On primordial groups for the Green ring

Alberto G. Raggi-Cárdenas^{∗} and Nadia Romero^{∗}

Instituto de Matemáticas, UNAM, A.P. 61-3, C.P. 58089, Morelia, Michoacán, México.

     E-mail: graggi@shi.matmor.unam.mx, nadia@matmor.unam.mx

Abstract

Consider the Mackey functor assigning to each finite group G the Green ring of finitely generated kG-modules, where k is a field of characteristic p > 0. Thévenaz foresaw in 1988 that the class of primordial groups for this functor is the family of k-Dress groups. In this paper we prove that this is true for the subfunctor defined by the Green ring of finitely generated kG-modules of trivial source.

Keywords: Green ring, primordial group, trivial source module.

1 Introduction

For a field k, the Green ring of the category of finitely generated kG-modules, a(kG), is by definition, spanned over ℤ by elements [M], one for each isomorphism class of finitely generated kG-modules and with structures given by [M] + [N] = [M ⊕ N] and [M][N] = [M ⊗_{k} N]. The subring generated by the kG-modules of trivial source (defined in Section 2.1) is denoted by a(kG, triv).

Assigning to each finite group G either of the two above-mentioned rings defines a globally defined Mackey functor, as described in Bouc [3] and Webb [9]. These functors are denoted by a(k ) and a(k , triv). Based on Section 3 of Webb’s paper [9], we suggest that the concept of primordial group can be defined for any globally defined Mackey functor. In terms of our two functors this concept is expressed in a familiar way: let M be either a(k ) or a(k , triv). Then a group G is called primordial for M if M(G)/T (G) ≠ 0 with

∗Partially supported by a grant from CONACyT, Project B0291 Funtores de tipo Burnside.

T (G) = ∑_{\begin{matrix}H↪G \\ H≇G \end{matrix}} tr_{H}^{G}M(H).

This definition also works for the Mackey functor G_{0}(k ), which assigns to G the Grothendieck group of finitely generated kG-modules. We write Prim(M) for the class of primordial groups for M.

Primordial groups were first studied by Dress [6] in the context of Green functors for a finite group G. Thévenaz [8] proved that for such a functor N, the closure under conjugation and subgroups of the primordials for N is the minimal set 𝒟 of subgroups of G satisfying N(G) = ∑_{H∈𝒟} tr_{H}^{G}N(H). Thévenaz also proved that if k is a field of characteristic p > 0, the primordial groups for ℚ ⊗ a(kG) are the p-hypoelementary subgroups of G (see Section 2 for definitions). Also, he conjectured that the primordials for G_{0}(kG) were the k-elementary subgroups of G, which was proved in 1989 by Raggi [7], and that an analogue of the Brauer-Berman-Witt Theorem (5.6.7, in Benson [1]) should hold for a(kG). The main result of this work, Theorem 3.2, states that this is true for the subring of trivial source modules.

Recent work in the subject of induction can be found in the papers of Boltje [2] and Coşkun [4]. It is important to mention that some results about primordial groups can be generalized to the context of globally defined Mackey functors, as for example Lemma 3.1 that shows a general behavior of primordial groups of subfunctors.

2 Preliminaries

From now on, we assume that k is a field of characteristic p > 0, all modules are finitely generated, and all groups are finite.

Recall that for a group G and a prime r, O^{r}(G) is defined as the smallest normal subgroup of G, such that G/O^{r}(G) is a r-group, and O_{r}(G) is the largest normal r-subgroup of G.

Definition 2.1. If q and r are primes, a group H is called q-hyperelementary if O^{q}(H) is cyclic, and r-hypoelementary if H/O_{r}(H) is cyclic.

Observe that H is q-hyperelementary if and only if H = C ⋊ Q with Q a q-group and C a cyclic group of order prime to q, and it is r-hypoelementary if and only if H = D ⋊ C where D is a r-group and C is cyclic of order prime to r. It is easy to prove that the classes of q-hyperelementary and r-hypoelementary groups are closed under subgroups and quotients.

Notation 2.2. We write ℤ_{m}^{∗} for the smallest non negative representatives of the multiplicative group of units modulo m (which we denote by (ℤ/mℤ)^{∗}).

Definition 2.3. Suppose H = C ⋊ Q is a q-hyperelementary group with C =< x > of order m prime to p. The group H is called k-elementary if the action of every y ∈ Q on x is given by yxy^{−1} = x^{a} with a ∈ I_{m}(k), where I_{m}(k) ⊆ ℤ_{m}^{∗} is the set of smallest non negative representatives of the image of Gal(k(ω)/k) under the injective morphism

Gal(k(ω)/k) ⟶ (ℤ/mℤ)^{∗}
σ ⟼ \overline{a}

if σ(ω) = ω^{a} with 1 ≤ a ≤ m − 1, (a, m) = 1 and ω is a primitive m-th root of unity.

We write E_{k} for the class of k-elementary groups.

Note that in the latter definition we can replace I_{m}(k) by I_{n}(k) where n is any multiple of m.

Definition 2.4. For a prime q, a group H is called q-Dress if O^{q}(H) is p-hypoelementary.

It is easy to see that O^{q}(H) is p-hypoelementary if and only if H/O_{p}(H) is q-hyperelementary. If in addition to this, H/O_{p}(H) is k-elementary, then H is called k-Dress for q.

The class of q-Dress groups is closed under subgroups and quotients, and it is denoted by Dr_{q}. The class of groups which are k-Dress for some prime will be denoted by Dr_{k}. We write Dr_{p}^{∗} for the class of k-Dress groups such that p divides the order of H/O_{p}(H); that is, H/O_{p}(H) = C ⋊ D is k-elementary with D being a nontrivial p-group.

Notation 2.5. Let G be a group and H a subgroup of G. If M is a kHmodule, we will write M ↑_{H}^{G} for the induced module kG ⊗_{kH} M. If N is a kG-module, the restriction of N to kH will be denoted by N ↓_{H}^{G}.

2.1 Trivial source modules

The following facts on trivial source modules are well known and can be found in Benson [1] and Curtis and Reiner [5].

Recall that for a kG-module it is equivalent to be (G, H)-projective and to be a direct summand of L↑_{H}^{G} for some kH-module L. If M is an indecomposable kG-module, there exists a subgroup D for which M is (G, D)-projective and D ⩽_{G} H for any subgroup H for which M is (G, H)-projective, such a group D is called a vertex of M. In this case, if L is an indecomposable kD-module such that M is a direct summand of L↑_{D}^{G}, then L is called a source of M. The module M is said to have trivial source if the field k is a source of M, which is equivalent to say that M is a direct summand of a permutation module.

It can be proved that any two vertices of M are conjugate in G, and that any two sources of M are conjugate by an element in N_{G}(D). Since we are assuming that k is a field of characteristic p, a vertex of M is a p-subgroup of G.

We prove the following property of trivial source modules, which will be used later.

Lemma 2.6. The tensor induction of a trivial source module is a trivial source module.

Proof. We denote the tensor induction from H to G by ↑_{H}^{G}^{⊗}. Let B be a kH-module of trivial source, then it is a direct summand of a permutation module, say ⊕_{a∈[H/K]}k = B ⊕ A. From the proof of Proposition 3.15.2 iii) in Benson [1], it is easy to see that the tensor induction of a permutation module is a permutation module. By the same proposition, on the right hand side we obtain B↑_{H}^{G}^{⊗} ⊕ A↑_{H}^{G}^{⊗} ⊕ X, where X is a sum of modules induced from proper subgroups of G.

The following corollary to the Green Indecomposability Theorem, 19.22 in Curtis and Reiner [5], will be used in the following sections, as well as the next lemma. Recall that k is a field of characteristic p.

Corollary 2.7 (19.23 in Curtis and Reiner [5]). Suppose that G is a p-group and H is an arbitrary subgroup. If L is an absolutely indecomposable kH module (i.e k′ ⊗ L is indecomposable for all k′ field extension of k), then L↑_{H}^{G} is an absolutely indecomposable kG-module.

Lemma 2.8. Let U be an indecomposable kH-module of trivial source with vertex containing O_{p}(H), and let M be an indecomposable kG-module for G ⩽ H such that U is a summand of M ↑_{G}^{H}. Suppose also that if |H/O_{p}(H)| is divisible by p, we have M of trivial source. Then

i) O_{p}(H) is contained in a vertex of M and it acts trivially on M.

ii) Any indecomposable summand V of M ↑_{G}^{H} has trivial source, O_{p}(H) is contained in a vertex of V and it acts trivially on V.

Proof. Let D be a vertex of U that contains O_{p}(H). If D_{1} and S are a vertex and a source of M, respectively, then O_{p}(H) ⊆ D_{1} ⊆ G (because U is a direct summand of S ↑_{D_{1}}^{H}). If |H/O_{p}(H)| is a p′-number then O_{p}(H) is a vertex of U and D_{1} ⊆ O_{p}(H), so we have O_{p}(H) = D_{1}. This implies that S is a source of U and that M has trivial source. With this we obtain that V is of trivial source, which is the first part of ii).

Since M is a summand of k ↑_{D_{1}}^{G}, then M ↓_{O_{p}(H)}^{G} is a direct summand of k ↑_{D_{1}}^{G}↓_{O_{p}(H)}^{G}. By the Mackey formula, the latter is isomorphic to ⨁_{a} k, where a runs over [G/D_{1}] so M ↓_{O_{p}(H)}^{G} is isomorphic to a sum of k. With this we prove i), and by the same argument, we prove that O_{p}(H) acts trivially on V.

Finally, if A is a vertex of V then V ↓_{O_{p}(H)}^{H}≅ ⨁ k is a summand of

k ↑_{A}^{H}↓_{O_{p}(H)}^{H}, which is isomorphic to ⨁_{b} k ↑_{O_{p}(H)∩A}^{O_{p}(H)}. By Corollary 2.7, we

have that k ↑_{O_{p}(H)∩A}^{O_{p}(H)} is indecomposable, then for some b we should have

k ≅ k ↑_{O_{p}(H)∩A}^{O_{p}(H)}, so O_{p}(H) is contained in A.

3 Primordial groups

Lemma 3.1.

i) Prim(a(k )) and Prim(a(k , triv)) are closed under subgroups and quotients.

ii) E_{k} ⊆ Prim(a(k )) ⊆ Prim(a(k , triv)) ⊆ ⋃_{q} Dr_{q}.

Proof. i) The proof is the same for both functors, so M represents any of them. First, let G be a primordial group for M, and H a subgroup of G. By 2.6 tensor induction provides a map from M(H) to M(G), and clearly it sends the class of the field k to itself. Suppose that k can be written as a linear combination of modules induced from proper subgroups of H, then, by iii) and iv) of Proposition 3.15.2 in Benson [1], its image is a linear combination of modules induced from proper subgroups of G. This contradicts that G is primordial for M.

Now we take G/K, a quotient of G. Consider the inflation from M(G/K) to M(G). Again, the class of k is invariant under inflation, so if it could be written as a linear combination of modules induced from proper subgroups of G/K, then, these could be seen as modules induced form proper subgroups of G, which is a contradiction.

ii) In order to prove the inclusions

E_{k} ⊆ Prim(a(k )) ⊆ Prim(a(k , triv))

we recall that for every group H, we have the following morphisms

a(kH) → G_{0}(kH) and a(kH, triv) ↪ a(kH),

the first one sends the class [T] in a(kH) to the class of T in G_{0}(kH). These are morphisms of unitary algebras and commute with induction. To represent any of them we write f_{H} : M(H) → N(H). Now suppose that H is primordial for N. Given the properties of f_{H}, if k can be written as a linear combination of modules induced from proper subgroups of H in M(H), then that can also be made in N(H), which is a contradiction. So H is primordial for M. Recall that Prim(G_{0}(k )) = E_{k}, as mentioned in the Introduction.

To prove the inclusion Prim(a(k , triv)) ⊆ ⋃_{q} Dr_{q}, we will write M for a(k , triv) and D for Prim(a(k , triv)). Let G be any group. We will use the following facts:

a) Using the Dress induction theorem for the Burnside ring, as stated in Yoshida’s paper [10], we obtain a generalization of this theorem for the Mackey functor M. Let p-Hypo be the class of p-hypoelementary groups, we have:

M(G) = ∑_{\begin{matrix}K≤G \\ K∈H(p-Hypo) \end{matrix}}tr_{K}^{G}M(K) + ⋂_{\begin{matrix}L≤G \\ L∈p-Hypo \end{matrix}} ker(res_{L}^{G})

where res_{L}^{G} is the restriction map from M(G) to M(L) and

H(p-Hypo) = {H group | ∃q prime with O^{q}(H) ∈ p-Hypo}.

It is not hard to prove that H(p-Hypo) = ⋃_{q} Dr_{q}. b) Observe that the Mackey functor a(k ) satisfies the Frobenius reci-

procity formulas

tr_{K}^{G}(m · res_{K}^{G}n) = (tr_{K}^{G}m) · n and tr_{K}^{G}((res_{K}^{G}n) · m) = n · (tr_{K}^{G}m)

for all m in a(kK) and n in a(kG), with K ≤ G.

c) Note also that D is the smallest class of groups closed under subgroups and quotients such that for every group G,

M(G) = ∑_{\begin{matrix}K≤G \\ K∈D \end{matrix}} tr_{K}^{G}M(K).

A proof of this is a slight modification of Thévenaz’s in [8].

We consider the inclusion a(kG, triv) ⊆ ℚ ⊗ a(kG) and we will write N for ℚ ⊗ a(k ). From the article of Thévenaz [8], we have that Prim(N) is the class p-Hypo and that

N(G) = ∑_{\begin{matrix}K≤G \\ K∈p-Hypo \end{matrix}} tr_{K}^{G}N(K).

So we have

1_{M(G)} = 1_{N(G)} = ∑_{\begin{matrix}K≤G \\ K∈p-Hypo \end{matrix}} tr_{K}^{G}n_{K}

where 1_{M(G)} represents the unity of M(G) and n_{K} is an element of N(K). From the formula in a), if m is any element m of ⋂\begin{matrix}L≤G \\ L∈p-Hypo \end{matrix} ker(res_{L}^{G}), we

have

m = 1_{M(G)} · m = ∑_{\begin{matrix}K≤G \\ K∈p-Hypo \end{matrix}}(tr_{K}^{G}n_{K}) · m

= ∑_{\begin{matrix}K≤G \\ K∈p-Hypo \end{matrix}} tr_{K}^{G}(n_{K} · res_{K}^{G}m) by b)

= ∑_{\begin{matrix}K≤G \\ K∈p-Hypo \end{matrix}} tr_{K}^{G}(n_{K} · 0) = 0.

So we have

M(G) = ∑_{\begin{matrix}K≤G \\ K∈H(p-Hypo) \end{matrix}}tr_{K}^{G}M(K).

Since H(p-Hypo) is closed under subgroups and quotients, using c) we obtain Prim(a(k , triv)) ⊆ ⋃_{q} Dr_{q}.

Theorem 3.2. Prim(a(k , triv)) = Dr_{k}.

The proof is given by Propositions 3.3 and 3.5.

With Proposition 3.5 we will conclude that every primordial group for a(k ) has to be k-Dress for some prime. On the other hand, the following proposition shows that every k-Dress group for a prime different from p is primordial for a(k ). As for the general case, the main difficulty arises from the fact that the techniques we use (namely Lemma 2.8) are ineffective for non trivial source modules.

Proposition 3.3.

i) Dr_{k} ∖ Dr_{p}^{∗} ⊆ Prim(a(k )).

ii) Dr_{p}^{∗} ⊆ Prim(a(k , triv)).

Proof. We prove i) and ii) simultaneously by contradiction. Let H be a k-Dress group. We have two cases:

• H/O_{p}(H) is not divisible by p; in this case we suppose H is not primordial for a(k ) to prove i).

• H/O_{p}(H) is divisible by p, so we assume H is not primordial for a(k , triv) to prove ii).

In both cases, k can be written as a linear combination of modules induced from proper subgroups of H

k ⊕ ( ⨁_{i} M_{i} ↑_{L_{i}}^{H} ) ≅ ⨁_{j} N_{j} ↑_{T_{j}}^{H} .

We show that in the first case we can assume that M_{i} and N_{j} are trivial source modules. Notice that if an indecomposable module of trivial source is a direct summand of one M_{i} ↑_{L_{i}}^{H} (or N_{j}↑_{T_{j}}^{H}), then by Lemma 2.8, M_{i} (or N_{j}) and all the indecomposable summands are trivial source modules. Therefore, by the Krull-Schmidt Theorem, we can assume all of the M_{i} and N_{j} are of trivial source. Since in the second case we already assume that they are trivial source modules, the following arguments are valid for both cases. From 2.8, we have that O_{p}(H) acts trivially on N_{j} and M_{i} and that they have a vertex containing O_{p}(H). We take the quotients

k ⊕ ( ⨁_{i} M_{i} ↑_{L_{i}/O_{p}(H)}^{H/O_{p}(H)} ) ≅ ⨁_{j} N_{j} ↑_{T_{j}/O_{p}(H)}^{H/O_{p}(H)} .

This isomorphism is an equality in a(k(H/O_{p}(H)), triv), which is contained in a(k(H/O_{p}(H))). Since H/O_{p}(H) is k-elementary, Lemma 3.1 yields a contradiction.

Lemma 3.4. If H is of smallest order being q-Dress and not k-Dress, then H is of the form

H =< x > ⋊ < y > where | < x > | = r, | < y > | = q^{n}

with r and q different primes and yxy^{−1} = x^{a} with a ∈ ℤ_{r}^{∗} ∖ I_{r}(k).

Proof. Being a quotient of H, H/O_{p}(H) is q-Dress but, since it is not kelementary and O_{p}(H/O_{p}(H)) = 1, then H/O_{p}(H) is not k-Dress. Therefore the minimality of H implies O_{p}(H) = 1. Hence, we have H = C ⋊ Q with C =< s > cyclic of order m and Q a q-group such that m not divisible by p and q. Now, as H is not k-elementary, there exists y ∈ Q such that ysy^{−1} = s^{a} with a ∈ ℤ_{m}^{∗} ∖ I_{m}(k), so H = C ⋊ C_{q^{n}}, where C_{q^{n}} is the cyclic of order q^{n} generated by y.

Now we write C = ∏_{i} C_{r_{i}}, where C_{r_{i}} is the r_{i}-Sylow subgroup of C and the r_{i} are all the primes that divide m, so

C ⋊ C_{q^{n}} = ∏_{i}(C_{r_{i}} ⋊ C_{q^{n}}).

In addition, if ω is a primitive m-th root of unity, we have the commutative diagram

\xymatrix{Gal(k(ω)/k) \ar[d]_{≅} \ar@{^{(}->}[r] & (ℤ/mℤ)^{∗} \ar[d]^{≅} \\ ∏_{i} Gal(k(ω^{m/r_{i}^{α_{i}}})/k) \ar@{^{(}->}[r] & ∏_{i}(ℤ/r_{i}^{α_{i}}ℤ)^{∗}}

where α_{i} is the largest positive integer such that r_{i}^{α_{i}} divides m. Thus, there exists i such that C_{r_{i}} ⋊ C_{q^{n}} is not k-elementary. Rewriting C_{r_{i}} = C_{r^{α}} we have H = C_{r^{α}} ⋊ C_{q^{n}}, where C_{r^{α}} =< x_{o} >, C_{q^{n}} =< y > and yx_{o}y^{−1} = x_{o}^{a} with a ∈ ℤ_{r^{α}}^{∗} ∖ I_{r^{α}}(k).

Finally, we take ζ a primitive r^{α}-th root of unity. We have

Gal(k(ζ)/k) ≅ Gal(k(ζ)/k(ζ^{r^{α−1}})) × Gal(k(ζ^{r^{α−1}})/k)

and (ℤ/r^{α}ℤ)^{∗} ≅ A_{r^{α−1}} × A_{r−1} where these groups have order r^{α−1} and r − 1 respectively. The morphism

\xymatrix{Gal(k(ζ)/k) \ar@{^{(}->}[r] & (ℤ/r^{α}ℤ)^{∗}}

takes Gal(k(ζ)/k(ζ^{r^{α−1}})) into A_{r^{α−1}} and Gal(k(ζ^{r^{α−1}})/k) into A_{r−1}, which is isomorphic to (ℤ/rℤ)^{∗}, so we have the commutative diagram

\xymatrix{Gal(k(ζ)/k) \ar@{->>}[d] \ar@{^{(}->}[r] & (ℤ/r^{α}ℤ)^{∗} \ar@{->>}[d] \\ Gal(k(ζ^{r^{α−1}})/k) \ar@{^{(}->}[r] & (ℤ/rℤ)^{∗}.}

Now, we have a^{q^{n}} ≡ 1 mod r^{α}, thus r does not divide the order of a modulo r^{α}, and we have a ∈ ℤ_{r}^{∗}. Since a is not in I_{r^{α}}(k), we have a ∈ ℤ_{r}^{∗} ∖ I_{r}(k). Taking x = x_{o}^{r^{α−1}} we have the result.

Proposition 3.5. Prim(a(k , triv)) ⊆ Dr_{k}.

Proof. The proof is by contradiction. We suppose there is a group H of smallest order in Prim(a(k , triv)) that is not k-Dress. Observe that the previous lemma is also valid if H satisfies a property that is preserved under subgroups and quotients and implies being q-Dress. Since this is the case for the property of being primordial for a(k , triv), the lemma gives us H = C ⋊ Q with C =< x > of order r and Q =< y > of order q^{n} with r and q different primes and yxy^{−1} = x^{a} with a ∈ ℤ_{r}^{∗} ∖ I_{r}(k).

We write 1_{H} to identify the field k as a kH-module. We shall prove that 1_{H} is a sum of modules induced from proper subgroups of H, contradicting the assumption of primordiality.

The image of 1_{Q} under the induction morphism 1_{Q}↑_{Q}^{H} is isomorphic, as vector spaces, to ⨁_{i=0}^{r−1} kx^{i} ⊗_{Q} 1_{Q}. If ω is a primitive r-th root of unity, then the k(ω)H-module k(w) ⊗ 1_{Q} ↑_{Q}^{H} as k(w)-vector space has the basis

{x^{i} ⊗_{Q} 1_{Q} | i = 0, …, r − 1}.

We can define another basis y_{t} := ∑_{i=0}^{r−1} ω^{−ti}(x^{i} ⊗_{Q} 1_{Q}) for t = 0, …, r − 1. To prove it is a basis, observe that the matrix

( \begin{matrix}1 & 1 & … & 1 & … & 1 \\ 1 & ω^{−1} & … & ω^{−j} & … & ω^{−(r−1)} \\ ⋮ & ⋮ & ... & ⋮ & ... & ⋮ \\ 1 & ω^{−(r−1)} & … & ω^{−j(r−1)} & … & ω^{−(r−1)^{2}} \end{matrix} )

has determinant equal to ∏_{i≠j}(ω^{−i} − ω^{−j}), which is different from 0 since ω is a r-th primitive root of unity.

H acts on this basis in the following way

xy_{t} = ∑_{i=0}^{r−1} ω^{−ti}(x^{i+1} ⊗_{Q} 1_{Q}) if j=i+1,

= ∑_{j=0}^{r−1} ω^{−t(j−1)}(x^{j} ⊗_{Q} 1_{Q}) = ω^{t}y_{t}

yy_{t} = ∑_{i=0}^{r−1} ω^{−ti}(yx^{i} ⊗_{Q} 1_{Q}) = ∑_{i=0}^{r−1} ω^{−ti}(x^{ai} ⊗_{Q} 1_{Q})

= ∑_{i=0}^{r−1} ω^{−tbi}(x^{i} ⊗_{Q} 1_{Q}) = y_{t′}

where b ∈ ℤ_{r}^{∗} is such that \overline{b}\overline{a} = 1 in (ℤ/rℤ)^{∗}, and 0 ≤ t′ ≤ r − 1 satisfies t′ ≡ tb mod r. From these relations we see that y_{0} is fixed under the action of H, so k(ω)y_{0} is k(ω)H-isomorphic to k(ω) and we have

k(ω) ⊗ 1_{Q} ↑_{Q}^{H} ≅ k(ω) ⊕ ( ∑_{t=1}^{r−1} k(ω)y_{t}).

It is clear that G = Gal(k(ω)/k) acts on k(ω) ⊗ 1_{Q} ↑_{Q}^{H}. Taking the fixed points of this action in the isomorphism above, gives us

1_{Q} ↑_{Q}^{H} ≅ 1_{H} ⊕ ( ∑_{t=1}^{r−1} k(ω)y_{t})^{G}.

Let σ be in G, we write b_{σ} ∈ I_{r}(k) for the integer through which σ is defined. We have

σy_{t} = ∑_{i=0}^{r−1} σ(ω^{−ti})(x^{i} ⊗ 1) = ∑_{i=0}^{r−1} ω^{−tb_{σ}i}(x^{i} ⊗ 1) = y_{s}

where 0 ≤ s ≤ r−1 and s ≡ tb_{σ} mod r. If u = ∑_{t=1}^{r−1} λ_{t}y_{t} is in (∑_{t=1}^{r−1} k(ω)y_{t})^{G}, then for each t ∈ ℤ_{r}^{∗} we must have σ(λ_{t}) = λ_{s}, where s ∈ ℤ_{r}^{∗} and s ≡ tb_{σ} mod r. From this we define the vector spaces

M_{l} := { ∑_{σ∈G} σ(λ_{l})y_{s} | s ≡ lb_{σ} mod r, λ_{l} ∈ k(ω)}

for each l ∈ ℤ_{r}^{∗}. Observe that M_{l_{1}} = M_{l_{2}} if and only if l_{1} ≡ l_{2}b_{σ} mod r for some σ ∈ G, this implies that

( ∑_{t=1}^{r−1} k(ω)y_{t})^{G} = ⨁_{l∈ℤ_{r}^{∗}/I_{r}(k)} M_{l}.

We shall prove that the right hand side of this equality is a sum of modules induced from proper subgroups of H. We have xM_{l} = M_{l} and yM_{l} = M_{l′} with l′ ∈ ℤ_{r}^{∗} and l′ ≡ lb mod r. Since a does not belong to I_{r}(k), neither does b, so M_{l} is never fixed under the action of y. Then M_{l} is not a kH-module. Taking the orbits of the action of y we have

( ∑_{t=1}^{r−1} k(ω)y_{t})^{G} = ⨁_{l∈ \frac{ℤ^{∗}_{r}/Ir(k)}{∼}} ( ⨁_{z∈[H/A]} zM_{l})

= ⨁_{l∈ \frac{ℤ^{∗}_{r}/Ir(k)}{∼}} (M_{l} ↑_{A}^{H} )

where A = Stab_{H}(M_{l}), and ∼ represents the action of y.

It is clear that A is a proper subgroup of H. Finally, observe that every M_{l} is a trivial source kA-module. If A = C_{r}⋊ < y^{d} >, then M_{l} is a direct summand of the induced module ( ∑_{σ∈G} ky_{s}) ↑_{<y^{d}>}^{A} , where s ∈ ℤ_{r}^{∗} and s ≡ lb_{σ} mod r. Since ky_{s} ≅ k, then M_{l} has trivial source.

Acknowledgements

We wish to thank Julio Aguilar Cabrera, whose Master thesis was of great help in writing this article, and Luis Valero Elizondo for helpful conversations.

References

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