A Tour of p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p$\end{document}-Permutation Modules and Related Classes of Modules

This survey provides an overview of numerous results on p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p$\end{document}-permutation modules and the closely related classes of endo-trivial, endo-permutation and endo-p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p$\end{document}-permutation modules. These classes of modules play an important role in the representation theory of finite groups. For example, they are important building blocks used to understand and parametrise several kinds of categorical equivalences between blocks of finite group algebras. For this reason, there has been, since the late 1990’s, much interest in classifying such modules. The aim of this manuscript is to review classical results as well as all the major recent advances in the area. The first part of this survey serves as an introduction to the topic for non-experts in modular representation theory of finite groups, outlining proof ideas of the most important results at the foundations of the theory. Simultaneously, the connections between the aforementioned classes of modules are emphasised. In this respect, results, which are dispersed in the literature, are brought together, and emphasis is put on common properties and the role played by the p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p$\end{document}-permutation modules throughout the theory. Finally, in the last part of the manuscript, lifting results from positive characteristic to characteristic zero are collected and their proofs sketched.


Introduction
The class of p-permutation modules is omnipresent in the modular representation theory of finite groups. To begin with, going back to the origins of representation theory, the theory of linear representations of finite groups investigates the structural connections between groups and automorphism groups of vector spaces. One of its most basic instance comes from the consideration of a group action on a finite set X, which is extended linearly to give a permutation representation on X. Elementary examples are given by the natural representation of the symmetric group S n or by the regular representation. Looking at representations of finite groups as modules over the group algebra, then a permutation representation corresponds to a permutation module, and, provided the base field has positive characteristic p, the indecomposable summands of the permutation modules are called the p-permutation modules.
This class of modules plays an important role in modular representation of finite groups and in the related block theory. For example, p-permutation modules are used to understand and parametrise several kinds of categorical equivalences between p-blocks of finite groups such as splendid Morita equivalences, basic equivalences, splendid Rickard equivalences, p-permutation equivalences, or the recently introduced functorial equivalences. They are also central objects to understand in the context of Alperin's weight conjecture.
The class of p-permutation modules was studied by Conlon [31] and Scott [71]. A further fruitful approach through invariant bases, the Brauer morphism and Galgebras is due to Puig; it appears in [15] by Broué. Thévenaz [74, §27] also provides a detailed introduction to this theory in the language of G-algebras, and Linckelmann [59,60] has the most up-to-date collection of results on the topic with detailed proofs mixing approaches.
There are several classes of modules which are closely related and which have intensively been studied over the past 5 decades. To start with, an endo-permutation module over a p-group is one whose endomorphism ring over the base field is a permutation module. This notion was introduced in 1978 by Dade in [34,35] and studied by many authors until their classification was completed, a quarter of a century later, by Bouc in [12]. However, the final results are due to the combined efforts [1,10,11,13,[28][29][30] of several authors in different combinations including Alperin, Bouc, Carlson, Dade, Mazza and Thévenaz. Crucial to this classification was the understanding of the class of endo-trivial modules, which are those (endo-permutation) modules, whose endomorphism ring is invertible in the stable module category of the group algebra. From 2006 on, once the classification of the endo-permutation had been completed, on the one hand, the focus was put on understanding and classifying endo-trivial modules over arbitrary finite groups. A complete classification has not been achieved yet, but many classes of finite groups could be treated, by many different authors (including Bessenrodt, Carlson, Grodal, Hemmer, Koshitani, Lübeck, Malle, Mazza, Nakano, Robinson, Thévenaz, and the author), using a variety of different methods, going from rank varieties to homotopy theory, passing through character theory, on top of standard module theory. On the other hand, endo-permutation modules can also be generalised to arbitrary finite groups, however, the concept of an endo-p-permutation module, i.e. one whose endomorphism ring is a p-permutation module, introduced by Urfer in [76,77] and developed by the author in [50,51] turns out to be more relevant. All these classes of modules are also important in block theory, as the p-permutation modules are.
Thévenaz has written a survey on endo-permutation modules [75] and there are three very good surveys describing the developments towards a classification of endotrivial modules since 2006: two brief surveys by Carlson [18,19] and a book by Mazza [65]. Our aim in this survey is therefore not to provide a detailed treatment, but an introduction to all of these concepts for non-experts, emphasising their common properties and the way they interact. Moreover, we outline the proofs of the results presented, whenever it is possible to obtain them using elementary arguments or the theory of vertices and sources, which may be thought of as the art of juggling with induction and restriction.
In Sect. 3 and Sect. 4 we give a introduction to permutation modules and ppermutation modules, outlining proof ideas of the most important results. Then, in Sect. 5 we review endo-permutation modules, endo-trivial modules and endo-ppermutation modules, also introducing a relative version of endo-trivial modules which has the advantage of encompassing all the latter classes of modules, through a common concept. Finally, in Sect. 7, we focus on relatively recent lifting results from positive characteristic to characteristic zero, which are less well-known. However, they are of great significance, as they allow for the use of ordinary character theory.

General Notation and Conventions
Unless otherwise stated, throughout this manuscript we adopt the following notation and conventions. All groups considered are assumed to be finite, and all modules over group algebras are assumed to be finitely generated left modules. We let p denote a positive prime number, G a finite group of order divisible by p, and P a finite p-group. We denote by Syl p (G) the set of all Sylow p-subgroups of G. We let O be a complete discrete valuation ring with field of fractions K and residue field k := O/J (O) of positive characteristic p, where J (O) =: p is the unique maximal ideal of O. We assume that k is a splitting field for G and its subgroups. At first we allow the case O = k. From Sect. 4 on, for simplicity, we will assume that the triple (K, O, k) is a splitting p-modular system for G and its subgroups. In order to state definitions or results, for which there is no difference between O and k, we let R ∈ {O, k}.
If H is a subgroup of G, then we write H ≤ G, and if x ∈ G then we set is the inflation of V from G/N to G, i.e. the action of g ∈ G on Inf G G/N (V ) is defined to be the action of the left coset gN on V . If M is an RG-module, then the first syzygy of M is by definition the kernel of a projective cover of M and is denoted by (M). If P is a p-group and Q ≤ P , then we denote by P /Q the relative syzygy operator with respect to Q, which by definition returns the kernel of a Q-projective relative cover of the given module. We refer to [17,72] for details on this notion and to §2.4 for Q-projectivity. In particular P /Q (R) is the kernel of a relative Q-projective cover of the trivial module R and with this notation = P /{1} . Finally, we recall that an RG-lattice is an RG-module which is free as an Rmodule. When R = O, we denote by OG-lat the category of all OG-lattices of finite R-rank, and when R = k, then we denote by kG-mod the category of finitedimensional kG-modules. Further standard notation, used in this manuscript but not introduced here, is as in [59,60,74,79].

Reduction Modulo p and Liftable Modules
Reduction modulo p (sometimes simply called reduction modulo p) is the functor k ⊗ O − : OG-lat −→ kG-mod mapping an OG-lattice L to k ⊗ O L ∼ = L/pL and with standard action on morphisms via the universal property of the tensor product. If L 1 , L 2 are OG-lattices and ϕ ∈ Hom OG (L 1 , L 2 ) is a morphism, then the reduction modulo p of ϕ is ϕ : L 1 /pL 1 −→ L 2 /pL 2 , x + pL 1 → ϕ(x) + pL 2 .
Notice that two non-isomorphic OG-lattices L 1 and L 2 may have isomorphic reductions modulo p. For example, given a p-group P , any two non-isomorphic OPlattices L 1 and L 2 of O-rank one are such that L 1 /pL 1 ∼ = k ∼ = L 2 /pL 2 . Clearly, reduction modulo p is always possible. In contrast, it is not always possible to go the other way around. Thus, a kG-module M is said to be liftable (to O or to OG) if there exists an OG-lattice L such that M ∼ = L/pL.

The Brauer Quotient
Let M be an OG-lattice. Given a subgroup H ≤ G denote by M H the set of H -fixed points of M. Then, given subgroups S, Q ≤ G such that S ≤ Q ≤ G, the relative trace map t Q S is defined to be the map

Vertices, Sources, Green Correspondence
Standard references for detailed expositions of Green's theory of vertices and sources are for example [33] or [59], and we recommend [79,Chap. 11] to anyone starting to learn about modular representation theory.
Given a subgroup H ≤ G, an RG-lattice M is said to be relatively H -projective A vertex of an indecomposable RG-lattice M is a subgroup Q ≤ G which is minimal subject to the property that M is relatively Q-projective. The set of all vertices of M is denoted by vtx(M). Given Q ∈ vtx(M), an RQ-source (or simply a source) of M is an RQ-lattice T such that M | Ind G Q (T ). Essential properties of vertices and sources, to have in mind in order to understand this text, are the following.
Properties (a) to (e) are standard and to be found with a proof in any textbook on modular representation theory, although the Green correspondence requires some work to be proved. Property (f) requires Higman's criterion and we refer to [59, Theorem 5.6.9].
Finally, we note that in this context, the central objects of study in this survey, the p-permutation RG-modules, are described by the following definition. However, we will first treat them through the more intuitive approach of permutation bases.

Permutation Modules
We start with a short review of permutation RG-modules. In this view, we begin with the basic definition of a permutation representation together with fundamental examples, at the foundations of representation theory of finite groups.

Permutation Representations
Given a finite group G and a finite G-set X, that is, a finite set X endowed with a left action G × X −→ X, (g, x) → g · x, we may construct the free R-module RX with basis X. More explicitly, extending the given action R-linearly yields an Rrepresentation of G, called the permutation representation of G on X.
Two fundamental examples are the following.
(1) If G = S n (n ∈ Z ≥1 ) is the symmetric group on n letters and X = {1, 2, . . . , n}, then ρ X is the natural representation of S n . (2) If X = G and the left action of G on X is just multiplication in G, then ρ X is the regular representation of G.

Permutation Modules
The RG-module RX corresponding to the R-representation ρ X of Sect. 3.1 is called the permutation RG-module on X. This leads to the following general definition.
Definition 3.1 An arbitrary RG-module is called a permutation RG-module if it admits an R-basis X which is invariant under the action of the group G.
It is clear that the basis X is then a finite G-set. Also, a permutation RG-lattice is R-free by definition, hence an RG-lattice.

Example 3.2 (a) An induced module of the form Ind
is clearly a permutation RGmodule on G; see also Example (2) in Sect. 3.1.
The following elementary observation shows that an arbitrary permutation RGmodule is isomorphic to a direct sum of induced modules of the form Ind G H (R) for various subgroups H ≤ G.

Observation 3.3
If RX is a permutation RG-module on X, then a decomposition of X as a disjoint union of G-orbits, say X = n i=1 X i (n ∈ Z ≥0 ), yields a decomposition of RX as a direct sum of RG-submodule as where each RX i is called a transitive permutation module. Furthermore, for each 1 ≤ i ≤ n, Similarly for inflation, just take the same invariant R-basis.
(b) The claim about conjugation can be proved as in Example 3.2. If M 1 and M 2 are permutation RG-modules on X 1 and X 2 respectively, then M 1 ⊕ M 2 is a permutation module on X 1 X 2 and M 1 ⊗ R M 2 is a permutation module on If M is a permutation RG-module on X, then X * is a permutation module on the dual basis to X. Lemma 3.6 If P is a p-group, then for any subgroup Q ≤ P , the permutation module Proof To prove the indecomposability of Ind P Q (R), we may assume that Then, it suffices to prove that the socle of Ind P Q (k) is indecomposable. Now, as P is a p-group and k is a splitting field for G, up to isomorphism, the only simple kP -module is the trivial module. Hence soc(Ind P Q (k)) is a direct sum of trivial submodules. This together with Frobenius reciprocity yields dim k soc(Ind P Q (k)) = dim k Hom kP (k, Ind P Q (k)) = dim k Hom kQ (Res P Q (k), k) = dim k Hom kQ (k, k) = dim k k = 1 , which forces soc(Ind P Q (k)) to be indecomposable. Next, as two induced modules Ind P Q 1 (R) and Ind P Q 2 (R) are isomorphic if and only if the subgroups Q 1 and Q 2 of P are conjugate, clearly Q ∈ vtx(Ind P Q (R)) and R is a source by definition.
As a result, we have parametrised the indecomposable permutation modules over p-groups.

Corollary 3.7
If P is a p-group, then the isomorphism classes of indecomposable permutation RP -modules are parametrised by the P -conjugacy classes of subgroups of P .

Definition and Characterisations
Coming back to Observation 3.3, we emphasise that the transitive permutation modules need not be indecomposable in general, although it is the case for p-groups. For instance, it is well-known that the PIMs of kG are the indecomposable summands of the group algebra kG ∼ = Ind G {1} (k) and each of them occurs with multiplicity equal to the dimension of its simple socle. Therefore, in general, it is necessary to investigate the indecomposable direct summands of the (transitive) permutation RG-modules. The following lemma is crucial to understand these summands.
is also a permutation RP -module for any M | L.
Proof By Observation 3.3 there exist n ∈ Z ≥0 and subgroups where each Ind P Q i (R) is indecomposable by Lemma 3.6. Thus, by the Krull-Schmidt theorem, if M | L, then Res G P (M) is isomorphic to the direct sum of some of the summands in the decomposition, hence is again a permutation RP -module.
This leads us to the following equivalent characterisations of the direct summands of the permutation RG-modules.
Proposition-Definition 4.2 (Characterisations of p-permutation modules) Let M be an RG-module and let P ∈ Syl p (G). Then, the following conditions are equivalent: If M fulfils any of these equivalent conditions, then it is called a p-permutation RGmodule.
At this point, we note that p-permutation RG-modules and trivial source RGmodules are essentially two different pieces of terminology for the same concept. Some authors tend to favour the use of the terminology p-permutation module to emphasise the existence of a P -invariant basis and reserve the terminology trivial source module for an indecomposable module with a trivial source (as introduced above). Other authors tend to favour the use of the terminology trivial source module to mean a direct sum of RG-modules with trivial sources, that is, our definition of a p-permutation module.
Proof (a)⇔(b): It is obvious that (a) implies (b). For the sufficient condition, notice that Res G g P (M) ∼ = g (Res G P (M)) for each g ∈ G, and recall that any p-subgroup is contained in a conjugate of P by the Sylow theorems. Thus, as by Properties 3.5 restriction and conjugation preserve permutation modules, requiring that Res G P (M) is Inducing this module to G, we deduce that M, being indecomposable, is isomorphic to a direct summand of Ind G R i (R) for some 1 ≤ i ≤ n. By the minimality of vertices it follows that R i = Q and that the trivial RQ-module R must be a source of M.
Furthermore, we claim that vtx(R) = Syl p (G). Indeed, if Q ∈ vtx(R) and P ∈ Syl p (G) is such that Q ≤ P , then R | Ind G Q (R) and so the Mackey formula yields where all summands Ind P P ∩ x Q (R) are indecomposable by Lemma 3.6.
(c) Lemma 3.6 shows that if G is a p-group, then any p-permutation module is in fact a permutation module. Hence, the concept of a p-permutation is reduced to the concept of a permutation module, and hence the former is not need for p-groups.

Green Correspondence for p-Permutation Modules
The Green correspondence provides us with a theoretical classification of all indecomposable p-permutation modules, vertex by vertex.  We emphasise that the characterisation of the indecomposable p-permutation RGmodules obtained via the Green correspondence is theoretically very powerful, however, does not provide us with a concrete description of such modules, because the first bijection in Assertion (c) above is not constructive. In general, the question of describing the structure of the indecomposable p-permutation RG-modules remains a difficult question, even for small groups, or modules belonging to blocks with small defect groups. One of the main issues being that p-permutation modules are determined by the source-algebra equivalence class of the block, but not by its Morita equivalence class. As a matter of fact, the question is rather complex already for cyclic blocks. A complete solution in this case can be found in [39].
Finally, we mention the following result of Okuyama's showing that simple modules with a trivial source have a simple Green correspondent.

Weight Modules and Alperin's Weight Conjecture
If Q ≤ G is a p-subgroup and S is a simple kN G (Q)-module with vertex Q, then the pair (Q, S) is called a weight of G with respect to Q. In this case, Q is a weight subgroup, S is a weight kG-module and the kG-Green correspondent g(S) of S is called a weight Green correspondent.
It is easy to see that any weight subgroup Q is p-radical subgroup, i.e. Q = O p (N G (Q)) is the largest normal p-subgroup of N G (Q). Also conjugation induces an equivalence relation on the set of all weights of G. Alperin originally stated his Weight Conjecture in [2] as follows. Weight modules are strongly related to p-permutation modules, as we see through the following observations.  Proof By the Burry-Carlson-Puig theorem, it is enough to prove that S | Res G N G (Q) (Ind G P (k)), which follows from the Mackey formula.
It follows that understanding the direct summands of a Sylow permutation module Ind G P (k) gives some control on the number of weights. For instance, this strategy was used by Cabanes in [16] to prove Alperin's Weight Conjecture for finite groups of Lie type in their defining characteristic. More precisely, he proves that in this case the image under the Brauer morphism of each indecomposable direct summand X of Ind G P (k) with vertex Q, i.e. the Green correspondent f (X) of X, is projective simple as a k[N G (Q)/Q]-module and hence a weight module.
In fact, the more general context in which the endomorphism algebra End kG (Ind G P (k)) is quasi-Frobenius was further examined by Naehrig in [67]. One of the main ideas is that under this assumption, where the last equality holds by [67, Theorem 3.1(d)] due to Green. This implies that in this special case Alperin's Weight Conjecture is true if and only if each indecomposable direct summand of Ind G P (k) is a weight Green correspondent.

Approach via the Brauer Quotient
A further fruitful approach to p-permutation modules through the Brauer morphism is due to Puig and appears in [15] by Broué. As a matter of fact, the Brauer construction applied to p-permutation modules is particularly well-behaved and provides us with alternative characterisations of vertices, sources and Green correspondents in this case.
Then, it is easy to check that when proving that M(Q) = x∈X Q k · br M Q (x) as k-vector space. (b) The set X \ X Q consists of the Q-orbits of X which are not singletons. Thus, since Q is a p-group, we have |X| ≡ |X Q | (mod p) and the claim follows from (a).
(c) The claim is also immediate from (a).
(d) Let P ∈ Syl p (N G (Q)) (which necessarily contains Q) and let Y be a Pinvariant R-basis of M. Then X is Q-invariant, so (a) applies. Now, as P normalises Q, certainly X Q is P -invariant. Thus the k-basis br M Q (X Q ) given by (a) is invariant under the action of the Sylow p-subgroup P /Q of N G (Q)/Q. The claim follows.  When working over the residue field k the Green correspondence for ppermutation modules has the following nice characterisations in terms of the Brauer quotient.
such that the following properties hold: Next, assume that M is indecomposable. Then M has a trivial source by Proposition-Definition 4.2, and therefore so does its kN G (Q)-Green correspondent f (M). By the definition of the Green correspondence, where X is a direct sum of indecomposable kN G (Q)-modules having a vertex strictly contained in Q. Hence X(Q) = 0 by Proposition 4.12, implying that It follows that the Brauer quotients can be used in order to determine the isomorphism type of p-permutation kG-modules.

Proposition 4.15
Let M and N be projective-free p-permutation kG-modules. Then the following assertions are equivalent: Proof It is clear that if M ∼ = N , then their Brauer quotients with respect to any nontrivial p-subgroup Q ≤ G are isomorphic. In order to prove the sufficient condition, assume that (b) holds. Clearly, by the Krull-Schmidt theorem, we may assume that M and N have no isomorphic direct summands. Let Q ≤ G be a p-subgroup which is maximal subject to M(Q) = {0}. As M is projective-free Q = {1}. Now, the maximality of Q implies that M and N have no direct summands with a vertex strictly containing Q. Choosing a direct sum decomposition M ∼ = n i=1 M i ⊕ Y , where each M i has vertex Q and none of the indecomposable direct summands of Y has vertex Q, then by Theorem 4.14. Similarly for N . Hence, we conclude from (b) that M and N have an indecomposable direct summand in common with vertex Q, which is a contradiction.

Endo-Permutation, Endo-Trivial, Endo-p-Permutation Modules
In this section we define and review several classes of RG-modules, which are closely related to the class of p-permutation modules, because their R-endomorphism ring is by definition a p-permutation RG-module. These come in different flavours, depending on further restrictions put on the latter p-permutation module.
We will see in Sect. 6 that all the OG-modules defined in this section are in fact automatically OG-lattices, i.e. free as O-modules. For this reason, talking about lattices or about modules does not make a difference here. Moreover, throughout this section, if not said otherwise, P denotes a p-group and G denotes a finite group of order divisible by p. Moreover, in order to understand the definitions below, we recall that the endomorphism algebra End R (M) of an RG-module M is naturally endowed with the structure of an RG-module through the action of G by conjugation, that is,

Endo-Permutation Modules over p-Groups
Endo-permutation modules were first introduced and thoroughly studied by Dade in his celebrated two-part paper [34,35]. They play a crucial role in modular representation theory of finite groups. To give a few examples, they appear naturally as sources of simple modules for p-soluble groups (see [74, §30]). They appear in Puig's characterisation of the source-algebra of nilpotent blocks in [69], or in Linckelmann's classification of blocks with cyclic defect groups up to source-algebra equivalence (see [60,Chap. 11]). They also appear in the theory of basic Morita equivalences and associated Picard groups, currently under intensive investigation by several authors working towards classifications of blocks up to Morita equivalence and the verification of Donovan's conjecture for various classes of small defect groups (see e.g. [9,37] and the references therein).
is a permutation RP -module. Moreover, an endo-permutation RP -module is called capped if it has a direct summand with vertex P .
Dade proved in [35,Theorem 6.6.] that all endo-permutation modules can be described from the knowledge of the indecomposable capped endo-permutation RPmodules. The characterisation of p-permutation modules via the Brauer quotient we gave in Sect. 4.4 yields the following characterisation of capped endo-permutation modules, essential throughout the theory.

Lemma 5.2
Let M be an endo-permutation RP -module. The following assertions are equivalent: Stability properties are the following.

Properties 5.3 (a)
The class of endo-permutation RP -modules contains the permutation RP -modules. (b) The class of (capped) endo-permutation RP -modules is closed under taking direct summands, R-duals, and finite tensor products over R. (c) The restriction, the inflation, the conjugation and the tensor induction of a (capped) endo-permutation module is again a (capped) endo-permutation module. (d) For any n ∈ Z the relative syzygy module n P /Q (R) is an endo-permutation RPmodule.
Note that the induction of an endo-permutation module is not necessarily an endopermutation module. Instead, the tensor induction is the correct operation to be used in this context. A classification of the capped endo-permutation kP -modules was achieved through the work of several authors in a long series of articles, starting with Dade's initial two-part article [34,35] in 1978 and ending with the work of Bouc [12] in 2006, with crucial steps achieved by Bouc and Thévenaz in [14] and by Carlson and Thévenaz in [28][29][30]. At this stage, we emphasise that Thévenaz has written a very detailed survey [75] on the classification of endo-permutation modules and its chronological developments. For this reason we do not give proofs and simply refer the reader to [75] for further details.

Proof Assertion (a) is straightforward from Properties 3.5 as End
The initial idea that enabled this classification, introduced by Dade in [34,35], is the fact that the class of capped endo-permutation RP -modules subject to a certain equivalence relation can be endowed with the structure of an abelian group, known nowadays as the Dade Group of P . The class of capped endo-permutation RP -modules is not closed under direct sums, so the direct sum cannot be used as a group operation, but the tensor product over R can, as we describe below. In Sect. 5.2 we will summarise the main milestones of this classification, which comes down to determining the structure of the Dade group. However, before we can proceed, we need to introduce several important subgroups of the Dade group.

Proposition-Definition 5.4 ([34]) (a) If M is a capped endo-permutation
In fact, one of the starting points of the classification was the following theorem of Lluis Puig, who, in the 1980's, introduced the notion of a Dade P -algebra. The connection with endo-permutation modules is the following. To start with, an RP -module M is an endo-permutation RP -module if and only if End R (M) is a permutation P -algebra, i.e. a P -algebra admitting a P -invariant R-basis. A Dade P -algebra (over R) is defined to be an R-simple permutation P -algebra A such that A(P ) = 0. Thus Lemma 5.2 yields: if M is a capped endo-permutation RP -module, then End R (M) is a Dade P -algebra. Conversely, any Dade P -algebra gives rise to a capped endo-permutation RP -module, unique over R = k. A very clear exposition of these facts is to be found in [74, §28 to §30]. The approach via Dade P -algebras lead in particular to the following fundamental result on the structure of the Dade group. The subgroups D tors R (P ) and D R (P ) are essential building blocks for the determination of the structure of the Dade group. Yet, another important building block is given by the subgroup of endo-trivial RP -modules, also introduced by Dade in [34,35]. In fact this notion was introduced independently by Alperin in [1], who called them invertible module, as they are invertible in the stable module category.
Note that if such an isomorphism exists, then F is the kernel of the trace map Moreover, we will often simply write End R (M) ∼ = R ⊕ (proj) instead of specifying a projective module F . The class of endo-trivial modules has less stability properties than the class of endo-permutation modules. Clearly, the inflation of an endo-trivial RP -module is not an endo-trivial module, as the inflation of a projective RP -module is not a projective module in general. However, the following properties hold.
If M and N are endo-trivial RP -modules, then so are the following RPmodules: follows from elementary properties of syzygy modules: To obtain Assertion (c), observe that if we decompose M as M = M 0 ⊕ M 1 , then as RG-modules and the claim follows from the Krull-Schmidt theorem, as R is nonprojective and can then only occur once as a direct summand of End R (M 0 ) or of End R (M 1 ), but not of both. It is also clear from the definitions that any endo-trivial RP -module is an endopermutation module, and it is capped by the characterisation of the capped modules in Lemma 5.2.
These properties allow us to define a group structure on the class of endo-trivial RP -modules, which can be identified with a subgroup of the Dade group.

The zero element is the class [R] of the trivial RP -module and the opposite of a class [M] is the class
Notice that the equivalence classes in D R (P ) are larger than in T R (P ) and the class of an endo-trivial RP -module may contain modules that are not endo-trivial.

The Structure of the Dade Group of a p-Group
The determination of the structure of the Dade group and of the group of endo-trivial modules was essentially realised over R = k. Their structure over R = O can then be deduced from the lifting results from k to O considered in Sect. 7. We record below the main steps which lead to the final classification. To start with, the abelian case was already understood by Dade when he started the theory.

Theorem 5.10 ([34, 35]) If P is an abelian p-group, then T k (P ) is cyclic generated by the class [ (k)] and
where each T (P /Q) is identified with a subgroup of D k (P ) via inflation, r is the number of non-cyclic quotients P /Q, and c is the number of cyclic quotients of P /Q of order at least 3.
The structure of the Dade group in finite and tame representation type plays a role in the final classification. If P ∼ = C p n (n ≥ 1) is cyclic of order p n (in multiplicative notation), then the structure of D k (P ) is clear from Dade's theorem above, namely When p = 2, there is a missing generator coming from the fact that T k ( If P is a dihedral, a semi-dihedral, or a generalised quaternion 2-group, then the structure of T k (P ) was obtained by Carlson and Thévenaz in [28, §5- §7]. In the same article, they prove the following general result about the structure of D k (P ) in these cases. This led to the determination of D k (P ) in tame representation type, using an induction argument, for which the starting point is the fact that D k ( The classification of all endo-trivial kP -modules, or equivalently the determination of the structure of T k (P ) was the next main step. It is mainly due to Carlson and Thévenaz again in [29,30]. In [30] they obtain the structure of the torsion part T tors k (P ) of T k (P ).
Therefore, it remains to consider the case in which P possesses elementary abelian p-subgroups of rank 2.
Another main step was the determination of the structure of the torsion part of D k (P ) in odd characteristic, obtained earlier by Bouc and Thévenaz. Using the results of [29], Bouc and Mazza [13] determined the structure of the Dade group of (almost) extra-special p-groups. Meanwhile, Bouc obtained crucial results on the tensor induction of relative syzygy modules in [10], and in [11] he made connections between the dual Burnside ring and the Dade group explicit. He also developed the machinary of biset functors, which, applied to the Dade group, lead to the final classification.
where the second summand is

Endo-Trivial Modules over Arbitrary Finite Groups
Endo-trivial modules over p-groups served as building blocks for Bouc's description of the Dade group and the classification of endo-permutation modules. Directly after this classification was achieved people turned their attention to endo-trivial modules over finite groups. Indeed, it is clear that Definition 5.7 does not use the fact that the group is a p-group and makes sense for an arbitrary finite group.
There are many reasons for wanting to understand these modules. Equivalently, we could say that an endo-trivial RG-module is an RG-module whose R-endomorphism ring is isomorphic to the trivial module in the stable module category of RG. Moreover, tensoring over R with an endo-trivial RG-module is a self-stable equivalence of Morita type, implying that this class of modules can be identified with an important part of the Picard group of the stable module category of RG. It was also proved by Bleher and Chinburg [8] that they are the modules whose deformation rings are universal (as opposed to versal).
On top of Thévenaz' survey [75] on endo-permutation modules already mentioned, there are three very good surveys describing the developments towards a classification of endo-trivial modules since 2006: two brief surveys by Carlson [18,19] and a book by Mazza [65], to which we refer the reader for a detailed treatment of the subject. For this reason, below we are only briefly going to describe stability properties and similarities of this class of modules with the other classes we have studied that far.

Example 5.17 (a)
The trivial module R and its syzygies n (R) (n ∈ Z) are endotrivial RG-modules, by the same argument we gave in the proof of Properties 5.8(b)(iv). (b) Any RG-module Z such that rk R (Z) = 1 is endo-trivial, as Z * ⊗ R Z ∼ = R. We write X R (G) for the set of all isomorphism classes of rank one RG-modules. This is a group for the tensor product ⊗ R over R.

Properties 5.18
Let M and N be endo-trivial RG-modules. Proof Assertions (b) to (e) will be proved in a more general context in Properties 5.38. Assertion (a) follows from the fact that, by definition, rk R (End R (M)) = rk R (M) 2 ≡ 1 modulo the R-rank of projective RG-module, which itself has R-rank divisible by |G| p .
As for p-groups, these properties lead to a group structure, which allows us to classify the endo-trivial RG-modules in an easier way. As we will see in Sect. 7 any endo-trivial kG-module is liftable to an endo-trivial OG-lattice. This fact allows us to assume that R = k, without loosing essential information. In fact, all articles concerned with classifications of endo-trivial modules assume that R = k. We sum up here some of the main results in this direction.

Proposition-Definition 5.19 ([22]) Two endo-trivial
To begin with, the group of endo-trivial modules is also finitely generated, and the rank of its free part can be characterised in terms of the local structure of the group G. As a consequence, in general, the difficult problem is to understand the torsion part T tors k (G) of the group T k (G). In this respect, the following characterisation through restriction to a Sylow p-subgroup and its normaliser are essential. For a subgroup H ≤ G, we may always consider the group homomorphism and when P ∈ Syl p (G), then we let K(G) := ker(res G P ) be its kernel. Clearly, in this way, we obtain a chain of inclusions It turns out that in many situations T tors k (G) = X k (G), or T tors k (G) = K(G), but this is not the general case and although recent work of Grodal [38] using homotopy theory brought many answers towards the structure of T tors k (G) its structure is still an open question in general. (c) If P is neither cyclic, nor generalised quaternion, nor semi-dihedral, then

Remark 5.22
The theorem tells us that in most cases, determining the torsion part of T k (G) comes down to determining which Green correspondents of the onedimensional kN G (P )-modules are endo-trivial, or in other words, determining which indecomposable p-permutation kG-modules with full vertex are endo-trivial.
This is in general a very hard question, and has generated a lot of work, by many different authors. We collect below the most important classes of finite groups for which the structure of T k (G) has been fully determined.

Endo-p-Permutation Modules over Arbitrary Groups
Considering an arbitrary finite group G, as seen in the previous subsections, the notion of a permutation module, which is good over a p-group, must be replaced by the notion of a p-permutation module in order to obtain similar behaviours and stability properties. Similarly, the notion of an endo-permutation module has to be replaced by the notion of an endo-p-permutation module in order to obtain a group structure similar to that of the Dade group of a p-group. Such modules were introduced by Urfer in his doctoral thesis [76] (in French) as described below. We refer to Urfer's article [77] for a published version in English, unfortunately not as complete as his thesis. Urfer defines endo-p-permutation modules over k, but the part of his work we present below holds over O as well. To begin with, we see that all the classes of kG-modules we have studied so far are subclasses of the class of endo-p-permutation modules. (b) If P is a p-group, then an RP -module is an endo-p-permutation RP -module if and only if it is an endo-permutation RP -module, as we have already observed in Example 4.4(c) that any p-permutation RP -module is a permutation RP -module in this case. (c) Any endo-trivial RG-module is an endo-p-permutation RG-module, as any projective RG-module is a p-permutation RG-module.
Stability properties of the class of endo-p-permutation RG-modules are the following.

Properties 5.26 (a) The R-dual and any direct summand of an endo-p-permutation
RG-module is an endo-p-permutation RG-module. Notice that, in contrast, direct sums and standard induction do not preserve the class of endo-p-permutation modules.
Proof For the R-dual, direct summands, tensor products over R, and conjugation the claims follow immediately from the equivalent characterisation of endo-ppermutation RG-modules in Remark 5.24(a) and the stability properties of ppermutation RG-modules under these operations from Properties 4.3. For restriction, there is nothing to do, and for tensor induction, restrict to p-subgroups, use Mackey's formula and the facts that conjugation, restriction, tensor induction and tensor product preserve endo-permutation modules. This proves Assertions (a), (b), and (c). To prove Assertion (d) observe that as M and N are relatively Q-projective, so is any direct summand of M ⊗ R N . Moreover, as Q ∈ vtx(M) and Q ∈ vtx(N ), Res G Q (M) and Res G Q (N ) are capped endo-permutation RQ-modules, and therefore so is proving that a vertex of L contains a conjugate of Q, and hence Q ∈ vtx(L), as required.
The main question that follows is of course, whether the class of endo-ppermutation modules can be endowed with a good equivalence relation in order to define a group structure similar to that of the Dade group of a p-group. There are in the literature two attempts to define such a group structure, the first one by Urfer in his doctoral thesis [76,77] and the second one by the author in [51], which we describe in the next subsection.
Urfer's construction is based on the following equivalence relation, generalising Dade's original approach to the definition of the Dade group in [34,35], and called compatibility.  This result leads us naturally to considering G-stable points of the Dade group. Indeed, as the Dade group D R (−) is in fact a Mackey functor (over Z), we may consider its G-stable points. (See [73] for an introduction to Mackey functors.) In other words, if Q ≤ G is a p-subgroup, then an element

Remark 5.28 If H ≤ G, then the induction Ind G H (M) of an endo-p-permutation RH -module to G is, in general, not an endo-p-permutation RG-module. However, the compatibility relation yields the following criterion: Ind G H (M) is an endo-p-permutation RG-module if and only if the endo-p-permutation R[ x H ∩ H ]modules Res
for every x ∈ G, where c x is conjugation by x ∈ G. Then, D R (Q) G-st denotes the subgroup consisting of the G-stable elements of D R (Q).

Theorem 5.29 ([77, Theorem 1.5]) Let M be an indecomposable RG-module with vertex Q and RQ-source S. Then, M is an endo-p-permutation module if and only if S is an endo-permutation RQ-module whose class [S] in the Dade group D R (Q) belongs to D R (Q) G-st .
Proof Assume first that M is an endo-p-permutation module. Then, by definition Res G Q (M) is endo-permutation, and so is S as direct summand of the latter by A first consequence of this theorem is the fact that for indecomposable modules with a common vertex, compatibility is detected locally by the sources of the modules. With the notion of compatibility Urfer defined the following group structure, which has never been given a name to. The first observation to make is that this new group structure generalises the constructions of the Dade group of a p-group to arbitrary finite groups.

Remark 5.32 (a) If P is a p-group, then D P (P ) ∼ = D R (P ) via the map sending the equivalence class [M] of an indecomposable endo-permutation RP -module M to its class [M] in D R (P ). (b) As the proof above and Proposition 5.30 show, an equivalence class in D Q (G)
may contain several indecomposable endo-p-permutation modules, namely all indecomposable endo-p-permutation modules with a common RQ-source.

Proof It is easy to verify that the map D Q (G)−→D R (Q) G-st , [M] →[Cap(Res G Q M)] is an isomorphism of groups, with inverse given by the map D R (Q) G-st
Finally, we emphasise that the group D Q (G) in fact classifies the sources of the indecomposable endo-p-permutation RG-modules with vertex Q, but not such modules themselves. In the next subsection, we show how to overcome this problem.

Relative Endo-Trivial Modules and the Generalised Dade Group of a Finite Group
Replacing projectivity by relative projectivity with respect to subgroups, or more generally with respect to modules, we can define the relative endo-trivial modules.
A detailed treatment of relative projectivity with respect to modules can be found Carlson's lecture notes [17] and in the author's work in [49][50][51]. We present below only the essential notions that allow us to define relative endo-trivial modules. Although adaptation to O are possible, some of the results we present below are specific to fields, hence we assume R = k.

Definition 5.34
Let V be a kG-module. Then, a kG-module M is termed relatively V -projective, or simply V -projective, provided there exists a kG-module N such that It is easy to see that projectivity relative to V is equivalent to projectivity relative to V * and it is also equivalent to projectivity relative to V * ⊗ k V ∼ = End k (V ). In other words, projectivity relative to modules generalises projectivity relative to subgroups.
(b) By Properties 2.1(c) ordinary projectivity is just {1}-projectivity in terms of projectivity relative to subgroups. Thus, it is clear from (a), that ordinary projectivity can be thought of as V -projectivity for V := Ind G {1} (k) ∼ = kG. In addition, a projective kG-module is V -projective for any kG-module V . Indeed, if M is projective, then there exists n ∈ N such that M | (kG) n . Thus, as We would like to avoid this situation, which is not interesting. Thus, we call a kGmodule V absolutely p-divisible provided p divides the k-dimension of all the indecomposable direct summands of V .

Remark 5.36
Considering kG-modules V which are absolutely p-divisible, we can naturally generalise the definition of an endo-trivial module. where (V -proj) denotes a V -projective kG-module.
Remark 5.36 tells us that the ordinary endo-trivial kG-modules, introduced in §5.1 and §5.3, are V -endo-trivial modules for V := kG. In addition, by Remark 5.36, the assumption on the absolute p-divisibility of V ensures that a V -endo-trivial kGmodule is not V -projective, and conversely.

Properties 5.38
Let V be an absolutely p-divisible kG-module. Let M and N be V -endo-trivial kG-modules. Then, the following assertions hold.  Proof (a) Because V is absolutely p-divisible, so is any V -projective kG-module, so considering dimensions the assumption that 1 (mod p). The second claim follows from the fact that any Qprojective kG-module has dimension divisible by |P : Q| (see [33, (19.26) Theorem]).
(b) The claim about the dual is immediate from the definition. For the second claim, by the assumption, we have (c) It is clear that the vertices of M are the Sylow p-subgroups of G as dim k (M) is coprime to p by (a). Moreover, we have The claim follows, because if k were not a direct summand of S ⊗ k S * , then M would be V -projective and therefore not V -endo-trivial.
Moreover, M is V -projective-free by assumption, thus the Krull-Schmidt theorem yields that M | Ind G P (k). In consequence, P being a vertex of M, k is a source of M.
With these stability properties, we can now copy the procedure we have seen for endo-trivial modules over p-groups in order to obtain a group structure from the class of V -endo-trivial modules, which generalises the group of endo-trivial kG-modules, based on the previous proposition. For further general properties of groups of V -endo-trivial kG-modules, we refer to [49,50]. Our aim here is to show how this construction can be used in order to define a group structure similar to that of the Dade group of a p-group for an arbitrary finite group of order divisible by p. In this view, we need to restrict our attention to endo-ppermutation modules, which are the equivalent of the capped endo-permutation modules over p-groups, and which we are going to regard as V -endo-trivial kG-modules for a well-chosen absolutely p-divisible kG-module V defined below.

Notation 5.40
Given P ∈ Syl p (G), we define F G := {Q P } to be the family of all proper subgroups of P and we set V (F G ) := Q∈F G Ind G Q (k).
Clearly, V (F G )-projectivity corresponds to projectivity relative to the family of all non maximal p-subgroups of G and hence does not depend on the choice of the Sylow p-subgroup P . It is therefore an absolutely p-divisible kG-module. With this notion we can come back to endo-p-permutation kG-modules. (d) End k (M) ∼ = k ⊕ N where N is a p-permutation kG-module, all of whose indecomposable direct summands have a vertex strictly contained in P .
Proving the equivalence of the conditions (a) to (d) is not difficult, but requires a series of technical results on V -projectivity and V -endo-trivial modules, which we have not presented here. We refer therefore to [51, Proposition 5.2] for a complete proof. However, we easily see that this subclass of the class of endo-p-permutation kG-modules has all the good stability properties, we may expect it to have. Notice that any ordinary endo-trivial module is strongly capped, and in particular, so is any one-dimensional kG-module. Therefore, up to identifications, the groups T k (G) and X k (G) can also be viewed as subgroups of D k (G) and we have a series of subgroup inclusions

Remark 5.44
If G is a p-group, then certainly the generalised Dade group we have constructed above is isomorphic to the Dade group of G as defined in Proposition-Definition 5.4. In this case, a strongly capped endo-permutation kP -module is simply a capped endo-permutation such that its cap has multiplicity one. So certainly, this definition can be made over R = O and we obtain a generalised Dade group D O (P ).
The structure of the generalised Dade group can be linked to Urfer's characterisation of the indecomposable endo-p-permutation modules with full vertex through the G-stable points of the Dade group of a Sylow p-subgroup via the short exact sequence given by the following theorem. Since (X) ∼ = X is finite and the Dade Group D k (P ) is finitely generated, so is the generalised Dade group. We refer to [50,51] for computations of the structure the generalised Dade group in some concrete examples; for instance groups with a cyclic Sylow p-subgroup, groups with a Klein-four Sylow 2-subgroup, p-nilpotent groups, or GL 3 (p) in its defining characteristic.
Finally, we note that as in the case of p-groups, the subgroup D k (G) of D k (G) generated by all the relative syzygy modules with respect to subfamilies of F G plays an important role in the structure of the generalised Dade group. For example, [51,Corollary 12.8] tells us that when p is odd and the normaliser N G (P ) of a Sylow p-subgroup of G controls fusion in G, then

OG-Modules Which Are Necessarily OG-Lattices
In §3, §4, and §5 we allowed ourselves to talk about OG-modules, when we actually meant OG-lattices. We now explain why this is not an issue. Permutation OGmodules are clearly O-free by definition, and so are p-permutation OG-modules. We prove below that endo-p-permutation OG-modules are also necessarily free when regarded as O-modules, that is, OG-lattices. In particular, so are endo-trivial and endo-permutation OG-modules.

Proposition 6.1 If M is an OG-module such that End O (M) is free when regarded as an O-module, then so is M when regarded as an O-module.
Proof Since O is a discrete valuation ring, it is in particular a principal ideal domain. Thus, by the structure theorem for finitely generated modules over principal ideal domains, the module M, regraded as an O-module, admits a direct sum decomposition of the form where r is a non-negative integer and s i , n i (1 ≤ i ≤ r) are positive integers. Now, we claim that if the torsion part of M is not trivial, then neither is the torsion part of End O (M). Indeed, for every 1 ≤ i ≤ r we have that is assumed to be O-free, this forces M to be O-free as well.

Corollary 6.2 Any endo-p-permutation OG-module is free when regarded as an Omodule. In particular, so is any endo-trivial or any endo-permutation OG-module.
Proof If M is an endo-p-permutation OG-module, then by definition End O (M) is a p-permutation OG-module, hence O-free and the claim follows from Proposition 6.1. Endo-trivial and endo-permutation OG-modules are endo-p-permutation modules. The claim follows.

Lifting from Positive Characteristic to Characteristic Zero
We now turn to lifting results. It turns out that most of the classes of kG-modules, which we have introduced so far in this survey consist of modules which are liftable to characteristic zero. Being liftable is a rather rare property and proving liftability of modules defined by a common property is in general a difficult task. It is the reason why the results presented in this section were obtained one-by-one over a long period of time. We present them with their proofs, provided the approach involved do not go beyond the methods and techniques introduced so far.
In fact, amongst finitely generated kG-modules very few classes of modules are known to be liftable to OG-lattices. The Fong-Swan theorem (see e.g. [36, Theorem 72.1]) asserts that all simple modules of p-soluble groups are liftable, and Hiß [40,41] studies groups whose Brauer-characters are liftable and gives a converse to Fong-Swan theorem result. It is also well-known that projective kG-modules lift to projective OG-modules in a unique way. Scott proved that this remarkable property can be generalised to p-permutation kG-modules, and, in turn, the latter implies that several other related classes of modules introduced in Sect. 5 are liftable.

Lifting p-Permutation kG-Modules
The lifting of p-permutation modules is due to Scott, who proved that the kendomorphism ring of a transitive permutation kG-module is liftable.
Hence the O-rank of Hom OG (L 1 , L 2 ) is |Q 2 \G/Q 1 |. The same argument with k instead of O, shows that the k-dimension of Hom kG (L 1 /pL 1 , L 2 /pL 2 ) is also |Q 2 \G/Q 1 | and surjectivity follows.
(b) As L is indecomposable, End OG (L) is a local ring. By (a), End kG (L/pL) is isomorphic to a quotient of End OG (L), hence is also local (see e.g. [59,Corollary 4.4.5]), and thus L/pL is indecomposable. Now, by Higman's criterion, there exists ϕ ∈ End OQ (L) such that Id L = tr G Q (ϕ) and Q is minimal with this property. Hence Id L/pL = tr G Q (φ) and thus Q contains a vertex of L/pL. On the other hand, using the fact that L has a trivial source by Proposition-Definition 4.2, certainly O | Res G Q (L), implying that k | Res G Q (L/pL). As k has vertex Q, we obtain that Q is contained in a vertex of L/pL, proving (b).
(c) Again by the characterisation of p-permutation OG-lattices in Proposition-Definition 4.2, it suffices to prove that the claim holds for an indecomposable p-permutation kG-module. So, let M be an indecomposable kG-module with vertex Q and trivial source. Then, M | Ind G Q (k) and there exists an idempotent ι ∈ End kG (Ind G Q (k)) such that M = ι(Ind G Q (k)), which is unique up to conjugacy (see e.g. [59, Corollary 4.6.10]). Now, by (a) the canonical map is surjective. Therefore, by the lifting theorem for idempotents, there exists an idempotent π ∈ End OG (Ind G Q (O)), unique up to conjugacy, such that ι is the reduction modulo p of π . Then L := π(Ind G Q (O)) is a direct summand of Ind G Q (O) such that L/pL ∼ = M, unique up to isomorphism, which is indecomposable with vertex Q by part (b). This leads to the following character-theoretic characterisations of p-permutations modules.  (c) Because S ∈ vtx(M), we have x ∩ g S = x for some g ∈ G if and only if x ∈ g S, as required.

Lifting Endo-Permutation kP -Modules
The question whether endo-permutation over p-groups form a class of liftable modules was open for a long time. As we will see in the next subsection, Alperin proved in 2001 that the subclass endo-trivial modules is. However, the result for endopermutation modules was only obtained in 2006 by Bouc as a consequence of their classification.
The first observation to make is as above that the reduction modulo p of endopermutation OP -modules is extremely well-behaved. The idea of the proof is essentially that it suffices to prove that the generators of the Dade group D k (P ) are liftable modules.
Proof (Sketch.) Any relative syzygy module P /Q (k) lifts to a relative syzygy module over O. In characteristic 2, when P is generalised quaternion, then straightforward calculations show that any indecomposable capped endo-permutation whose class lies in D k (P ) tors ex is liftable as well by straightforward calculations. Thus, it follows from Theorem 5.15 that any element of D k (P ) is liftable, proving (a). Since any endo-permutation kP -module can be described in terms of the indecomposable capped ones, (b) follows.
Going further, it is easy to describe the kernel of π p , and we aim to prove that π p is in fact a split morphism. Next, we explain why studying the Dade group of G over k is equivalent to Puig's approach via the Dade group of Dade P -algebras, but not over O. We already explained in §5.1, that the endomorphism algebra End R (M) of an endo-permutation RP -module M is naturally endowed with the structure of a so-called Dade P -algebra (i.e. an O-simple permutation P -algebra whose Brauer quotient with respect to P is non-zero). Furthermore, there exists also a version of the Dade group, denoted by D alg R (P ), obtained by defining an equivalence relation on the class of all Dade Palgebras rather than capped endo-permutation RP -lattices, where multiplication is given by the tensor product over R. We refer to [74, §28-29] for this construction. This induces a canonical homomorphism which is surjective by [74,Proposition 28.12]. The identity element of D alg R (P ) being the class of the trivial P -algebra R, it follows that the kernel of d R is isomorphic to X R (P ) when R = O, whereas it is trivial when R = k. Now, reduction modulo p also induces a group homomorphism Also, finding a group-theoretic section of π p is equivalent to finding a group-theoretic section of d O .
In odd characteristic, the explicit construction of such a group-theoretic section for d O is due to Puig and can be found in print in [74,Remark 29.6]. Below, we translate this construction from Dade P -algebras to endo-permutation modules. In characteristic 2 the question of the existence of such a section was open for a long time and eventually proved in [58], relying on Bouc's classification of endo-permutation kPmodules.
Warning: For the remainder of this subsection we need to identify the Dade group D R (P ) with the generalised Dade group of P over R, what we are allowed to do by Remark 5.44. So the elements of the classes in D R (P ) are not just capped endopermutation RP -modules, but strongly capped endo-permutation RP -modules. This is one of the key arguments used in [58] which makes the construction of a section in characteristic 2 possible. We prove this theorem in several steps.
To prove Assertion (a), we need to consider determinants. Recall that given an OP -lattice L, we may consider the composition of the underlying representation of P with the determinant homomorphism det : GL(L) −→ O × . This is a linear character of P and is called the determinant of L. If the determinant of L is the trivial character, then we say that L is an OP -lattice of determinant 1. It is immediate that the O-dual L * of an OP -lattice L of determinant 1 also has determinant 1 as the action of g ∈ P on ϕ ∈ L * is given by (g·ϕ)(x) = ϕ(g −1 x) for all x ∈ L, and hence det(g, L * ) = det(g −1 , L) = det(g, L) −1 . Similarly, the tensor product L⊗ O N of two OP -lattices L and N of determinant 1 also has determinant 1 since the determinant of a tensor product satisfies the well-known property det(g, Now, amongst the lifts of a strongly capped endo-permutation kP -module M, there always exists one which has determinant 1 (see e.g. [74,Lemma 28.1]), on the one hand because we assume that k is large enough, and on the other hand because dim k (M) is prime to p as M is strongly capped (and not just capped!). This lift of determinant 1 is unique, up to isomorphism, and will be written M . It follows from the remarks above that * Then, to obtain Theorem 7.6(a) when p ≥ 3, it suffices to prove the following lemma.   ) ], proving that σ p is well-defined, and it is clear that it is a group homomorphism again because having determinant 1 is preserved by the tensor product over O. Finally, it is straightforward that π p • σ p is the identity on D k (P ).
Let us now turn to characteristic 2.

Lifting Endo-Trivial kG-Modules
It was proved by Alperin in 2001 that endo-trivial modules over p-groups are liftable. The corresponding result for arbitrary groups was obtained in 2016, as described below.
Lemma 7.10 Let L be an OG-module. Then, L is endo-trivial if and only if L/pL is an endo-trivial kG-module.
Proof The necessary condition is clear. Indeed, if L is endo-trivial, then End O (L) ∼ = O ⊕ X, where X is a projective OG-module. Therefore, where X/pX is a projective kG-module by Theorem 7.1(b). Conversely, assume that L/pL is endo-trivial, i.e. (L/pL) * ⊗ k L/pL ∼ = k ⊕ Y for some projective kGmodule Y , and the rk O L = dim k L/pL is coprime to p. It follows that L * ⊗ O L ∼ = O ⊕ X for some OG-lattice X and the Krull-Schmidt theorem yields X/pX ∼ = Y , proving that X is projective, and hence L is endo-trivial.

Theorem 7.11 ([4, Theorem]) If P is a p-group, then any endo-trivial kP -module lifts to an endo-trivial OP -module.
Alperin's proof is based on the key fact that the image of a representation of a pgroup lies in the special linear group. We state below a slightly more general version of this result and its proof. implying that sl n (k) is projective, when regarded as a kG-module via ρ.
Next, taking a pull-back X 2 of ρ and the homomorphism induced by reduction modulo p from SL n (O)/SL n (O, 2) → SL n (k), with kernel SL n (O, 1)/SL n (O, 2) ∼ = sl n (k), yields a group extension This extension splits because H r (G, sl n (k)) = Ext r kG (k, sl n (k)) = 0 for each r ≥ 1 as sl n (k) is an injective kG-module. As a consequence, ρ lifts to a homomorphism ρ 2 : G → SL n (O)/SL n (O, 2). Inductively, for every m > 2, we can construct a homomorphism  where q is the quotient homomorphism. So,ρ O • q lifts ρ. Therefore, we may assume that G ≤ GL n (k). Now, set G 1 := GC with C := {aI n | a n = det(g) for some g ∈ G} and G 0 := G 1 ∩ SL n (k). So, the situation is as follows: GL n (k) Clearly, G 1 is a central product of G with C, and of G 0 with C. As G ≤ G 1 has p -index the embedding G 1 ≤ GL n (k) defines an endo-trivial module, and in turn restricting from G 1 to G 0 the embedding G 0 ≤ SL n (k) defines an endo-trivial kG 0module as G 0 ≤ G 1 has p -index, too. The latter kG 0 -module lifts to an endo-trivial OG 0 -module by Theorem 7.12, and denoting the corresponding representation by ψ we have ψ(G 0 ) ≤ SL n (O). Now, reduction modulo p induces a bijection between the group of p -roots of unity in O and the group of roots of unity in k, sending ψ(G 0 ) ∩ Z(SL n (O)) onto G 0 ∩ Z(SL n (k)). The inverse defines a lift of C into {aI n | a ∈ O × } ≤ GL n (O), which agrees with ψ on G 0 ∩ Z(SL n (k)) and which we also denote by ψ . Then is a faithful representation of G 1 which lifts G 1 ≤ GL n (k). Again, as |G 1 : G| and |G 1 : G 0 | are prime to p, this defines an endotrivial OG-module lifting the initial representation of G.
This lifting result opened the door to a fruitful approach to the theory of endotrivial modules through ordinary character theory, based on the following results. Due to publication delays the following result, characterising endo-trivial modules which are at the same time p-permutation modules, appeared earlier than the previous one, although it is a consequence. Proof Since M is a p-permutation module, we know from Lemma 7.2(b) that χ M (x) is a non-negative integer for each p-element x ∈ G. First, assuming that M is endotrivial, by Proposition 7.14 this forces χ M (x) = 1 for each non-trivial p-element x ∈ G.
Conversely, assume that χ M (x) = 1 for each non-trivial p-element x ∈ G. Notice that we may also assume that dim k (M) > 1, as any one-dimensional kG-module is endo-trivial. Now, as dim k (M) ≡ χ M (x) = 1 (mod p) for any non-trivial pelement x ∈ G (see e.g. [ where N 1 , . . . , N r (r ∈ Z ≥1 ) are non-trivial indecomposable p-permutation kGmodules. At the level of characters we havē χ M · χ M = 1 G + χ N 1 + · · · + χ N r and it follows immediately that χ N i (x) = 0 for each 1 ≤ i ≤ r. Hence, vtx(N i ) = {{1}} for each 1 ≤ i ≤ r by Lemma 7.2(c). In other words, N i is projective for each 1 ≤ i ≤ r, proving that M is endo-trivial.
The aforementioned results led to a classification of the simple endo-trivial modules for the finite quasi-simple groups based on ordinary character theory in [52,53,62,63], partially involving computer algebra methods and computation with GAP4, CHEVIE, or MAGMA. Character-theoretical arguments were also used to determine the structure of the group T k (G) of endo-trivial modules, and in particular of its torsion subgroup T tors k (G), for certain finite quasi-simple groups of Lie type in [52], for the sporadic groups and their covering groups in [55], for the Schur covers of the alternating and symmetric groups in [54], or four some groups with dihedral or semi-dihedral Sylow 2-subgroups in [44][45][46].

Lifting Endo-p-Permutation kG-Modules
The question whether endo-p-permutation kG-modules also form a class of liftable modules is natural. It was already raised by Urfer in his PhD thesis, but only answered 12 years later in [57], again exploiting strongly capped endo-p-permutation modules and the generalised Dade group of a finite group rather than Urfer's original approach. The proof we give below also provides us with an alternative proof for the liftability of endo-trivial modules over arbitrary finite groups, provided it is known that endotrivial modules over p-groups are liftable.
We start by showing that the reduction modulo p of endo-p-permutation OGmodules is extremely well-behaved.

Lemma 7.16
Let L be an endo-p-permutation OG-module and consider L/pL its reduction modulo p. Then, the following assertions hold: (b) Observe that it follows from Theorem 7.1(a), that the canonical map k ⊗ O U G −→ (k ⊗ O U) G is a kG-isomorphism for any p-permutation kG-module U . (See also [57,Lemma 3.1].) Therefore, writing A = m i=1 U i as a direct sum of indecomposable p-permutation OG-modules, we obtain that the canonical homomorphism is an isomorphism of k-algebras.
(c) It is clear that L/pL is decomposable if L is, hence it remains to prove the necessary condition. Setting A := End O (L) as in (b), it is enough to prove that End kG (L/pL) = (k ⊗ O A) G is a local algebra. Write ψ : A G −→ A G /pA G for the canonical homomorphism given by reduction modulo p. By Nakayama's Lemma pA G ⊆ J (A G ), so that any maximal left ideal of A G contains pA G . Therefore This is a skew-field, as required, since we assume that L is indecomposable. Hence L/pL is also indecomposable. Then there is also a decomposition End k (L/pL) ∼ = k ⊗ O End O (L) ∼ = U 1 /pU 1 ⊕ · · · ⊕ U n /pU n .
As L is an endo-p-permutation OG-module, U i is a p-permutation module for each 1 ≤ i ≤ n. Thus, by Scott's theorem (Theorem 7.1), for each 1 ≤ i ≤ n, the module U i /pU i is indecomposable and the vertices of U i and U i /pU i are the same. Now, by Properties 5.26(d), each U i as a vertex contained in Q and one of them has vertex Q. Therefore U i /pU i has a vertex contained in Q for each 1 ≤ i ≤ n and one of them has vertex Q, and it follows that Q ∈ vtx(L/pL).

Lifting Fusion Stable Endo-Permutation kG-Modules and Brauer-Friendly kG-Modules
Finally, we mention two lifting results, of the same flavour as the ones presented above, concerning two further classes of modules strongly linked to endopermutation modules and involving certain stability conditions with respect to a group or to a block fusion system. (For the definition of a fusion system we refer to the book [5] by Aschbacher-Kessar-Oliver devoted to this topic.) The first one by Kessar and Linckelmann is concerned with fusion stable endo-permutation kGmodules. To understand the statement of this theorem, we need to give the definition of the Dade group of a fusion system on a finite p-group introduced by Linckelmann and Mazza in [61]. Let P be a finite p-group and F be a saturated fusion system on P . If Q ≤ P is a subgroup, ϕ ∈ Hom F (Q, P ) and M is an RP -module, denote by Res ϕ (M) the RQ-module which is equal to M as an R-lattice and with u ∈ Q acting on m ∈ M as ϕ(u) · m. Then, the class [V ] ∈ D R (P ) of a capped endo-permutation RP -module V is called F -stable if the endo-permutation RQ-modules Res ϕ (V ) and Res P Q (V ) have isomorphic caps, for any subgroup Q ≤ P and any morphism ϕ ∈ Hom F (Q, P ). The Dade group of F can then be defined as the following subgroup of the Dade group of P :  Assuming that G and H are two finite groups and k is a splitting field for G × H , Kessar and Linckelmann used Theorem 7.18 in order to prove that a Morita equivalence (resp. a stable equivalence of Morita type) between two blocks of kG and kH induced by an indecomposable (kG, kH )-bimodule M with endo-permutation source V can be lifted to a Morita equivalence (resp. a stable equivalence of Morita type) between the corresponding blocks of OG and OH induced by an (OG, OH )-bimodule L with endo-permutation source W such that k ⊗ O L ∼ = M and k ⊗ O W ∼ = V . See [43,Theorem 1.13].
Finally we come to the liftability of Brauer-friendly kG-modules. Brauer-friendly RG-modules were introduced by Biland in his doctoral thesis, also in French. See [7] for a published version in English. We do not define these modules formally in this manuscript, as their definition is rather technical and goes beyond the methods and objects we have introduced so far. However, we note that any indecomposable Brauerfriendly RG-module has an endo-permutation source which is subject to a certain stability condition with respect to the fusion system of the block of RG containing the module, in the spirit of the F -stability introduced above. Brauer-friendly RGmodules are not necessarily G-stable and provide us with natural examples of RGmodules with endo-permutation source which are not endo-p-permutation modules. Watanabe [78,Theorem 4.1] proves that an indecomposable Brauerfriendly kG-module lying in a block of kG lifts to a Brauer-friendly OG-module belonging to the corresponding block of OG, provided the fusion system of the block is saturated.

Glossary
To finish with, we summarise the different classes of modules and group structures introduced in this survey. We recall that p is a prime number, G denotes a finite group of order divisible by p, P a p-group and R ∈ {O, k}.

Modules:
• A permutation RG-module is an RG-module admitting an R-basis which is (globally) invariant under the action of the group G. See Definition 3.1. and/or for corrections and comments on preliminary versions of this manuscript. Notwithstanding, the content of this paper has very much been influenced by Jacques Thévenaz, who first taught me about these notions. Robert Boltje, Serge Bouc, Michael Geline, Jürgen Müller, Lucas Ruhstorfer and the Deutsche Bahn also significantly contributed in a way or another to the content of this survey.
Funding Note Open Access funding enabled and organized by Projekt DEAL.

Declarations
Competing Interests The authors declare no competing interests.
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