, we have solved the problem for all 2-groups G such that F G = {C 4 , Q 8 , D 8 }, in particular for all metacyclic 2-groups. Note that by [4,Theorem 5.1] any 2-group every subgroup of which is generated by two elements is metacyclic.

A nonabelian group of order 27
Let p be an odd prime number and G p 3 be the group generated by σ, τ and the relations σ p 2 = τ p = 1 and τ σ = σ p+1 τ. (8.1) This is the only nonabelian group of order p 3 containing a cyclic subgroup of index p. The center Z of G p 3 is the cyclic group generated by σ p , and G p 3 /Z is elementary abelian of order p 2 . Therefore, G p 3 is extraspecial. Let U be the elementary abelian subgroup of G p 3 generated by σ p and τ ; it is the unique maximal elementary abelian subgroup of G p 3 . The quotient group G p 3 /U is the cyclic group of order p generated by the class of σ.
We will solve the system of equations of Lemma 8.1 when p = 3, i.e., for the group G 27 of order 27, generated by σ, τ and the relations σ 9 = τ 3 = 1 and τ σ = σ 4 τ. (8.4) The elementary abelian subgroup U considered above is generated by σ 3 and τ . We assume the existence of x ∈ R such that N U (x) = 1. The center Z of G 27 is the cyclic group generated by σ 3 ; it is contained in U . Therefore, if we set x 0 = (1 + τ + τ 2 )(x) ∈ R, we have N Z (x 0 ) = 1.
The element w can be expressed (as a noncommutative polynomial with integer coefficients) in terms of the norm one element x, the elements of U , and the values b(σ), b(τ ) given in Lemma 8.2. As a consequence of Proposition 4.4, the element a = b(σ) + (1 − σ)(w) is U -invariant and N G 27 /U (a) = 1. Therefore, N G 27 (y) = 1 for y = ax or y = xa by Proposition 4.1.
As a consequence of Section 2, we have solved the problem for all 3-groups G such that F G = {C 9 , G 27 }.