The Dade group of a metacyclic p-group

Nadia Mazza^{∗} Institut de Mathématiques Université de Lausanne CH-1015 Lausanne

September 30, 2002

Abstract

The Dade group D(P) of a finite p-group P, formed by equivalence classes of endopermutation modules, is a finitely generated abelian group. Its torsion-free rank equals the number of conjugacy classes of non-cyclic subgroups of P and it is conjectured that every nontrivial element of its torsion subgroup D^{t}(P) has order 2, (or also 4, in case p = 2). The group D^{t}(P) is closely related to the injectivity of the restriction map Res : T (P) → ∏_{E} T (E) where E runs over elementary abelian subgroups of P and T (P) denotes the group of equivalence classes of endo-trivial modules, which is still unknown for (almost) extra-special groups (p odd). As metacyclic p-groups have no (almost) extra-special section, we can verify the above conjecture in this case. Finally, we compute the whole Dade group of a metacyclic p-group.

1 Introduction

In the p-modular representation theory of finite groups, the family of endo-permutation modules seems to occupy a key position. These modules appear indeed as sources of simple modules for p-solvable groups (cf. [Pu2]) and also in the local analysis of derived equivalences between blocks (cf. [Ri]).

Let p be a prime, P a finite p-group and k a field of characteristic p (which we can suppose to be algebraically closed). All the kP -modules we consider are finitely generated. Recall that an endo-permutation kP -module M is a kP -module such that the P -algebra End_{k} M is a permutation kP -module (i.e. there exists a P -invariant k-basis). If M is indecomposable, we say it has vertex P if M is not projective relative to any proper subgroup of P. If M is an endo-permutation module with at least one indecomposable summand M_{0} with vertex P, then this summand is unique, up to isomorphism, and, following Dade, we call it the cap of M. This allows us to define an equivalence relation on these modules:

M ∼ N ⟺ M_{0} ≅ N_{0}.

∗This work is a part of a doctoral thesis in preparation at the University of Lausanne, under the supervision of Prof. Jacques Thévenaz

We write D(P) for the set of equivalence classes of such kP -modules. It is an abelian group for the composition given by [M] + [N] = [M ⊗ N]. The unit element is the class of the trivial kP -module k and the inverse of [M] is −[M] = [M^{∗}], where M^{∗} = Hom_{k}(M, k) is the dual of M.

J. Alperin proved recently that, for any P -set X, the relative syzygy Ω_{X}^{1}(k) of the trivial module, that is the kernel of the linear map from kX to k which sends each x ∈ X to 1, is an endo-permutation kP -module. Moreover, if X is a transitive P -set, then Ω_{X}^{1}(k) is indecomposable with vertex P (cf. [Al2]). To simplify the notations we write Ω_{X} for the class of Ω_{X}^{1}(k) in D(P).

When dealing with endo-permutation modules, we can’t avoid the special family of endo-trivial modules. Recall that these are precisely the kP -modules M such that End_{k} M ≅ k ⊕ L, for a free kP -module L. They all have an indecomposable direct summand with vertex P and are obviously endo-permutation modules. So we can consider the same relation as before and look at the corresponding group T (P), which turns out to be a subgroup of D(P) (note that the classes in T (P) contain, in general, less elements than the corresponding ones in D(P)). The following theorem is due to Dade. It classifies all endo-permutation modules (up to equivalence) in the abelian case and shows that T (P) is of relevant importance when studying D(P).

Theorem 1.1 [Dade, [Da]] Let P be an abelian p-group.

1. T (P) is generated by Ω_{P}. Thus T (P) is trivial, if |P | ≤ 2, cyclic of order 2, if P is cyclic of order ≥ 3, and infinite cyclic otherwise.

2. D(P) is isomorphic to the direct sum of the groups T (P/Q), where Q runs over the set of subgroups of P.

More than 20 years after Dade’s result, the question concerning an arbitrary finite p-group P is still open. Only partial results have been obtained in the general case. First, L. Puig proved that D(P) is always finitely generated (cf. [Pu1]). More recently, J. Alperin determined the torsion-free rank of T (P) (cf. [Al2]) and S. Bouc and J. Thévenaz calculated the torsion-free rank of D(P) (cf. [BoTh]). Following [CaTh], the problem of determining the torsion subgroup D^{t}(P) comes down to finding a detecting family F of groups for which the restriction map Res : T (P) → ∏_{E∈F} T (E) is injective. That would then allow us to recover all torsion elements of D(P) through restrictiondeflation maps to all sections of P belonging to F (see Theorem 10.1 in [CaTh]). By now, for an odd p, we have to take for F all elementary abelian groups of rank 1 or 2 and also all extraspecial groups of exponent p. However, as it is hoped that we can take for F just the elementary abelian groups of rank 1 or 2, we will discuss here the case of p-metacyclic groups for odd p, where no extraspecial group of exponent p appears in the set of all sections of P. The final aim is to prove the following result.

Theorem 1.2 Let p be an odd prime and P be a metacyclic p-group. Then we have an isomorphism of abelian groups:

α_{P} : D(P) ⟶ ∏_{H∈[X /P]} T (H/Φ(H)) ≅ ℤ^{r} × (ℤ/2ℤ)^{s},

where Φ(H) is the Frattini subgroup of H, for all subgroup H of P, and where [X /P] = [S/P]∪[C/P] is a set of representatives of conjugacy classes of non trivial subgroups of P, divided into S/P, the set of conjugacy classes of non-cyclic subgroups of P, of cardinality r, and C/P, the set of conjugacy classes of non-trivial cyclic subgroups of P, of cardinality s.

Moreover, the set {Ω_{P/Φ(H)}, H ∈ [S/P]} form a basis of the torsion-free part and the set {Teninf_{H/Φ(H)}^{P} Ω_{H/Φ(H)}, H ∈ [C/P]} a basis (over ℤ/2ℤ) of the torsion subgroup D^{t}(P) of D(P).

Note that it will also prove that the injection of Theorem 4.1 of [BoTh] is, in fact, a bijection.

Let us finally recall some of the notions we are going to use. A group is called metacyclic if it is an extension of a cyclic group by a cyclic group. Let P be a finite p-group. We write Q ≤ P, if Q is a subgroup of P, and we write Q ≤_{P} R, if Q and R are subgroups of P such that Q ≤ ^{g}R, for a g ∈ P, and where ^{g}R = gRg^{−1}. If Q ≤ P, we have an obvious restriction map Res_{Q}^{P}, from D(P) to D(Q), and we can also consider the tensor induction Ten_{Q}^{P} from D(Q) to D(P). If, moreover, Q is normal in P, we have an inflation map Inf_{P/Q}^{P} from D(P/Q) to D(P) and a deflation map Def_{P/Q}^{P} from D(P) to D(P/Q). This map is obtained as follows. If M is an endo-permutation kP -module, then we consider the Brauer construction A[Q] = A^{Q}/(∑_{R<Q} A_{R}^{Q}) of the P -algebra A = End_{k} M, where A^{Q} = End_{kQ} M is the set of fixed points of A under the action of Q, and ∀R < Q, A_{R}^{Q} denotes the image of the relative trace from A^{R} to A^{Q}. As P is a p-group, we know that A[Q] ≅ End_{k} N for a unique (up to isomorphism) endo-permutation k[P/Q]-module N. So we define Def_{P/Q}^{P}([M]) = [N] ∈ D(P/Q). If R ◁ Q < P, we write Defres_{Q/R}^{P} instead of Def_{Q/R}^{Q} ∘ Res_{Q}^{P}, and Teninf_{Q/R}^{P} instead of Ten_{Q}^{P} ∘ Inf_{Q/R}^{Q}. We assume the reader to be familiar with all the properties of these notions and with basic results of representation theory of finite groups.

2 Structure of metacyclic p-groups

Let P be a metacyclic p-group (p odd) such that the sequence 1 → C_{p^{n}} → P → C_{p^{m}} → 1 is exact. So we can choose u, v ∈ P such that u^{p^{n}} = 1, v^{p^{m}} ∈ <u> (including m = 0, in case P is cyclic) and ^{v}u = u^{p^{l}+1}, for an integer l such that 0 < l ≤ n (we have l = n if P is abelian) and where vu = vuv^{−1}. So |P | = p^{m+n}, <u> ◁ P and any element of P can be expressed as u^{a}v^{b}, with 0 ≤ a < p^{n} and 0 ≤ b < p^{m}. It is easy to verify that <u^{p^{n−1}}> ≤ Z(P). Indeed, this is the only subgroup of <u> of order p. Moreover, since <u> ◁ P, one has that <u> ∩ Z(P) ≠ {1}. If P is not cyclic, we have the following result.

Lemma 2.1 Let P be a non-cyclic metacyclic p-group for an odd prime p. Then P has a unique elementary abelian subgroup of rank 2.

Proof . Let E be an elementary abelian subgroup of P of rank 2. Then E ∩<u> ≠ {1}, because E/E ∩ <u> is cyclic (it is isomorphic to the subgroup E<u>/<u> of the cyclic group P/<u>). Thus, E contains u^{p^{n−1}}. The image of E in P/<u> is non-trivial, thus E contains an element of the form u^{a}v^{p^{m−1}}, and in fact E = <u^{p^{n−1}}, u^{a}v^{p^{m−1}}> (since <u^{p^{n−1}}> < <u^{p^{n−1}}, u^{a}v^{p^{m−1}}> ≤ E). Set w = v^{p^{m−1}}. The action of w on <u> is given by ^{w}u = u^{r}, where r = sp^{n−1} + 1, for some integer s > 0 : if w and u commute, take s = p. This will be the case in particular if n = 1, since this implies that P is abelian.

Moreover, there is an integer d such that w^{p} = u^{d}, and dis a multiple of p if P is non-cyclic (otherwise d is prime to p and there exists an integer e, prime to p, such that de ≡ 1 (mod p^{n}), but then u = u^{de} = w^{pe} ∈ <v>). Then

(u^{a}w)^{p} = u^{a}(^{w}u)^{a}(^{w^{2}}u)^{a} ⋯ (^{w^{p−1}}u)^{a}w^{p} = u^{a(r+r^{2}+···+r^{p−1})+d} = u^{a \frac{r^{p}−1}{r−1} +d}.

Finally,

\frac{r^{p} − 1}{r − 1} = (sp^{n−1})^{p−1} + (\begin{matrix}p \\ 1 \end{matrix}) (sp^{n−1})^{p−2} + ⋯ + (\begin{matrix}p \\ p − 2 \end{matrix}) sp^{n−1} + (\begin{matrix}p \\ p − 1 \end{matrix}) .

Since (sp^{n−1})^{p−1} ≡ 0 (mod p^{n}) (even if n = 1, with the above convention on s), and since, for p odd, the binomial coefficients are all multiple of p, it follows that \frac{r^{p}−1}{r−1} ≡ p (mod p^{n}). Hence, (u^{a}w)^{p} = u^{ap+d} and this is equal to 1 if and only if ap + d is a multiple of p^{n}. Equivalently, a is congruent to −\frac{d}{p} modulo p^{n−1}, and E = <u^{p^{n−1}}, u^{− \frac{d}{p}} v^{p^{m−1}}>. Conversely, the previous argument shows that the group E defined by this formula is elementary abelian of rank 2. Hence P has an elementary abelian subgroup of rank 2, and it is unique.

□

We can deduce immediately from Lemma 2.1 the following consequence, which is fundamental for the motivation of this paper.

Corollary 2.2 Let P and p be as in the previous lemma. Then P has no extraspecial section of exponent p.

Proof . Any non-trivial extraspecial p-group of exponent p has many elementary abelian subgroups of rank 2. As any subgroup and any quotient of a metacyclic group is still metacyclic, it follows that any section of P is a metacyclic group and thus it cannot be an extraspecial group of exponent p.

□

Remark 2.3 Note that this lemma doesn’t apply to metacyclic 2-groups, as, for instance, the dihedral group of order 8 is metacyclic but has 2 elementary abelian subgroups of rank 2.

Here is another property of non-cyclic metacyclic p-groups which will be of relevant importance for the proof of Theorem 1.2.

Lemma 2.4 Assume P to be non-cyclic and let H be a non-cyclic subgroup of P. Then H is uniquely determined by Φ(H). In other words, if H and K are two non-cyclic subgroups of P, with Φ(K) = Φ(H), then K = H.

Proof . Consider N = N_{P} (Φ(H)). The quotient group N/Φ(H) is also metacyclic and so has a unique elementary abelian subgroup of rank 2, by the previous lemma and the fact that it is non-cyclic (since it contains H/Φ(H) ≅ C_{p} × C_{p} because H is metacyclic, hence generated by two elements, and non-cyclic). So N has a unique subgroup containing Φ(H) and such that the quotient is elementary abelian of rank 2. Thus H is unique.

□

We conclude this section with an immediate consequence of this lemma.

Corollary 2.5 Under the same hypothesis of Lemma 2.4, we have N_{P} (H) = N_{P} (Φ(H)).

Proof . Φ(H) is a characteristic subgroup of N_{P} (H). Thus we have N_{P} (H) ≤ N_{P} (Φ(H)). Conversely, if u ∈ N_{P} (Φ(H)), then Φ(^{u}H) = (^{u}Φ(H)) = Φ(H) implies H = ^{u}H by Lemma 2.4.

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3 The Dade group of a metacyclic p-group

Let P be a metacyclic p-group (p odd) and consider the homomorphism of abelian groups

Ψ_{P} = ∏_{Q∈[X /P]} Defres_{Q/Φ(Q)}^{P} : D(P) ⟶ ∏_{Q∈[X /P]} D(Q/Φ(Q)),

where [X /P] is a set of representatives of the conjugacy classes of the non-trivial subgroups of P. We can write X /P as the disjoint union of S/P and C/P, where S/P is the set of conjugacy classes of non-cyclic subgroups of P, and C/P is the set of conjugacy classes of non-trivial cyclic subgroups of P.

Proposition 3.1 The map Ψ_{P} is injective.

Proof . For all Q ∈ [X /P], the quotient Q/Φ(Q) is cyclic of order p, if Q is cyclic, and elementary abelian of rank 2 otherwise. So we have respectively D(Q/Φ(Q)) = T (Q/Φ(Q)) ≅ ℤ/2ℤ or D(Q/Φ(Q)) ≅ ⨁_{R<Q} T (Q/R) ≅ ℤ ⊕ (ℤ/2ℤ)^{p+1}. As P doesn’t have any extraspecial section of exponent p (cf. Lemma 2.2), Theorem 10.1 of [CaTh] implies that the map

∏_{H/K∈[Y/P]} Defres_{H/K}^{P} : D(P) ⟶ ∏_{H/K∈[Y/P]} D(H/K) is injective,

where Y denotes the set of all elementary abelian sections of P of rank 1 or 2. Let us prove the result by induction on |P |. If |P | = p, then S = ∅, C = {P } and so Ψ_{P} = Id_{D(P)} is injective. Assume |P | > p and let a ∈ Ker(Ψ_{P}). We must prove that Defres_{H/K}^{P}(a) = 0, ∀ H/K ∈ Y. Let H < P. Then it is clear that Res_{H}^{P}(a) ∈ Ker(Ψ_{H}). Thus, by induction hypothesis, we have Res_{H}^{P}(a) = 0 and so Defres_{H/K}^{P}(a) = 0. If H = P and K ◁P is such that P/K is elementary abelian of rank 1 or 2, then we must show that Def_{P/K}^{P}(a) = 0. If P/K is elementary abelian of rank 2, then K = Φ(P), and Def_{P/K}^{P}(a) = 0 by assumption. Suppose |P : K| = p. Then Φ(P) < K < P

and we have Def_{P/K}^{P}(a) = Def_{P/K}^{P/Φ(P)} ( Def_{P/Φ(P)}^{P}(a)) = Def_{P/K}^{P/Φ(P)}(0) = 0, by hypothesis on a.

□

In order to prove the injectivity of the map given in Theorem 1.2, we first need two more lemmas.

Lemma 3.2 Let P be a non-cyclic metacyclic p-group (p odd). Then, the map

∏_{1<Q<P} Res_{Q}^{P} : D^{t}(P) ⟶ ∏_{1<Q<P} D^{t}(Q) is injective.

Proof . Proposition 3.1 implies that ∏_{1<Q≤P} Defres_{Q/Φ(Q)}^{P} : D^{t}(P) ⟶ ∏_{1<Q≤P} D^{t}(Q/Φ(Q))

is injective. Moreover, ∏_{Φ(Q)<R<Q} Res_{R/Φ(Q)}^{Q/Φ(Q)} : D^{t}(Q/Φ(Q)) ⟶ ∏_{Φ(Q)<R<Q} D^{t}(R/Φ(Q)) is injective for any non-cyclic subgroup Q of P, by Lemma 6.1 of [BoTh]. So, there is a commutative square

D^{t}(P) ⟶ ∏_{1<S<P} D^{t}(S)
↓  ↓
∏_{1<Q≤P} D^{t}(Q/Φ(Q)) ⟶ ∏_{Φ(Q)<R<Q} D^{t}(R/Φ(Q)),

where the upper horizontal map is ∏_{1<S<P} Res_{S}^{P}, the bottom map is ∏_{Φ(Q)<R<Q} Res_{R/Φ(Q)}^{Q/Φ(Q)}, which is injective by Lemma 6.1 of [BoTh], and the left vertical map is ∏_{1<Q≤P} Defres_{Q/Φ(Q)}^{P}, which is injective by the previous proposition. Hence the top map is injective.

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Lemma 3.3 Let P be a metacyclic p-group (p odd) and C the set of all non-trivial cyclic subgroups of P. Then, the map ∏_{[C∈C/P]} Defres_{C/Φ(C)}^{P} : D^{t}(P) ⟶ ∏_{C∈[C/P]} T (C/Φ(C)) is injective.

Proof . Note that this map is well defined, as C/Φ(C) is cyclic of order p and so D(C/Φ(C)) = T (C/Φ(C)) is cyclic of order 2, for all C ∈ C. Let us proceed by induction on |P | and start with |P | = p. Then C = {P } and the above map is the identity map. Assume now |P | > p. If P is cyclic, then, the lemma coincides with Proposition 3.1, and so there is nothing to prove. If P is not cyclic, then, by induction and the above lemma, it follows that ∏_{C∈C} Defres_{C/Φ(C)}^{P} maps D^{t}(P) into ∏_{C∈C} T (C/Φ(C)). As Ker ( Defres_{Q/Φ(Q)}^{P} ) = Ker ( Defres_{^{g}Q/Φ(^{g}Q)}^{P} ), ∀Q ≤ P, we can restrict this map to the conjugacy classes of non-trivial cyclic subgroups without loosing the injectivity and so the lemma is proved.

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Notice that this lemma gives us an upper bound for the rank of the torsion subgroup of D(P). We will show later that this rank is, in fact, equal to this upper bound. Let us turn back now to the injectivity question. The result of Proposition 3.1 can be improved in the following sense. If P is an abelian group, we can consider the “projection” ρ_{P} : D(P)↠T (P), defined thanks to the isomorphism given in Theorem 1.1, where the reverse isomorphism is equal to

⨁_{Q<P} Inf_{P/Q}^{P} : ⨁_{Q<P} T (P/Q) ⟶ D(P).

Let P be a metacyclic p-group (p odd) and let α_{P} be the homomorphism of abelian groups defined as the composition ( ∏_{Q∈[X /P]} ρ_{Q/Φ(Q)}) ∘ Ψ_{P}.

Theorem 3.4 The map α_{P} : D(P) ⟶ ∏_{Q∈[X /P]} T (Q/Φ(Q)) is injective.

Proof . Ψ_{P} maps D(P) into ∏_{Q∈[S/P]} D(Q/Φ(Q)) × ∏_{C∈[C/P]} D(C/Φ(C)). If C ∈ C, then C/Φ(C) is cyclic of order p. So D(C/Φ(C)) = T (C/Φ(C)) and ρ_{C/Φ(C)} = Id. If Q ∈ S, then Q/Φ(Q) is elementary abelian of rank 2 and it follows from Theorem 1.1 that we have an isomorphism of abelian groups D(Q/Φ(Q)) ≅ T (Q/Φ(Q)) × ∏_{Φ(Q)<R<Q} T (R/Φ(Q)). Now, for each Φ(Q) < R < Q, we choose x ∈ R \ Φ(Q) and we set C_{R} = <x>. Then R = C_{R}Φ(Q), since Φ(Q) < C_{R}Φ(Q) ≤ R, and Φ(C_{R}) = <x^{p}> = C_{R} ∩ Φ(Q), since x^{p} ∈ Φ(Q) and since C_{R} ∩ Φ(Q) < C_{R}. Thus, C_{R}/Φ(C_{R}) ≅ R/Φ(Q) and we get an isomorphism of abelian groups D(Q/Φ(Q)) ≅ T (Q/Φ(Q)) × ∏_{Φ(Q)<R<Q} T (C_{R}/Φ(C_{R})), using the isomorphism of abelian groups

Iso_{R/Φ(Q)}^{C_{R}/Φ(C_{R})} between C_{R}/Φ(C_{R}) and R/Φ(Q), and where the subgroups C_{R} are cyclic, chosen as above. Finally, D(P) maps into

∏_{Q∈[S/P]} (T (Q/Φ(Q)) × ∏_{Φ(Q)<R<Q} T (C_{R}/Φ(C_{R}))) × ∏_{C∈[C/P]} T (C/Φ(C)).

Since Iso_{R/Φ(Q)}^{C_{R}/Φ(C_{R})} ∘ Res_{R/Φ(Q)}^{Q/Φ(Q)} ∘ Defres_{Q/Φ(Q)}^{P} = Iso_{R/Φ(Q)}^{C_{R}/Φ(C_{R})} ∘ Defres_{R/Φ(Q)}^{P} = Defres_{C_{R}/Φ(C_{R})}^{P}, for all Φ(Q) < R < Q and for all Q ∈ S, we can omit the factors T (C_{R}/Φ(C_{R})) appearing in the above formula (because we get the same image in some T (C/Φ(C)), for the corresponding conjugate subgroup C ∈ [C/P] of C_{R}) and the map is still injective. Hence, α_{P} is injective.

□

To prove the surjectivity of α_{P}, we are going to choose a subset of D(P) which is mapped onto a set of generators of the target. In order to do this, we must first define an order on the set [X /P] of representatives of the conjugacy classes of the non-trivial subgroups of P. Consider separately the representatives [C/P] of the conjugacy classes of non-trivial cyclic subgroups and [S/P] of the non-cyclic subgroups. Note that S = ∅ iff P is cyclic, and C ≠ ∅, ∀ P ≠ {1}. On both sets, there is a “natural” order ⪯, induced by the inclusion (that is H ⪯ K, if H ≤_{P} K, and H and K are both in C, or both in S) and we can extend them to total orders, still denoted by ⪯, on [C/P] and on [S/P]. Let now H and K be in [X /P] and write H ⪯ K if exactly one of the following condition is satisfied:

1. H ∈ [C/P] and K ∈ [S/P].

2. H, K ∈ [C/P] and H ⪯ K.

3. H, K ∈ [S/P] and H ⪯ K.

Consider the following set of elements of D(P) and order it according to the increasing order of the subgroups which appear in the subscripts.

B = { Teninf_{C/Φ(C)}^{P} Ω_{C/Φ(C)}, C ∈ [C/P]} ⋃ {Ω_{P/Φ(Q)}, Q ∈ [S/P]}.

Let us prove that B form a basis of the abelian group D(P), viewed as the direct sum of a free ℤ-module of rank |[S/P]| and a 𝔽_{2}-vector space of dimension |[C/P]|. Let C ∈ [C/P] and consider Teninf_{C/Φ(C)}^{P} Ω_{C/Φ(C)}. It is a torsion element so the component of its image by α_{P} in any T (Q/Φ(Q)) for Q ∈ [S/P] must be zero as T (Q/Φ(Q)) ≅ ℤ. Moreover, in Theorem 6.2 of [BoTh], it is proved that Defres_{C′/Φ(C′)}^{P}(Teninf_{C/Φ(C)}^{P} Ω_{C/Φ(C)}) = δ_{C,C′}Ω_{C/Φ(C)}, ∀ C, C′ ∈ [C/P]. Let H ∈ [S/P] (and hence assume that P is not cyclic, i.e. S is not empty) and prove that for all K ∈ [X /P] such that H ⪯ K we have Defres_{K/Φ(K)}^{P} Ω_{P/Φ(H)} = δ_{H,K}Ω_{H/Φ(H)}. By definition of the order, the condition H ⪯ K implies K ∈ [S/P]. We have Defres_{K/Φ(K)}^{P} (Ω_{P/Φ(H)}) =

Def_{K/Φ(K)}^{K} ( Res_{K}^{P} (Ω_{P/Φ(H)})). Let us recall some useful results of [Bo]: if X is a finite P -set, K

and N two subgroups of P with N ◁ P, then Def_{P/N}^{P} Ω_{X} = Ω_{X^{N}}, where X^{N} denotes the set of fixed point of X under the action of N. We also have Res_{K}^{P} Ω_{X} = Ω_{Res^{P}_{K} X}. Then, using Mackey’s formula and the isomorphism of P -sets Ind_{K}^{P}{∗} ≅ P/K, it follows that

Res_{K}^{P}(P/Φ(H)) = ( ∐_{g∈[K\P/Φ(H)], g∈N} \underbrace{K/(K ∩ Φ(H))}_{(I)}) ∐ ( ∐_{g∈[K\P/Φ(H)], g∉N} \underbrace{K/(K ∩ ^{g}Φ(H))}_{(II)}),

where N = N_{P} (Φ(H)) = N_{P} (H). If H = K, then all the elements of (I) are fixed by Φ(H). But there is no fixed point in (II). Indeed, if

x(H ∩ ^{g}Φ(H)) ∈ (H/H ∩ ^{g}Φ(H))^{Φ(H)} = {x(H ∩ ^{g}Φ(H)) | y^{x} ∈ H ∩ ^{g}Φ(H), ∀ y ∈ Φ(H)},

then Φ(H) = Φ(H)^{x} ≤ H ∩ ^{g}Φ(H) ≤ ^{g}Φ(H). Thus Φ(H) = ^{g}Φ(H), i.e. g ∈ N. It follows that Defres_{H/Φ(H)}^{P} Ω_{P/Φ(H)} = Ω_{H/Φ(H)} ∈ D(H/Φ(H)), as Ω_{H/Φ(H)}^{1}(k) is a direct summand of the kernel

of the map (⨁_{g∈[N/H]} k[H/Φ(H)] → k) (and so is “the” cap of this kernel). If |H| < |K|, then

there is no fixed point at all and so Defres_{K/Φ(K)}^{P} Ω_{P/Φ(K)} = 0. If |H| = |K|, then H and K are two non conjugate non-cyclic subgroups of P having the same order. That implies Φ(H) and Φ(K) have same order, but are not conjugate, by Lemma 2.4. It follows that X^{Φ(K)} = ∅ and so Defres_{K/Φ(K)}^{P} Ω_{P/Φ(H)} = 0. Let us sum up all these calculations: we proved that α_{P} maps B onto a set of generators of the target group and in terms of matrix, taking the set {Ω_{H/Φ(H)}, H ∈ [X /P]} as basis for the target and B for the source, we get an upper triangular matrix with ones on the diagonal. So we have proved the following result.

Theorem 3.5 Let P be a metacyclic p-group (p odd). Then,

α_{P} = ∏_{H∈[X /P]} (ρ_{H/Φ(H)} ∘ Defres_{H/Φ(H)}^{P} ) : D(P) ⟶ ∏_{H∈[X /P]} T (H/Φ(H))

is an isomorphism of abelian groups.

Moreover, the set {Ω_{P/Φ(H)}, H ∈ [S/P]} forms a basis of the torsion-free part and the set {Teninf_{H/Φ(H)}^{P} Ω_{H/Φ(H)}, H ∈ [C/P]} a basis (over ℤ/2ℤ) of D^{t}(P).

Remark 3.6 Finally some remarks on endo-permutation modules.

1. If P is a metacyclic p-group (p odd), it is immediate, from Lemma 2.1, that T (P) = <Ω_{P} > and so it is cyclic of order 2, if P is cyclic, and infinite cyclic, otherwise.

2. Take the same notations as in Theorem 3.5. The matrix of the map α_{P}, in the given ordered basis, is upper triangular with 1 on the diagonal. But, in general, it is not the identity matrix. Indeed, consider for instance a non-cyclic metacyclic p-group P of order ≥ p^{3} and E < P the unique elementary abelian subgroup of P. Then Φ(E) = {1} and we have α_{P} (Ω_{P/Φ(E)}) = Ω_{E} + ∑_{C∈[C/P]}(Ω_{C}).

3. Let us finish with a remark about splendid equivalences between blocks. Recall that a kP - module M has an endo-split permutation resolution X_{M} if there exists abounded complex X_{M} of permutation kP -modules and an isomorphism M ≅ H_{0}(X_{M}), such that H_{n}(X_{M}) = 0, ∀ n ≠ 0 and the complex End_{k}(X_{M}) is split. It follows from Theorem 3.5, using Lemma 2.3.7 of [Bo] and Lemma 7.3 of [Ri], that if P is a metacyclic p-group (p odd), then every endo-permutation kP -module M with a direct summand of vertex P has an endo-split permutation resolution X_{M}. Thus, Theorem 7.8 of [Ri] can be applied to p-nilpotent groups G with a metacyclic p-Sylow subgroup P. In other words (assume k to be a splitting field for G), if kB is the principal block of kG, M an endo-permutation module belonging to kB and X_{M} an endo-split permutation resolution of M, then X_{M} is a splendid tilting complex inducing a splendid derived equivalence between the derived categories D^{b}(kP) and D^{b}(kB).

References

[Al1] J. L. Alperin, Local representation theory, Cambridge studies in advanced mathematics 11, Cambridge University Press, 1986.

[Al2] J. L. Alperin, A construction of endo-permutation modules, J. Group Theory 4 (2001), 3–10.

[Bo] S. Bouc, Tensor induction of relative syzygies, J. reine angew. Math. (Crelle), 523 (2000), 113–171.

[BoTh] S. Bouc and J. Thévenaz, The group of endo-permutation modules, Invent. Math. 139 (2000), 275–349.

[CaTh] J. Carlson, J. Thévenaz, Torsion endo-trivial modules, Algebr. Represent. Theory 3 (2000), 303–335.

[CR] C. W. Curtis, I. Reiner, Representation theory of finite groups and associative algebras, Pure and applied mathematics 11, John Wiley and Sons, 1962.

[Da] E. C. Dade, Endo-permutation modules over p-groups, I, II, Ann. Math. 107 (1978), 459–494, 108 (1978), 317–346.

[Pu1] L. Puig, Affirmative answer to a question of Feit, J. Algebra 131 (1990), 513–526.

[Pu2] L. Puig, Notes sur les P -algèbres de Dade, Unpublished manuscript, 1988.

[Ri] J. Rickard, Splendid equivalences: derived categories and permutation modules, Proc. London math. soc. 3 (1996), 331–358.