1 Introduction

The(absolutely)p-divisible kG-module is introduced by Benson and Carlson[1].It is a tool to study the decomposition of the tensor product of two kG-modules,and a tool to study nilpotent elements in the Green ring.Different from many other kinds of kG-modules,its definition is independent of many classical aspects for the group algebra kG,but only essentially depended on the prime number p,and its class is a big one,all(relative)projective kG-modules are p-divisible.

The endotrivial kG-module is named by Dade[2].It is a building block for the endopermutation modules which are the sources for the irreducible modules of many finite groups(see[3]),and it forms an important part for the Picard group of self-equivalences of the stable module category.In this paper,based on the p-divisible kG-module,we extend the ordinary endotrivial kG-module and the relative endotrivial kG-module to the p-endotrivial kG-module(see[2]–[4]).

The(indecomposable)p-endotrivial kG-module here,at the same time,is a special kind of the splitting trace module(see[5]),that is,the kG-module V such that the trace map Tr:End(V)→k is split,and Auslander and Carlson[5]proved that the tensor of the splitting trace module V with the almost-split sequence for the trivial kG-module k cannot be split.

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Here we focus on the group Tp(G)of p-endotrivial kG-modules;on the one hand,we study the restriction map for this group,and prove that if the subgroup H contains the normalizer of a Sylow p-subgroup of G,for example,H is a strongly embedded subgroup of G,then Tp(G)Tp(H)(Theorem 2.8),and K(G)K(H)(Theorem 2.9);on the other hand,by using Tp(G)we obtain a generalized Dade group Dp(G)for the finite group G,and prove that Dp(G)can be embedded into Tp(G)as a subgroup(Theorem 3.3).Our results extend the results for the group T(G)of endo-trivial kG-modules and the results for the Dade group D(P)for the finite p-group P.

Throughout the paper,we fix a prime number p,a finite group G such that p||G|,and an algebraic closed field k of characteristic p.All modules are finitely generated,and p divides the order of any finite group involve in a p-endotrivial kG-module.For the necessary terminologies in the paper the reader can consult[6].

2 The Group Tp(G)of p-endotrivial kG-modules

For a prime number p and a finite group G with p||G|,we say that a kG-module V is a p-divisible kG-module if the dimension of any indecomposable direct summand of V is divisible by p.

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The terminology of the p-divisible kG-module is introduced to be an absolutely p-divisible kG-module(see[1]);con fined to the algebraic closed field k,any indecomposable kG-module is already absolutely indecomposable therein,so the p-divisible kG-module is also the absolutely p-divisible kG-module therein.

Remark 2.1 The class of p-divisible kG-modules is a big one,any(relative)projective kG-module is p-divisible(see[7],Exer.23.1),but the trivial kG-module k is not p-divisible.The direct summand of a p-divisible kG-module,the direct sum of two p-divisible kG-modules,the tensor product of a p-divisible kG-module and a kG-module,remain to be p-divisible(see[1],Proposition 2.2).Sometimes we denote a p-divisible kG-module with p-divisible for short.

Definition 2.1 Let V be a kG-module.If the endomorphism module of V can be regarded as the direct sum of the trivial module k and a p-divisible kG-module,that is,

where U is a p-divisible kG-module,then we say that V is a p-endotrivial kG-module,or V is p-endotrivial.

Remark 2.2 (1)Here,for the endomorphism module Endk(V),g·f:=gfg−1,g∈ G,f∈Endk(V),and for any(indecomposable)p-endotrivial kG-module V,k|EndkG(V),the trace map(Tr:Endk(V)→k)is a split surjection,so the tensor product of V with the almost-split sequence for the trivial kG-module k must fail to split(see[5]).

(2)The p-endotrivial kG-module extends the(relative)endo-trivial kG-module(see[2]and[4]).The trivial kG-module k is the most simple p-endotrivial kG-module,but any p-divisible kG-module cannot be p-endotrivial.

Proposition 2.1 Let V be a p-endotrivial kG-module and W be a p-divisible kG-module.Then V⊕kW is a p-endotrivial kG-module.Moreover,under kG-module isomorphism,V is the direct sum of p-divisible kG-modules and its uniquely indecomposable direct summand U which is p-endotrivial,and any vertex of U is a Sylow p-subgroup of G.

Proof.Since the canonical kG-module isomorphism

Remark 2.3 Up to isomorphism,for each[V]of Tp(G),there is a unique indecomposable p-endotrivial kG-module cap(V)in[V].

On the contrary,let U be an indecomposable non-p-divisible direct summand of V.Firstly,if the vertex P of U is a proper p-subgroup of G,then p|dimk(U)(see[7],Exer.23.1),contradiction.So the vertices of U are Sylow p-subgroups of G.

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Secondly,

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hence

and Tp(G)Tp(H).

Finally,if there exists another X which is also an indecomposable non-p-divisible direct summand of V and(U⊕kX)|V,then(k⊕kk)|Endk(V),contradiction.Hence,under kG-module isomorphism,V has a unique indecomposable direct summand which is p-endotrivial.

Proposition 2.2 shows that the p-endotrivial kG-module has the “similar” cap phenomenon as that of the endo-permutation module(see[7]and[8]),here we call U in Proposition 2.1 the cap of V.Two p-endotrivial kG-modules S and V are said to be equivalent if their caps are isomorphic,exactly,they are equivalent whenever S∗⊗kV k⊕k(p-divisible)(see[5],Proposition 4.1).We write[V]for the equivalence class of a p-endotrivial module V,and Tp(G)for the non-empty set of equivalence classes of p-endotrivial kG-modules.

and the tensor product with W is p-divisible(see[1],Proposition 2.2),we see thatEndk(W)⊕k Homk(V,W)⊕kHomk(W,V)is p-divisible,hence,V ⊕kW is p-endotrivial.

Definition 2.2 For any[U],[V]∈ Tp(G),let[U]+[V]=[U⊗kV].Then the tensor product⊗kfor kG-modules induces an abelian group structure on Tp(G),we call(Tp(G),+)the group of p-endotrivial kG-modules.

Indeed,the above is well-defined.Let M,N,V be p-endotrivial kG-modules.If[M]=[N],then

持续长度是该模型的一个基本参数.持续长度与自由旋转链的处理方法有异曲同工之妙,因为二者都采用了后键依次在前一键上投影并依此前推至所需键的计算方法.自由旋转链因为需要所有键之间进行两两点乘,因此其平均范围广,需要大量样本的统计平均方可采用投影法进行平均处理.持续长度涉及的是各个键在第一个键方向上投影,一旦键的方向与第一个键相反,它在z轴方向的投影就是负值.经平均处理后,负值就不再出现,代之以收敛于0的投影值.因此,持续长度的计算同样应该是大量连接单元的统计平均值.

Hence

and

The well-defined abelian group Tp(G)is an extension of the group T(G)of(relative)endotrivial kG-modules(see[4],[8]and[9]).

Proposition 2.2 Let[U]and[V]belong to Tp(G),m,n∈Z.Then we have the following conclusions:

Proof.(1)Since

and

we see that

where for each g with 1g∈[HG/H],(gU)is p-divisible since one can check that p||H:H∩gH|.So(I(U))is p-endotrivial by Proposition 2.1,and then (U)is also p-endotrivial by Lemma 2.1.

that is,

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(2)Let U=cap(U) ⊕kU1,where U1is p-divisible.On the one hand,since Ωn(U1)is also p-divisible,we have

On the other hand,Ωn(cap(U))is indecomposable by Proposition 11.7.1(5)of[6],and from(1)we see that Ωn(cap(U))is p-endotrivial.Hence

and

(3)By(2),we see that

(4)The first part is from the fact

For the second part,since Ωn(k)is also p-endotrivial,we have

by Proposition 11.7.2 of[6].

(5)We define the map[Ωn]as follows:

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By(1)and(3)one can check that[Ωn]is well-defined.Moreover,by(4)we see that[Ωn]is also an invertible map with the inverse map[Ω−n].

The proof is completed.

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Proposition 2.3 Let 0→U→W→V→0 be a short exact sequence of kG-modules,where W is projective.Then[U]∈ Tp(G)if and only if[V]∈ Tp(G),and in this case,

Proof.Since

we see that[U]∈ Tp(G)if and only if[V]∈ Tp(G),and[U]=[Ω(V)],[V]=[Ω−1(U)].

Lemma 2.1 Let G≥H and V be a kG-module.Then we have the following conclusions:

(1)If is p-endotrivial,then V is p-endotrivial;

(2) If H contains a Sylow p-subgroup of G and V is p-endotrivial,thenis p-endotrivial.

Proof.(1)We know that p does not divide dimk((V))and so it does not divide dimk(V),hence k is a direct summand of Endk(V)by Corollary 4.7 of[5].

Let Endk(V)=k⊕kX for some kG-module X.Then

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We see that(X)is a p-divisible kH-module(Krull-Schmidt Theorem),and X is a p-divisible kG-module,it means that V is p-endotrivial.

(2)Let Endk(V)=k⊕kX for some p-divisible kG-module X.Then we have

we prove by apagoge that (X)is p-divisible,so (V)is a p-endotrivial kH-module.Indeed,if(X)is not p-divisible,then k|Endk((X))by Corollary 4.7 of[5],henceAt the same time,

by Frobenius Reciprocity,and(k)⊗kEndk(X)is also p-divisible,so (k)is pdivisible,it contradicts with the order of (k).

Let P be a Sylow p-subgroup of G and V be a kG-module.If (V)k⊕kU,where U is a p-divisible kP-module,then we say that V is a p-critical kG-module.Obviously,any kG-module with trivial Sylow restriction and any critical kG-module are p-critical(see[10]and[11]).

Let P be a Sylow p-subgroup of G,and G≥H≥P.Lemma 2.1 shows that the restrictions of modules induce a map from Tp(G)to Tp(H),denoted by,as follows:

We can see that

It means thatis a group isomorphism provided by Green correspondence,and

Proposition 2.4 Let P be a Sylow p-subgroup of G,G≥H≥P.Thenis a group homomorphism,and ≤K(G).Particularly,if G≥H≥NG(P),thenis an injective group homomorphism.

Proof.We see that

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That is,is a group homomorphism.

If V∈then

and as the proof of Lemma 2.6(2),

That is,V∈K(G).

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Particularly,if G≥H≥NG(P)and[V]∈Tp(G),by Lemma 2.1,we set (cap(V))=U⊕k(p-divisible),where U is the cap of(cap(V)).By Proposition 2.1,cap(V)and U have a common vertex P.It means that U must be the Green correspondent f(cap(V))of cap(V)by Burry-Carlson-Puig Theorem(see Theorems 11.6.4 and 11.6.9 of[6]),but the map f in Green correspondence is a bijection from the isomorphism classes of indecomposable kG-modules with a vertex P to that of indecomposable kH-modules with the same vertex P,so it also means thatis an injective group homomorphism.

We recall that a group H is a strongly p-embedded subgroup of G if p divides|H|but does not divide|H∩xH|for each x∈G−H.Note here that any strongly p-embedded subgroup H of G contains the normalizer in G of a Sylow p-subgroup of G and such H always exists whenever some Sylow p-subgroup of G is a trivial intersection(T.I.set).

Theorem 2.1 Let G≥H≥P,where P is a Sylow p-subgroup of G.If H≥NG(P),in particular,H is strongly p-embedded in G,thenis a group isomorphism provided by Green correspondence,that is,

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that is,U is p-endotrivial.

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Proof.Firstly,by Proposition 2.4,we see thathere is an injective group homomorphism,that is,for any[V]∈Tp(G),

where U=cap((V))and U is the Green correspondent of V.

Secondly,for any indecomposable p-endotrivial kH-module U,(U)is p-endotrivial.Indeed,

It means that

Thirdly,for any indecomposable p-endotrivial kH-module U,the Green correspondent of U is cap((U)),and then it is p-endotrivial.Indeed,the Green correspondent of U is the unique indecomposable direct summand of (U)with a vertex P,but at the same time cap((U))has a vertex P by Proposition 2.1,hence,it is cap(I(U)).

Finally,on the one hand,for any p-endotrivial kH-module U,from the above discussion we have

We denote with K(G).Hence,K(G)consists of all equivalence classes of pcritical kG-modules.The kernel of the restriction map on T(G)is important for the structure of T(G)(see[9]and[11]).

On the other hand,for any p-endotrivial kG-module V,we have

and then

Hence,we see that

by Proposition 2.1 and Lemma 2.1.It means that

Summing up the above,we conclude thatis a group isomorphism provided by Green correspondence between indecomposable kG-modules and indecomposable kH-modules with the same vertex P,and Tp(G)Tp(H).The proof is completed.

Theorem 2.2 Let G≥H≥P,where P is a Sylow p-subgroup of G.Then

restricted on K(G)as follows

is a group homomorphism.Furthermore,if G≥H≥NG(P),in particular,H is strongly p-embedded in G,then this map is a group isomorphism provided by Green correspondence,that is,

and K(G)K(H).

Proof.Let[V]∈K(G).Then

and

that is,(V)∈K(H).Hence,the above mapfrom K(G)to K(H)is welldefined,and similarly to the proof of Theorem 2.1,one can check that this map remains to be surjective.So all results of this theorem follow from Proposition 2.4 and Theorem 2.1.

3 The Generalized Dade Group Dp(G)

Proposition 3.1 Let V be an endo-p-permutation kG-module.If V is p-endotrivial,then V has a unique indecomposable direct summand whose vertex is a Sylow p-subgroup of G,and moreover,Endk(V)k⊕kX,where each indecomposable direct summand of X is a trivial source kG-module whose vertex is a proper p-subgroup of G.

Proof. Let V=V1⊕kY,where V1is the cap of V,and Y is p-divisible.Then any vertex of V1is a Sylow p-subgroup of G by Proposition 2.1.

The vertex of any indecomposable direct summand W of Y is a proper p-subgroup of G.Indeed,W is both a p-divisible kG-module and an endo-p-permutation kG-module(see[8]).If the Sylow p-subgroup P of G is a vertex of W,then(W)is a p-divisible kP-module by the proof of Lemma 2.1,and is a capped endo-permutation kP-module(see Corollary 28.9 of[7]).It means that the source module M of W is a p-divisible kP-module,and is an indecomposable capped endo-permutation kP-module.Hence,dimk(M)≡ ±1(mod p)by Corollary 28.11 of[7],that is,p does not divide dimk(W),contradiction.So the unique indecomposable p-endotrivial direct summand of V,that is,the cap of V is the uniquely indecomposable direct summand whose vertex is a Sylow p-subgroup of G.

Let Endk(V)=k⊕kX,where X is p-divisible.Then any indecomposable direct summand N of X is a p-divisible kG-module,and is a p-permutation kG-module(see Proposition 0.2 of[12]).Similarly to the above,we see that any vertex of N is a proper p-subgroup of G,too.That is,each indecomposable direct summand of X is a trivial source kG-module and its vertices are proper p-subgroups of G.

Let P be a finite p-group.Then any two of indecomposable direct summands with a vertex P of the capped endo-permutation kP-modules are necessarily isomorphic(see Corollary 28.9 of[7]),and we obtain an equivalence relation for capped endo-permutation kP-modules,this equivalence relation correspondents to the isomorphism classes of their indecomposable direct summands with a vertex P,and the equivalence classes endowed with the multiplication induced from the tensor product,form a group,written D(P),D(P)is the so-called Dade group for the finite p-group P(see Remark 29.6 of[7]).

From Proposition 3.1 we see that there exists a subclass of endo-p-permutation kG-modules,that is,the class of kG-modules which are both an endo-p-permutation kG-module and a p-endotrivial kG-module,for any such kG-module V in this subclass,similarly to the capped endo-permutation kP-module,there exists a uniquely indecomposable direct summand with a Sylow p-subgroup as its vertex,that is,cap(V)of V.Moreover,by borrowing the equivalence relation for p-endotrivial kG-modules,that is,U∼V if both U and V are endo-p-permutation kG-modules and[U]=[V]in Tp(G),we obtain a similar equivalence relation for the above subclass,and write also[V]for the equivalence class of an endo-p-permutation kG-module V,and denote all such equivalence classes with Dp(G).We conclude with the following definition.

Definition 3.1 Let U and V be endo-p-permutation kG-modules.If they are p-endotrivial endowed with the operator as follows:

then Dp(G)becomes an additive group(Dp(G),+)and we say it the generalized Dade group for G.

Remark 3.1 Indeed,if G is a finite p-group P,then we can see from the below that Dp(G)is the ordinary Dade group D(P)for P(see Theorem 3.1).

Furthermore,we write X(G)for the abelian group of all isomorphism classes of onedimensional kG-modules,where the group law is induced by⊗k,too.X(G)can also be identified with the group of linear characters of G,and is isomorphic to the p′-part of G/[G,G],and hence is a p′-group.

Theorem 3.1 There are canonical injective group homomorphisms so that

are embeddings of groups,in particular,

where P is a finite p-group.Proof. If χ is a one-dimension kG-module,the canonical trace map χ∗⊗kχ → k must be split,and then χ∗⊗k χ k,that is,χ is endo-trivial.Hence,X(G)⊆ T(G),which means that X(G)can be regarded as a subclass of T(G),under the similar equivalence relation and the canonical injection.

Any indecomposable endo-trivial kG-module U is p-endotrivial,and at the same time it is an endo-p-permutation kG-module such that(U)is a capped endo-permutation kS-module,where S is a Sylow p-subgroup of G.Indeed,U is p-endotrivial,and (U)is endo-trivial.We set Endk((U))=k⊕kY,where Y is a projective kS-module,and then Y is 1-projective(see Proposition 11.6.2 of[6]).Since the indecomposable k-module is just k itself,we see that any indecomposable direct summand of Y has the trivial source,so Y is a permutation kS-module,and (U)is an endo-permutation kS-module,and then U is an endo-p-permutation kG-module.Since S is a vertex of U by Proposition 2.1,there exists an indecomposable direct summand with a vertex S,such as the source module of U,it means that(U)is capped.

Let[U1]and[U2]belong to T(G).Any indecomposable endo-trivial kG-module is both a p-endotrivial kG-module and an endo-p-permutation kG-module;moreover,if[U1]=[U2]in T(G),then the unique indecomposable p-endotrivial direct summand of U1and that of U2are isomorphic,and then[U1]=[U2]in Dp(G).So we can view T(G)as a subclass of Dp(G),T(G)⊆Dp(G)under the canonical injection.

From Proposition 3.1 and the definition of Dp(G)we see that Dp(G)⊆Tp(G),by the similarly canonical injection as the above.

We have X(G)⊆T(G)⊆Dp(G)⊆Tp(G),and at the same time,their“+”are the same way,so the above canonical injections are canonical embeddings,that is,

Now,we prove that

Indeed,on the one hand,by Proposition 3.1,any end-permutation kP-module V which is also p-endotrivial must be a capped end-permutation kP-module,it means that Dp(P)⊆D(P).On the other hand,if V is an indecomposable capped endo-permutation kP-module,we see that p does not divide dimk(V)by Corollary 28.11 of[7],hence,Endk(V)=k⊕kV1(see Corollary 4.7 of[5]),where k has multiplicity 1 as the direct summand of Endk(V)and V1is a trivial source kP-module.Since P is a finite p-group,any indecomposable direct summand of V1is of the form(k)by Green indecomposability theorem(see Corollary 23.6 of[7]),and since k has multiplicity 1,R must be the proper p-subgroup of P.Hence,(k)is p-divisible and V is p-endotrivial,it means that D(P)⊆Dp(P).

Following the above discussion,we see that

We are done.

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