University of California Santa Cruz Fusion Systems And

Home , Group isomorphism, Opposite group

UNIVERSITY OF CALIFORNIA SANTA CRUZ

FUSION SYSTEMS AND BISET FUNCTORS VIA GHOST ALGEBRAS A dissertation submitted in partial satisfaction of the requirements for the degree of

DOCTOR OF PHILOSOPHY

MATHEMATICS

Shawn Michael O’Hare

June 2013

The Dissertation of Shawn Michael O’Hare is approved:

Professor Robert Boltje, Chair

Professor Geoﬀ Mason

Professor Martin Weissman

Tyrus Miller Vice Provost and Dean of Graduate Studies Copyright c by Shawn Michael O’Hare 2013 Table of Contents

Abstract v

Dedication vi

Acknowledgments vii

Introduction1 Some Notation ...... 4

1 Background7 1.1 Fusion System Basics ...... 7 1.2 Bisets ...... 10 1.3 Double Burnside Groups ...... 15 1.4 Two Ghost Groups ...... 17 1.5 Subgroups of Burnside Groups ...... 21 1.6 Biset Categories ...... 24

2 Biset Categories 27 2.1 A Special Class of Groups ...... 28 2.2 Fusion Preserving Isomorphisms ...... 31

3 Characteristic Idempotents 36 3.1 Calculating the Characteristic Idempotent ...... 36 3.2 Bideﬂation of Characteristic Idempotents ...... 42

4 A Generalized Burnside Functor 47 4.1 Pseudo-rings ...... 47 Condensation and Decondensation ...... 49 4.2 Decondensation of the Burnside Functor ...... 51 4.3 The Action of Elementary Subgroups ...... 53 4.4 Subfunctors ...... 57

References 67

iii A Single Burnside Rings 70 A.1 Sets with a Group Action ...... 70 A.2 Operations on Sets with a Group Action ...... 76 A.3 The Single Burnside Ring ...... 78

Notation 86

Index 88

iv Abstract

Fusion Systems and Biset Functors via Ghost Algebras

Shawn Michael O’Hare

In this work we utilize ghost groups of Burnside groups introduced by Boltje and Danz in order to investigate fusion systems of finite groups, double Burnside modules, and biset functors. We give an expression for the coefficients of the characteristic idempotent ωF associated to an arbitrary fusion system F, and demonstrate that bideflation does not generally preserve this idempotent when F is unsaturated. Motivated by the theory of p-completed classifying spaces, we study when two groups are isomorphic in the left-free p-local biset category B¡p , and prove that two groups G and H are isomorphic in this category when there exists an isomorphism between their Frobenius p-fusion systems. Finally, we consider a process mirroring Green’s theory of idempotent condensation and demonstrate that a generalized Burnside functor is the decondensation of the usual Burnside functor, and that this decondensation preserves the subfunctor lattice.

v To my family, for their encouragement and support.

vi Acknowledgments

First and foremost I am indebted to Robert Boltje, without whose insight and patience this thesis would not exist. I have learned a tremendous amount from each of my committee members, and they have helped me develop as a mathematician. Martin Flashman’s enthusiasm for the subject convinced me to pursue it in earnest. David Kinberg has my profound thanks for recognizing and fostering my intellectual curiosity when I was a teenager. I am also grateful for my friends in Santa Cruz, many of them also colleagues in the Mathematics Department. They have enriched my life to an ineﬀable degree. My early collaborations with Corey Shanbrom instilled me with the requisite tenacity to come as far as I have. Rob Laber and David Deconde have helped me expand my knowledge base and focus the direction of my future studies. I am in debt to Victor Dods and Wyatt Howard for their many hours of stimulating conversation.

vii Introduction

Double Burnside groups serve as the common core linking together all the objects we study in this thesis. For a finite group G, the set of isomorphism classes of finite G-sets is a commutative monoid under the coproduct of sets, and the Grothendieck group B(G) of this monoid is enriched with a commutative ring structure via the direct product of sets. We call B(G) the single Burnside ring of G. Given another finite group H and commutative ring R, we define the double Burnside R-module RB(G, H)

to be R ⊗Z B(G × H). There is a natural correspondence between (G × H)-sets and (G, H)-bisets, which are sets endowed with a left G-action and a right H-action that commute, given by viewing a left (G × H)-set X as a (G, H)-biset via gxh := (g, h−1)x for g ∈ G, h ∈ H, x ∈ X. When G = H, the tensor product of bisets induces a generally non-commutative ring structure on B(G, G), which we then call the double Burnside ring of G. While the single and double Burnside rings of G encode much representation theoretic information about G, their multiplicative structures can be complicated. One successful remedy is to embed the Burnside rings into other rings, called ghost rings, with more tractable multiplicative structures. This is classically done by taking marks, i.e., computing fixed points. Our study of fusion systems and biset functors primarily exploits ghost groups for the double Burnside group introduced by Boltje and Danz in [BD12; BD13]. Fusion systems are of interest to abstract group theorists, modular representation theorists, and topologists. There is hope that the techniques involved in classifying saturated fusion systems could streamline the classification of finite simple groups. As- sociated to every p-block of a finite group is a saturated fusion system, and these fusion systems provide a nice context in which to investigate aspects of modular representation theory. The Martino-Priddy conjecture [MP96], proved by Oliver [BLO03; Oli04; Oli06], asserts that two groups have homotopy equivalent p-completed classifying spaces if and

1 only if their Frobenius p-fusion systems are equivalent. Ragnarsson and Stancu [RS13] create a bijection between saturated fusion systems on a p-group S and the set of characteristic idempotents in the bifree dou- ∆ ble Burnside Z(p)-algebra Z(p)B (S, S). Boltje and Danz [BD12] construct a ghost ring B˜∆(S, S) for the bifree double Burnside ring B∆(S, S), a mark homomorphism ρ: B∆(S, S) → B˜∆(S, S), and are able to extend the correspondence in [RS13] to the set of all fusion systems on S and a set of certain explicitly described idempotents in ˜∆ ∆ the rational bifree ghost algebra QB (S, S). While the image of Z(p)B (S, S) under ρ ˜∆ ˜∆ is contained in Z(p)B (S, S), the idempotents in Z(p)B (S, S) do not necessarily correspond to saturated fusion systems. One of our initial hopes was to furnish a completely algebraic proof that a fusion system on S is saturated if and only if its corresponding ∆ characteristic idempotent ωF is an element of Z(p)B (S, S), which was initially proved by topological means in [RS13]. As of this writing, our goal remains unachieved. The basic theory of saturated fusion systems can be reformulated in terms of characteristic idempotents [RS13, Section 8]. In particular, the characteristic idempotents of saturated fusion systems are well-behaved with respect to factoring, in the sense that that if F is a saturated fusion system on S and F/T is a quotient system over a strongly F-closed subgroup T , then the characteristic idempotents of F and F/T are S related by bideﬂation, i.e., BidefT (ωF ) = ωF/T . We show in Section 3.2 that this result does not extend generally by constructing an unsaturated fusion system F and quotient S system F/T such that BidefT (ωF ) 6= ωF/T . A convenient setting for our studies are various subcategories of the R-biset

category BR, and categories related to them by means of ghost groups. The category BR has as objects the class of finite groups (or, to get an equivalent category, a transversal for the isomorphism classes of finite groups) and for two objects G and H we define

HomBR (G, H) := RB(H,G). Morphism composition is induced from the tensor product of bisets. Both [Rag07, Theorem A] and [RS13, Section 9] relate the stable homotopy of classifying spaces of ﬁnite groups to the left-free p-local biset category B¡p , which is a

subcategory of the Z(p)-biset category with the same object class, but whose morphisms are generated by transitive bisets corresponding to p-groups. To better understand stable equivalences, it would be useful to determine whether B¡p has the same isomorphism classes as the bifree p-local biset category B∆p , a subcategory of B¡p whose morphisms are generated by bifree bisets. In Chapter2 we use the N-grading of ghost algebras

2 introduced in [BD12, Section 6] to explore when two groups are isomorphic in B¡p . In Section 2.2 we prove that if there is a fusion preserving isomorphism between the Frobenius p-fusion systems of G and H, then G and H are isomorphic in B∆p , further relating fusion systems to stable homotopy of classifying spaces. Many natural operations that occur in the representation theory of ﬁnite groups– for example, restriction and induction–arise as functors involving bisets. As these operations appear in a variety of contexts, such as group cohomology, the algebraic K-theory of group rings, and algebraic number theory, the abstract study of these operations is suﬃciently motivated. The formalism of operations like restriction and induction is en-

coded by biset functors, which are R-bilinear functors from some subcategory of BR to the category of R-modules. Their theory was the main tool in the classification of endo-permutation modules for a p-group [Bou06]. Biset functors have also been lever- aged successfully to determine the unit group of the single Burnside ring of a p-group [Bou07]. Bouc shows in [Bou10] that the simple modules of the double Burnside algebra FB(G, G) for a finite group G and field F are evaluations at G of simple F-biset functors. In the same text, he determines the subfunctor lattice of the Q-Burnside functor, the Yoneda -biset functor Hom (1, −) corresponding to the trivial group. Note in par- Q BQ ticular that the evaluation of the Q-Burnside functor at G can be canonically identified with the single Burnside Q-algebra QB(G), hence the name. We consider a process that mirrors Green’s theory of idempotent condensation [Gre07] and demonstrate that a generalized -Burnside functor–the Yoneda functor Hom (1, −) where A contains Q AQ Q

the biset category BQ as a non-full subcategory–is the decondensed Burnside functor. By an argument analogous to Bouc’s, we show in Chapter4 that decondensation preserves the subfunctor lattice of the Q-Burnside functor. Technically, we argue within the framework of admissible modules over twisted category algebras. Every Q-biset functor can be identiﬁed with an admissible B-module, where B is the category algebra of the biset category over Q. Boltje and Danz [BD13] show that B embeds into a Q-algebra A˜, whose admissible modules correspond to generalized Q-biset functors. In analogy to the situation of ghost groups for Burnside groups, we hope the study of admissible A˜-modules will shed light on biset functors.

3 Some Notation

In this section we detail some of the notational conventions we will employ.

Set Theory

Throughout, let A and B denote sets and f : A → B and g : B → A functions.

• ∅ is the empty set.

• A ⊆ B denotes subset inclusion and A ⊂ B denotes strict subset inclusion.

•| A| denotes the cardinality of A.

• g ◦ f denotes the composition of g with f, with f acting ﬁrst.

• A q B denotes the set coproduct of A and B, i.e., the disjoint union. Likewise, A × B denotes the product, i.e., Cartesian product.

• Z and Q denote the sets of integers and rationals, respectively.

Group Theory

Let G and H denote groups.

• H ≤ G denotes that H is a subgroup of G, while H < G denotes that H is a proper subgroup of G.

• If H ≤ G, then [G : H] denotes the index of H in G.

• 1G, or simply 1, denotes the trivial subgroup of G.

• 1G, or simply 1 denotes the identity element of G.

• For g ∈ G, g−1 denotes the inverse of g.

• H ¢ G denotes that H is a normal subgroup of G.

• For x ∈ G, xG := {xg | g ∈ G} and Gx := {gx | g ∈ G}.

• For subsets U, V ⊆ G, UV := {uv | u ∈ U, v ∈ V }.

•h x, y, . . . i denotes the subgroup generated by the elements x, y, etc.

4 • When H ≤ G, G/H := {gH | g ∈ G} denotes the set of left cosets of H in G. Likewise, H\G := {Hg | g ∈ G} denotes the set of right cosets of H in G.

• For a prime p, Sylp(G) denotes the set of Sylow p-subgroups of G.

g −1 g −1 • For g ∈ G and U ≤ G, we set U := gUg and U := g Ug. Also, cg : G → G denotes the group automorphism x 7→ gxg−1.

• For the subgroups H, Q, R of G set HomH (Q, R) := {cx : x ∈ NH (Q, R)} where x NH (Q, R) := {x ∈ H : Q ≤ R}, i.e., the transporter in H of Q into R.

• When H and K are subgroups of G, H\G/K := {HxK | x ∈ G} denotes the set of (H,K)-double cosets in G. We let [H\G/K] denote a set of representatives for the (H,K)-double cosets in G.

• If H and K are subgroups of G, then H ≤G K denotes that H is G-subconjugate −1 to K, i.e., that there is some g ∈ G such that gHg ≤ K. We write H =G K to signify that H and K are conjugate within G.

• If H ≤ G, then [H]G denotes the set of subgroups of G which are G-conjugate to H.

• For a group G, SG denotes the subgroup lattice of G, on which G acts via conjuga-

tion. By T (SG) we mean a transversal of the orbit space SG/G, i.e., a collection of subgroups of G consisting of precisely one subgroup per conjugacy class.

• For a set X, Sym(X) denotes the symmetric group of X, i.e., the set of all bijections from X to itself together with function composition as the binary operation.

Rings and Modules

• Given a set of prime numbers π, we let Z(π) denote the localization of Z with respect to the set Z \ ∪p∈πpZ. In the case that π = {p} for a single prime p, we

write Z(p). Of course, Z(p) is the usual localization of Z at the prime ideal (p). It consists of all reduced rationals whose denominators are not divisible by p.

• If X is a set and R a ring, then RX denotes the free R-module with basis X.

• If M is an R-module for a ring R and S is a ring containing R, then SM := S⊗R M denotes the extension of scalars.

5 Categories

We will typically use calligraphic fonts such as C to denote categories. Let C and D denote categories.

• HomC(A, B) denotes the set of morphisms in C from A to B.

• The identity morphism of an object X is denoted 1X .

• If X is an object in C, then the C-isomorphism class of X, or simply the isomorphism class of X when C is understood, is the collection of objects in C which are isomorphic to X, i.e., the objects Y in C for which there exist morphisms f : X → Y and g : Y → X such that g ◦ f is the identity morphism of X and f ◦ g is the identity morphism of Y .

• Func(C, D) denotes the category of covariant functors from C to D.

• For a ring R, R-Mod and Mod-R denote the category of left R-modules and right

R-modules, respectively. If M and N are left R-modules, we set HomR(M,N) :=

HomR- Mod(M,N).

• If X is an object in C, then HomC(X, −) denotes the Yoneda functor of X. The values that this Yoneda functor takes will depend on what algebraic structure the morphism sets of C possess.

6 Chapter 1

Background

1.1 Fusion System Basics

Finite group theory in general and the classification of finite simple groups in particular provides the setting for the early study of fusion in finite groups. Suppose one is interested, for instance, in determining whether a group G has a normal p-complement. Recall that a normal p-complement in G is a normal subgroup K of G with index equal to the order of some Sylow p-subgroup of G. It turns out that G will have such a normal p-complement if each Sylow p-subgroup controls its own fusion. While Burnside and Frobenius studied the fusion of p-elements in finite groups, an inkling of the more general concept of fusion systems appears in [Sol74a; Sol74b; Sol73], where Solomon noticed a pattern to the fusion of involutions in a Sylow 2- subgroup P of Spin7(3) that did not appear to be induced via an inclusion of P as a Sylow 2-subgroup in some larger finite group. Around 1990 Puig, who was interested in modular representation theory, gave an axiomatic definition for an abstract fusion system on a finite p-group P . At the time, much of his work went unpublished, but many results were later collected in [Pui06]. Briefly, the objects are the subgroups of P and the morphism sets model a system of conjugations among subgroups of P induced by the inclusion of P in some ambient group, without specific reference to that ambient group. An important class of fusion systems are the saturated fusion systems, which model the Sylow inclusion of P in some group, i.e., the ambient group has P as a Sylow p-subgroup. Saturated fusion systems are proving to be useful in a number of fields. In

7 modular representation theory, they arise as invariants of blocks, and provide a useful framework in which to reformulate and approach the Alperin weight conjecture. Abstract group theorists hope that future techniques developed for classifying the saturated 2- fusion systems can streamline the classification of finite simple groups. Broto, Levi, and Oliver have popularized saturated fusion systems with homotopy theorists as a model for studying the p-completed classifying space of a finite group. This introductory chapter on fusion systems collects together the relevant def-

initions we need in order to investigate the characteristic idempotent ωF associated to a fusion system F. For an accessible introduction to the subject we recommend [Cra11; Lin07]. Those wishing to discover more about the topological side of fusion systems should consult [AKO11].

1.1 Deﬁnition (Fusion system of G on P ). Let G be a ﬁnite group and P ∈ Sylp(G).

The category FP (G) has as objects all subgroups of P and for two objects Q, R we deﬁne

HomFP (G)(Q, R) := HomG(Q, R).

Composition of morphisms in FP (G) is the usual composition of group homomorphisms.

We call FP (G) the fusion system of G on P or sometimes the Frobenius p-fusion system 0 of G. It is easy to see that if P ∈ Sylp(G), then FP (G) and FP 0 (G) are isomorphic categories.

1.2 Deﬁnition (Fusion system on P ). Let P be a ﬁnite p-group. A fusion system on P is a category F determined by the following data:

1. Obj(F) := SP , the set of subgroups of P .

2. All morphisms in F are injective group homomorphisms, and composition is given by the usual homomorphism composition.

3. HomP (Q, R) ⊆ HomF (Q, R) for each pair of objects Q and R. In particular, if Q ≤ R ≤ P , then the inclusion map Q,→ R is in F.

4. For each pair of objects Q and R and all ϕ ∈ HomF (Q, R), the induced group isomorphism ϕ: Q → ϕ(Q), u 7→ ϕ(u)

8 and its inverse ϕ−1 : ϕ(Q) → Q, ϕ(u) 7→ u

are morphisms in F. This, together with the previous axiom, implies that the set of morphisms in F is closed under restricting the domain and codomain. We will often be economical with notation and write ϕ−1 to mean ϕ−1. Given two objects ∼ Q and R in F, we will write Q =F R to denote that they are isomorphic in the category F.

1.3 Deﬁnition. Let F be a fusion system on P and suppose Q ≤ P . We say Q is fully 0 0 ∼ 0 F-normalized provided that |NP (Q)| ≥ |NP (Q )| for all Q ≤ P such that Q =F Q . 0 0 Similarly, Q is fully F-centralized provided that |CP (Q)| ≥ |CP (Q )| for all Q ≤ P such ∼ 0 that Q =F Q . For ϕ ∈ HomF (Q, P ) deﬁne

−1 Nϕ := y ∈ NP (Q) | ϕ ◦ (cy|Q) ◦ ϕ ∈ AutP (ϕ(Q)) .

−1 Here we view ϕ◦cy ◦ϕ as a group automorphism of ϕ(Q). In other words, Nϕ consists of the y ∈ NP (Q) for which there exists some z ∈ NP (ϕ(Q)) such that ϕ ◦ (cy|Q) =

(cz|ϕ(Q)) ◦ ϕ. It is easy to see that if Q ≤ R ≤ NP (Q) and ϕ extends to an F-morphism

ψ : R → P , then R ≤ Nϕ.

1.4 Definition (Saturated fusion system). Let F be a fusion system on a finite p-group P . We say that F is saturated provided that it satisfies the following two conditions.

1. (Sylow axiom). If Q ≤ P is fully F-normalized, then Q is fully F-centralized and

AutP (Q) ∈ Sylp(AutF (Q)).

2. (Extension axiom). For every Q ≤ P and every ϕ ∈ HomF (Q, P ) such that ϕ(Q)

is fully F-centralized, ϕ extends to Nϕ.

1.5 Remark (Diﬀerent axioms for fusion systems). Linckelmann [Lin07] provides a diﬀerent set of axioms for a saturated fusion system.

1. (L-Sylow). AutP (P ) ∈ Sylp(AutF (P )).

2. (L-Extension). For every Q ≤ P and every ϕ ∈ HomF (Q, P ) such that ϕ(Q) is

fully F-normalized, ϕ extends to Nϕ.

9 Utilizing [Lin07, Propositions 2.5-2.7], one can show that these saturation axioms are equivalent to the axioms we gave. It is easier to verify that a fusion system is saturated using the second set of axioms, but when we want to prove properties of saturated fusion systems we often use the seemingly stronger axioms given in our deﬁnition.

1.6 Lemma. Let G be a group with P ∈ SylP (G) and Q ≤ P .

1. Q is fully FP (G)-normalized if and only if NP (Q) ∈ SylP (NG(Q)).

2. Q is fully FP (G)-centralized if and only if CP (Q) ∈ SylP (CG(Q)).

Proof. See [Lin07].

1.7 Proposition. The Frobenius p-fusion system of G is saturated.

Proof. See [Lin07].

Not all saturated fusion systems are of the form FP (G). If F is a fusion system on P and is not of the form FP (G) we say that F is an exotic fusion system. Ruiz and Viruel [RV04] classiﬁed all saturated fusion systems on the extraspecial group of order p3 and exponent p and found three exotic fusion systems when p = 7. Also, there are saturated fusion systems on Sylow 2-subgroups of Spin7(q) (for odd prime q) that are exotic.

1.2 Bisets

Situations involving a group acting via permutations on a set are ubiquitous. Within the context of group theory, one is often interested in groups acting on finite sets. For a finite group G, the single Burnside ring B(G) is the Grothendieck ring of the commutative semiring of isomorphism classes of finite G-sets with respect to disjoint unions and direct products. In essence, B(G) is a ring theoretic encoding of how G acts on finite sets. It is a fundamental object of study in the representation theory of finite groups since it maps surjectively onto the representation ring of G consisting of permutation representations. Many algebraic objects associated to G, such as character rings, can be realized as B(G)-modules. In this way, the behavior of Burnside rings

10 under induction, restriction, conjugation, etc., sheds light on how the associated objects behave under similar operations. We develop some of the basic theory of group actions and single Burnside rings in appendixA. The double Burnside ring B(G, G) is the Grothendieck ring of the generally non-commutative semiring of isomorphism classes of finite (G, G)-bisets with respect to disjoint union and the tensor product of bisets. They are at the core of the theory of biset functors, which have proved to be very useful tools for investigating the representation theory of finite groups, as pointed out in this thesis’s introduction. In order to described double Burnside rings we first need to develop some elementary facts concerning bisets. Our treatment can be economical since a majority of the results in this thesis use the machinery developed in [BD13]. For a more in-depth treatment of bisets we recommend [Bou10, Chapter 2]. Throughout, G and H denote groups, usually finite.

1.8 Definition (Biset). Let G and H be groups. A (G, H)-biset is a set X equipped with left G-action and a right H-action that commute in the sense that (g·x)·h = g·(x·h) for all x ∈ X, g ∈ G, and h ∈ H. Any (G, H)-biset structure on X gives rise to left (G × H)-set structure on X via (g, h) · x := g · x · h−1 for all x ∈ X, g ∈ G, and h ∈ H. Conversely, a (G × H)-set structure on X induces a (G, H)-biset structure on X via g · x · h := (g, h−1) · x for all x ∈ X, g ∈ G, h ∈ H. Thus, the study of (G, H)-bisets amounts to the study of left (G × H)-sets. If X and Y are (G, H)-bisets, a map f : X → Y is a (G, H)-biset morphism or is (G, H)-equivariant provided f(g · x · h) = g · f(x) · h for all g ∈ G, h ∈ H, x ∈ X. Note that f is (G, H)-equivariant if and only if it is G × H-equivariant when we view X and Y as (G × H)-sets, according to the prescription above. In the case that f is bijective we say that X and Y are isomorphic (G, H)-bisets, which we denote by X =∼ Y . The opposite biset of a (G, H)-biset X is the (H,G)-biset X◦ whose underlying set is X equipped with the (H,G)-biset structure defined by h · x◦ · g := g−1 · x · h−1 for g ∈ G, h ∈ H, and x ∈ X. If L ≤ G × H, the opposite subgroup L◦ of H × G is defined by L◦ := {(h, g) ∈ H × G | (g, h) ∈ L}. There is a (H,G)-biset isomorphism (G × H/L)◦ =∼ (H × G)/L◦. A (G, H)-biset is transitive provided it is transitive as a (G×H)-set. From A.5

11 it follows that any transitive (G, H)-biset is isomorphic to G × H/L for some subgroup L of G × H. Throughout this thesis we assume that the (G, H)-biset structure on G × H/L is induced from the usual left multiplication of G × H. In other words, g · (u, t)L · h := (gu, h−1t)L for g, u ∈ G and h, t ∈ H. Given a (G, H)-biset X and a point x ∈ X, the point stabilizer of x is the stabilizer subgroup stabG×H (x). That is

stabG×H (x) := {(g, h) ∈ G × H | (g, h) · x = x} = {(g, h) ∈ G × H | g · x = x · h} .

1.9 Deﬁnition (Sections). Let G and H be ﬁnite groups. A section of G is a pair (U, V ) of subgroups of G such that V ¢ U. If (U, V ) is a section then so is (gU, gV ) for any g ∈ G, hence G acts on the set of its sections via conjugation.

Given a subgroup L of G × H, we let p1 : G × H → G and p2 : G × H → H denote the canonical projection morphisms. Further, deﬁne

k1(L) := {g ∈ G | (g, 1) ∈ L} ,

k2(L) := {h ∈ H | (1, h) ∈ L} ,

q(L) := L/(k1(L) × k2(L)).

For i ∈ {1, 2}, note that ki(L) ¢ pi(L) and so we have canonical projection morphisms

πi : pi(L) → pi(L)/ki(L). An element (x1, x2) ∈ L is in ker(πi ◦ pi) if and only if xi ∈ ki(L), so ker(πi ◦ pi) = k1(L) × k2(L) for i ∈ {1, 2}. The maps πi ◦ pi induce group isomorphisms p¯i : q(L) → pi(L)/ki(L), and so in particular there is a group isomorphism −1 ηL :=p ¯1 ◦p¯2 : p2(L)/k2(L) → p1(L)/k1(L). There is a bijection [Bou10, Lemma 2.3.25] between the set SG×H of subgroups of G × H and the set of quintuples (B, A, η, D, C) where (B,A) is a section of G, (D,C) is a section of H, and η : D/C → B/A is a group isomorphism.

1.10 Deﬁnition (Bifree). Let G and H be groups. A (G, H)-biset X is left-free provided it is free as left G-set. Similarly, X is right-free provided it is free as a right H-set. In the case that X is left-free and right-free, it is bifree. The class of bifree (G, H)- sets in general strictly contains the class of free (G × H)-sets, as 1.12 demonstrates.

1.11 Deﬁnition. Suppose G and H are ﬁnite groups, U is a subgroup of H, and that ϕ: U → G is a group homomorphism. A left-free subgroup of G × H is any subgroup of

12 the form ¡(ϕ(U), ϕ, U) := {(ϕ(u), u) ∈ G × H | u ∈ U}. In the case that ϕ is a group isomorphism we write ∆ϕ(U) := ∆(ϕ(U), ϕ, U) := ¡(ϕ(U), ϕ, U) and refer to such subgroups as the twisted diagonal subgroups of G × H or as bifree subgroups of G × H.

When G = H and ϕ: U → U is the identity map, we simply write ∆(U) := ∆ϕ(U).

By ¡G×H and ∆G×H we mean the subsets of SG×H consisting of left-free and bifree subgroups, respectively. This naming convention is justiﬁed by the following proposition, the proof of which is routine.

1.12 Proposition. Let G and H be ﬁnite groups and suppose L is a subgroup of G × H. Then:

1. L ∈ ¡G×H if and only if k1(L) = 1.

2. L ∈ ∆G×H if and only if k1(L) = 1 and k2(L) = 1.

3.A (G, H)-biset X is left-free if and only if k1(stabG×H (x)) = 1 for all x ∈ X.

Similarly, X is right-free if and only if k2(stabG×H ) = 1 for all x ∈ X.

1.13 Definition (Elementary subgroups and bisets). There are five basic bisets which are the analogs of the five basic operations on sets with a group action (A.18). Indeed, the elementary operations on sets with a group action can be realized as the biset tensor product functors corresponding to the five elementary bisets. Let G and H be finite groups.

G 1. Suppose H ≤ G. Induction from H to G is the subgroup IndH := ∆(H) of G × H. G Restriction from G to H is the subgroup ResH := ∆(H) of H × G.

2. Suppose ϕ: H → G is a group isomorphism. Isogation of ϕ is the subgroup

Isoϕ := ∆(G, ϕ, H) of G × H.

3. Given a normal subgroup N ¢ G, Inﬂation from G/N to G is the subgroup G ◦ InfG/N := ¡(G/N, π, G) of G × (G/N), where π : G → G/N is the canonical G projection. Deﬂation from G to G/N is the subgroup DefG/N := ¡(G/N, π, G) of (G/N) × G.

Given L ∈ SG×H , the (G, H)-biset corresponding to L is G × H/L. Corresponding to each of the ﬁve elementary subgroups deﬁned above is a transitive biset, and we call these

13 ﬁve types of transitive bisets the elementary bisets. We conﬂate the notation between the elementary subgroup and the biset corresponding to it. That is,

G 1. IndH := G × H/∆(H), viewed as a (G, H)-biset.

G 2. ResH := H × G/∆(H), viewed as an (H,G)-biset.

3. Isoϕ := G × H/∆(G, ϕ, H), viewed as a (G, H)-biset.

G ◦ 4. InfG/N := G × (G/N)/ ¡ (G/N, π, G) , viewed as a (G, G/N)-biset.

G 5. DefG/N := (G/N) × G/ ¡ (G/N, π, G), viewed as a (G/N, G)-biset.

1.14 Definition (∗-product of subgroups). Let G, H and K be finite groups and suppose L ≤ G × H and M ≤ H × K. Define

L ∗ M := {(g, k) ∈ G × K | ∃h ∈ H :(g, h) ∈ L, (h, k) ∈ M} .

Viewing M as a relation from K to H and L a relation from H to G, L ∗ M is the usual composition of relations.

The next lemma tells us how to decompose a subgroup of G × H into the ﬁve elementary subgroups deﬁned in 1.13.

1.15 Lemma (Decomposition of subgroups). Let G and H be ﬁnite groups and sup-

pose L ≤ G × H corresponds to the quintuple (P1,K1, η, P2,K2). Then

G P1 P2 H L = Ind ∗ Inf ∗ Isoη ∗ Def ∗ Res . P1 P1/K1 P2/K2 P2

G P1 P2 H Proof. Set D := Ind ∗ Inf ∗ Isoη ∗ Def ∗ Res . Clearly, D ≤ P1 × P2. For P1 P1/K1 P2/K2 P2 g ∈ P1 and h ∈ P2,

(g, h) ∈ D ⇐⇒ η(hK2) = gK1

⇐⇒ ∃(x, y) ∈ L : xK1 = gK1 ∧ yK2 = hK2 ⇐⇒ ∃(x, y) ∈ L :(x1g, 1) ∈ L ∧ (1, y−1h) ∈ L ⇐⇒ (g, h) ∈ L,

as desired.

14 1.16 Definition (Tensor product of bisets). For finite groups G, H and K, suppose that X is a (G, H)-biset and Y is an (H,K)-biset. The direct product X × Y carries a (G, K)-biset structure in the obvious way, i.e, g · (x, y) · k := (g · x, y · k) for g ∈ G, x ∈ X, y ∈ Y , and k ∈ K. Moreover, X × Y has a left H-action defined by h · (x, y) := (xh−1, hy) for h ∈ H, x ∈ X, y ∈ Y . As these three actions commute, the fiber product

X ×H Y := X × Y/H inherits a (G, K)-biset structure. We call X ×H Y the biset tensor product of X and Y over H. One can show [Bou10, Lemma 2.3.14] that the tensor product of bisets is associative and distributes across the coproduct.

1.17 Lemma (Decomposition of bisets). Let G and H be ﬁnite groups and suppose

L is a subgroup of G × H that corresponds to the quintuple (P1,K1, η, P2,K2). Then

∼ G P1 P2 H G × H/L = Ind × Inf × Isoη × Def × Res P1 P1/K1 P2/K2 P2 P1 P1/K1 P2/K2 P2

as (G, H)-bisets.

Proof. See [Bou10, Lemma 2.3.26]

1.18 Lemma (Mackey formula for bisets). Let G, H, and K be ﬁnite groups. For subgroups L ≤ G × H and M ≤ H × K, there is a (G, K)-biset isomorphism

G × H H × K X G × K × =∼ , L H M L ∗ (h,1)M h∈[p2(L)\H/p1(M)]

where [p2(L)\H/p1(M)] is a set of representatives for the (p2(L), p1(M))-double cosets of H.

Proof. See [Bou10, Lemma 2.3.24].

1.3 Double Burnside Groups

As a Z-module, the double Burnside group B(G, G) is just the single Burnside group B(G × G) (confer A.3). The tensor product of bisets induces a generally non- commutative ring structure on B(G, G).

1.19 Deﬁnition (Double Burnside group). Let G and H be ﬁnite groups and R a commutative ring. The double Burnside group B(G, H) is the abelian group B(G × H),

15 i.e., the Grothendieck group of the commutative monoid of isomorphism classes of ﬁnite (G×H)-sets with respect to set coproduct. This means that B(G, H) is a free Z-module with standard basis {[G × H/L] | L ∈ T (SG×H )}, where T (SG×H ) is a transversal for the G × H-conjugacy classes of subgroups of G × H, and RB(G, H) := R ⊗Z B(G, H) is a free R-module with the same basis. The tensor product of bisets induces an R-bilinear map deﬁned on a spanning set by

− ·H −: RB(G, H) × RB(H,K) → RB(G, K), ([X], [Y ]) 7→ [X ×H Y ], where X is a (G, H)-biset. Y is an (H,K)-biset, and [X] denotes the image of the isomorphism class of X in B(G × H). In the case that G = H = K, this map endows RB(G, G) with an associative, generally non-commutative unital ring structure with multiplicative identity [G] := [G × G/∆(G)]. When equipped with this ring structure, we call B(G, G) the (integral) double Burnside ring and RB(G, G) the double Burnside R-algebra. Henceforth, whenever we mention the ring structure of B(G, G) we will mean the multiplication induced by the tensor product of bisets.

The Mackey formula for bisets (1.18) tells us how − ·H − behaves on basis elements, and in particular informs us a bit about the ring structure of B(G, G).

1.20 Corollary. Let G, H and K be ﬁnite groups. For subgroups L ≤ G × H and M ≤ H × K, one has

G × H H × K X G × K · = , L H M L ∗ (h,1)M h∈[p2(L)\H/p1(M)]

where [p2(L)\H/p1(M)] is a set of representatives for the (p2(L), p1(M))-double cosets of H.

Proof. This is immediate from 1.18.

As another application of the Mackey formula for bisets, we can diagonally embed the single Burnside ring B(G) into the double Burnside ring B(G, G).

1.21 Proposition. Let G be a ﬁnite group and H a subgroup. The following map is a ring embedding:

∆: B(G) → B(G, G), ∆([G/H]) := [G × G/∆(H)],

16 where [G/H] is a standard basis element. Proof. Given subgroups H and K of G, note that for x ∈ G,

(x,1) x x ∆(H) ∗ ∆(K) = ∆(H ∩ K, cx,H ∩ K).

Thus from 1.20 X G × G ∆([G/H]) · ∆([G/K]) = G ∆(H) ∗ (x,1)∆(K) x∈[H\G/K] X G × G = x x , ∆(H ∩ K, cx,H ∩ K) x∈[H\G/K]

(1,x−1) x x x x but since ∆(H ∩ K) = ∆(H ∩ K, cx,H ∩ K), we have [G × G/∆(H ∩ K)] = x x [G × G/∆(H ∩ K, cx,H ∩ K)], so X G × G = ∆(H ∩ xK) x∈[H\G/K] = ∆[G/H] · [G/K].

This establishes that ∆ is a ring homomorphism. It is injective since ∆([G/H]) =

∆([G/K]) ⇐⇒ [G × G/∆(H)] = [G × G/∆(K)] ⇐⇒ ∆(H) =G×G ∆(K) ⇐⇒ H =G K ⇐⇒ [G/H] = [G/K].

1.4 Two Ghost Groups

The classical mark homomorphism ρG,G : B(G, G) → ZSG×G (confer A.25) G×G extends to a Q-vector space isomorphism ρG,G : QB(G, G) → (QSG×G) . Thus, the Q-algebra structure of QB(G, G) induces a Q-algebra structure on the G×G-ﬁxed point G×G subspace (QSG×G) . A natural question to ask is whether this Q-algebra structure extends to all of QSG×G. Boltje and Danz [BD13] construct Q-algebra structures on QSG×G related via Möbius inversion on SG×G that allow QB(G, G) to be realized as a condensed algebra eQSG×Ge for an idempotent e ∈ QSG×G. In this section we recapitulate some of the constructions from [BD13].

1.22 Definition. Let G be a finite group. Define the following additive maps: X α: B(G) → ZSG, [X] 7→ stabG(x), x∈X X ζ : ZSG → ZSG,U 7→ V. V ≤U

17 P SG It’s clear that ζ is a Z-module isomorphism with inverse U 7→ V ≤U µU,V · V . Given a commutative ring R, after extended scalars from Z to R we obtain an R-module morphism α: RB(G) → RSS and and R-module isomorphism ζ : RB(G) → RSG. It happens that the classical R-module mark homomorphism

X H ρ: RB(G) → RSG, a 7→ |a | · H

H∈SG factors through α and ζ.

1.23 Proposition (3.2, [BD13]). Let G be a ﬁnite group and R a commutative ring. The following diagram of R-module homomorphisms commutes:

α G RB(G) (RSG) RSG

ζ ρ ζ G (RSG) RSG

Here ρ denotes the classical mark homomorphism. If |G| is a unit in R, then G α: RB(G) → (RSG) is an R-module isomorphism. Additionally, ζ : RSG → RSG G G and ζ :(RSG) → (RSG) are R-module isomorphisms.

1.24 Notation. As in A.25, for a subgroup U of a ﬁnite group G, the orbit sum of U in ZSG + X −1 [U]G := gUg gNG(U)∈G/NG(U) G is an element of (ZSG) . Given a transversal T for the collection of G-conjugacy classes of subgroups of G, the set + [U]G | U ∈ T

G is a Z-basis for (ZSG) called the standard basis. Let G and H be ﬁnite groups and suppose R is a commutative ring. We will

write ρG,H : RB(G, H) → RSG×H , αG,H : RB(G, H) → RSG×H and ζG,H : RSG×H →

RSG×H when referring to the R-module homomorphisms deﬁned in 1.22, with G replaced by G × H.

1.25 Definition (Idempotent elmenets). Let G be a finite group and R a commutative P ring. Define eG := g∈G ∆(G, cg,G) ∈ RSG×G and set e˜G := ζG,G(eG).

18 An important observation is that αG,G : QB(G, G) → QSG is almost multiplicative when QSG×G is taken as a the monoid algebra over (SG×G, ∗), with ∗ the relational composition deﬁned in 1.14. By twisting with a 2-cocycle κ, αG,G becomes κ multiplicative. More generally, product − ∗H − on subgroups that respects the tensor product of bisets under the α maps. By following with the ζ maps, we obtain another κ product − ∗˜H −.

κ κ 1.26 Definition (The products ∗H and ∗˜H ). Let G, H, and K be finite groups and R a commutative ring such that |G| and |H| are units in R. For L ≤ G × H, M ≤ H × K and N ≤ G × K define |k (L) ∩ k (M)| κ(L, M) := 2 1 ∈ R, |H| and two R-bilinear maps:

κ κ − ∗H −: RSG×H × RSH×K → RSG×K , (L, M) 7→ L ∗H M := κ(L, M) · (L ∗ M), κ X L,M − ∗˜H −: RSG×H × RSH×K → RSG×K , (L, M) 7→ aN · N, N≤G×K

L,M where the coeﬃcient aN ∈ R is given by

L,M X SG×H ×SH×K 0 0 aN := µ(L0,M 0),(L,M) · κ(L ,M ). 0 0 (L ,M )∈SG×H ×SH×K , N≤L0∗M 0

S ×S Here, µ G×H H×K denotes the Möbius function on the poset SG×H × SH×K equipped with the usual direct product partial order, i.e., (L0,M 0) ≤ (L, M) if and only if L0 ≤ L and M 0 ≤ M. The results of [BD13, Section 3] establish that:

• κ is a 2-cocycle, i.e., for ﬁnite groups G, H, K, J and subgroups L ≤ G × H, M ≤ H × K, and N ≤ K × J we have

κ(L ∗ M,N) · κ(L, M) = κ(L, M ∗ N) · κ(M,N).

κ • αG,K (a ·H b) = αG,H (a) ∗H αH,K (b) for elements a ∈ RB(G, H), b ∈ RB(H,K).

κ κ • ζG,K (a ∗H b) = ζG,H (a) ∗˜H ζH,K (b) for a ∈ RSG×H , b ∈ RSH×K .

κ κ • (RSG×G, ∗G) and (RSG×G, ∗˜G) are R-algebras with respective multiplicative iden- P tities |G|∆(G) and |G| H≤G ∆(H), and respect idempotents eG and e˜G. More- κ κ over, ζG,G :(RSG×G, ∗G) → (RSG×G, ∗˜G) is an R-algebra isomorphism.

19 G×H κ κ • αG,H (RB(G, H)) = (RSG×H ) = eG∗GRSG×H ∗H eH and also ρG,H (RB(G, H)) = κ κ e˜G ∗˜G QSG×H ∗˜H e˜H . In particular,

G×G κ κ αG,G : RB(G, G) → (RSG×G) = eG ∗G RSG×G ∗G eG, G×G κ κ ρG,G = ζG,G ◦ αG,G : RB(G, G) → (RSG×G) =e ˜G ∗˜G RSG×G ∗˜G e˜G

are R-algebra isomorphisms.

L,M κ The coeﬃcients aN that appear in the expression L ∗˜H M can be simpliﬁed to a degree.

1.27 Deﬁnition. Let G, H, and K be ﬁnite groups and suppose we have subgroups L ≤ G × H, M ≤ H × K, N ≤ G × K. The set

XN := {(L, M) ∈ SG×H × SH×K | N ≤ L ∗ M} is is a convex subposet of the direct product poset SG×H × SH×K . Note that

L,M X XN 0 0 aN = µ(L0,M 0),(L,M) · κ(L ,M ). 0 0 (L ,M )∈XN

Further, deﬁne

YN := {(L, M) ∈ XN | p1(L) = p1(N), p2(L) = p1(M), p2(M) = p2(N)} and for any (L, M) ∈ YN deﬁne

L,M 0 0 0 0 YN := (L ,M ) ∈ YN | (L ,M ) ≤ (L, M) .

1.28 Theorem (4.2, [BD13]). Let L ≤ G × H, M ≤ H × K, and N ≤ G × K for ﬁnite groups G, H, and K. Then as an element of a commutative ring R,  0 if (L, M) ∈/ ,  YN aL,M = N P YN 0 0  µ(L0,M 0),(L,M) · κ(L ,M ) otherwise.  0 0 L,M (L ,M )∈YN

κ In particular it follows that L∗˜H M is an R-linear combination of subgroups N ≤ G×K satisfying p1(N) = p1(L) and p2(N) = p2(M) and is zero unless p2(L) = p1(M).

20 1.29 Proposition (4.4, [BD13]). For ﬁnite groups G, H, K, suppose that L ≤ G×H and M ≤ H × K are either both left-free or both right-free. Then   1 · L ∗ M if p (L) = p (M), κ  |H| 2 1 L ∗˜H M = 0 otherwise.

1.30 Corollary. For ﬁnite groups G and H, suppose L ≤ G × H and V ≤ H are subgroups. Then  0 if V 6= p (L), κ  2 L ∗˜H (V × 1) = L, p2(L)×1 a · (p1(L) × 1) if V = p2(L).  p1(L)×1

Proof. This is just a special case of 1.28 where K = 1 and M = V ×1 for some subgroup κ κ V ≤ H. If V 6= p2(L), then L ∗˜H (V × 1) = 0. Recall that L ∗˜H (V × 1) is a Q-linear combination of subgroups N ≤ G × 1 satisfying p1(N) = p1(L), but the only such N is p1(L) × 1. This establishes the result.

1.5 Subgroups of Burnside Groups

1.31 Definition. Let (D, S) be a pair consisting of a class D of finite groups and a class S of subsets SG,H ⊆ SG×H for each G, H ∈ D. We say (D, S) satisfies condition I provided that for groups G, H, K ∈ D:

1. SG,H is closed under G × H-conjugation and taking subgroups.

2. For L ∈ SG,H and M ∈ SH,K we have L ∗ M ∈ SG,K .

3. ∆(G) ∈ SG,G.

◦ The pair (D, S) satisﬁes condition II provided that for all G, H ∈ D, (SG,H ) = SH,G.

1.32 Deﬁnition (The Burnside group BS (G, H)). Let (D, S) be a pair as in 1.31. For G, H ∈ D, the subgroup of B(G, H) spanned by the transitive bisets [G × H/L] such S that L ∈ SG,H is denoted by B (G, H). Similarly, the subgroup of ZSG×H spanned by S SG,H is denoted by ZSG×H .

21 Two particular cases of interest are when SG,H is either ¡G×H or ∆G×H for G, H ∈ D. The left-free (respectively, bifree) double Burnside group is deﬁned by ¡ S ∆ S B (G, H) := B (G, H) when SG,H = ¡G×H (respectively, B (G, H) := B (G, H)

when SG,H = ∆G×H ). Finally, suppose that F is a fusion system on a p-group S and let S(F) denote the set of twisted diagonal subgroups ∆(ϕ(U), ϕ, U) such that ϕ: U → ϕ(U) is an F- morphism. When D = {S} and S = {S(F)}, we write BF (S, S) := BS (S, S). It’s easy to verify in this case that (D, S) satisﬁes conditions I and II. Conversely, if some

arbitrary pair ({S} , S) satisﬁes condition I and II, and moreover SS,S ⊆ ∆S×S, then the set of group homomorphisms induced from S form a fusion system F(S) on S.

1.33 Remark. Given a pair (D, S) as in 1.32 that satisﬁes condition I, the Mackey formula for the multiplication of basis elements (1.20) implies that the tensor product of bisets induces R-bilinear maps

S S S − ·H −: RB (G, H) × RB (H,K) → RB (G, K)

for a commutative ring R and groups G, H, K ∈ D.

When (D, S) satisﬁes condition I and SG,H ⊆ ¡G×H for each pair of ﬁnite groups G, H ∈ D, and |G| is a unit in R for each G ∈ D, the classical mark homomor-

phisms ρG,H restrict to give R-module isomorphisms

S S S G×H ρG,H : RB (G, H) → (RSG×H )

κ that respect the R-bilinear maps ∗˜H , in the sense that

S S κ S ρG,H (a ·H b) = ρG,H (a) ∗˜H ρH,K (b)

for a ∈ RBS (G, H), b ∈ RBS (H,K), and G, H, K ∈ D. In particular, we have an R-algebra isomorphism

S S S κ G×G ρG,G : RB (G, G) → (RSG×G, ∗˜G) .

This recapitulates [BD12, Theorem 4.7], although in that work the mark homomorphism and multiplication have additional scaling factors, which we now deﬁne.

1.34 Definition. Suppose (D, S) satisfies condition I and SG,H ⊆ ¡G×H for each pair of finite groups G, H ∈ D. Given groups G, H, K in D, define the R-bilinear map

S S S κ − ·H −: RSG×H × RSH×K → RSG×K , (L, M) 7→ |CH (p2(L))| · L ∗˜H M

22 and R-module homomorphism

L S 0 S S X |a | (ρG,H ) : RB (G, H) → RSG×H , a 7→ L. |CG(p1(L))| L∈SG×H

By [BD12, Theorem 4.7],

S 0 S 0 S 0 (ρG,H ) (a ·H b) = (ρG,H ) (a) ·H (ρH,K ) (b)

S S for a ∈ RB (G, H), b ∈ RB (H,K). By [BD12, Lemma 4.5], the R-bilinear map −·H − restricts to the underlying Z-submodules of ﬁxed points, that is, we have an injective bilinear map

S G×H S H×K S G×K − ·H −:(ZSG×H ) × (ZSH×K ) → (ZSG×K ) .

S G×G S In particular, (ZSG×G) serves as an integral ghost ring for B (G, G). If |G| is S S G×G invertible in R, then RB (G, G) and ((RSG×G) , ·G) are isomorphic R-algebras. ˜S S G×H To ease notation somewhat, we set RB (G, H) := (RSG×H ) . When G = S H, we view RB˜ (G, G) as an R-algebra under the multiplication − ·G −.

S 1.35 Remark (An N-grading on B (G, G)). Suppose (D, S) satisﬁes condition I and

that SG,H ⊆ ¡G×H for each G, H ∈ D, and set ∆(S) := SG,H ∩ ∆G×H . Let R denote a

commutative ring. For a non-negative integer n, SG,H,n denotes the set of the L ∈ SG,H S such that the composition length of k2(L) is equal to n. The R-submodule RSG×H,n of ˜S S G×H RSG×H spanned by SG,H,n is G × H-invariant, so RBn (G, H) := (RSG×H,n) is an S R-submodule of RB˜ (G, H). Since SG,H is a disjoint union of the subsets SG,H,n, we have ˜S M ˜S RB (G, H) = RBn (G, H). n∈N This direct sum decomposition respects the tensor product of subgroups [BD12, Lemma ˜S ˜S 6.2], and so we obtain an N-grading of the R-algebra RB (G, G). In particular, RB0 (G, G) = ∆(S) RB˜n (G, G). S S L For n ∈ N, let RBn (G, H) be the set of a ∈ RB (G, H) such that |a | = 0 unless L ∈ SG,H,n. We obtain a direct sum decomposition

S M S RB (G, H) = RBn (G, H), n∈N

23 S 0 and the mark homomorphism (ρG,H) restricts to an R-module isomorphism

S 0 S ˜S (ρG,H ) : RBn (G, H) → RBn (G, H) provided that |G × H| is a unit in R. As the mark homomorphism respects the tensor S product of bisets, when G = H we have an N-grading of the R-algebra RB (G, G).

1.6 Biset Categories

Throughout this section G denotes a transversal for the isomorphism classes of ﬁnite groups. Biset functors are studied in much more detail in [Bou96; Bou10].

1.36 Deﬁnition (Biset category). For a commutative ring R, the R-biset category BR has as objects the set G, and for two objects G, H we deﬁne HomBR (G, H) := RB(H,G), with morphism composition induced from the tensor product of bisets. That is,

− ◦ −: HomBR (H,K) × HomBR (G, H) → HomBR (G, K) is given by − ·H −RB(K,H) ×H RB(H,G) → RB(K,G). Given a pair (D ⊆ G, S) as in 1.32 that satisﬁes condition I we obtain a (not S necessarily full) subcategory BR of BR whose object class is D and whose morphisms are the Burnside R-submodules RBS (G, H), for G, H ∈ D. ˜S Under the same notation and assumptions above, we obtain categories BR deﬁned in the same way but whose morphism sets are the ghost R-modules RB˜S (G, H).

1.37 Deﬁnition (The category A). Given a transversal G for the isomorphism classes of ﬁnite groups, we let A denote the category whose object class is G and whose morphism sets are given by

HomA(G, H) := ZSH×G, with composition L ◦ M := L ∗ M for L ∈ HomA(H,K) and M ∈ HomA(G, H).

1.38 Deﬁnition (Twisted category algebras). Given a small category C and a commutative ring R, the category algebra RC is the free R-module with basis Mor(C) equipped with an R-algebra structure induced from morphism composition in C. That is, for C-morphisms a and b, we deﬁne a · b := a ◦ b provided that a ◦ b exists and 0 otherwise.

24 A 2-cocycle α of C with values in R× is a map α that assigns to any pair of morphisms a and b such that a ◦ b exists a unit α(a, b) in R satisfying: for any three morphisms a, b, c in C such that a ◦ b and b ◦ c exist, α(a ◦ b, c)α(a, b) = α(a, b ◦ c)α(b, c).

The twisted category algebra RαC is the free R-module whose basis is the collection of C-morphisms and whose R-algebra structure is deﬁned by  α(a, b) · (a ◦ b) if a ◦ b exists, a · b := 0 otherwise. In the case that C is the category A introduced in 1.37 and α is the 2-cocycle κ introduced in 1.26, we set A := QκA. Additionally, we set B := QB. Note in particular that A and B have the Q-module decompositions M A = QSG×H G,H∈G M B = QB(G, H). G,H∈G κ The Q-bilinear maps − ∗˜H − also endow A with a diﬀerent Q-algebra structure via   κ a ∗˜H b if a ∈ QSG×H , b ∈ QSH×K , a ∗˜κ b := 0 otherwise.

When we view the Q-module A as a Q-algebra in this way we denote it by A˜. The maps ˜ ζG,H induce a Q-algebra isomorphism ζ : A → A.

1.39 Notation. Given a preadditive category C and a commutative ring R, we let CR denote the category with the same object class as C, but whose morphism sets are the R-modules

HomCR (X,Y ) := R ⊗Z HomC(X,Y ), where X and Y are objects.

1.40 Deﬁnition (Biset functor). For us, a (covariant) R-biset functor is an R-linear functor F : BR → R- Mod, where F is said to be R-linear provided the induced map

HomBR (G, H) → HomR(F (G),G(H)), f 7→ F (f) is R-linear. Associated to every R-biset functor is an RB-module M MF := F (G) G∈G

25 where a ∈ RB(G, H) acts on m ∈ F (K) via  0 if K 6= H, a · m := [F (a)](m) if K=H. L Conversely, if M is an RB-module such that M = G∈G 1G · M, where 1G is the multiplicative identity [G × G/∆(G)] of RB(G, G), we obtain an R-biset functor FM by setting FM (G) := 1G · M and sending a ∈ HomBR (H,G) to the “multiplication by a” map FM (a): 1H · M → 1G · M, 1H · m 7→ a · m.

The R-Burnside functor is the Yoneda functor HomBR (1, −) that assigns to each object G the single Burnside R-algebra RB(G), after canonically identifying RB(G, 1) with RB(G). We will primarily be interested in the case when R = Q. The above constructions apply to any category C whose morphism sets have an R-module structure such that the composition of morphisms is R-bilinear. In particular,

Q-linear functors from AQ to Q-Mod are in correspondence to a certain class of A- modules, and consequently to a certain class of A˜-modules. One principle object of our investigations is the Yoneda functor := Hom (1, −) and the associated A˜- QS AQ module.

26 Chapter 2

Biset Categories

Motivated by the question of whether the left-free p-local biset categroy B¡p has the same isomorphism classes as B∆p , we show this is indeed the case for the class of finite groups that can be written as a direct product of their Sylow p-subgroup and a Hall p’-subgroup. More generally, we establish that an isomorphism in B¡p implies an isomorphism in Be∆p . In section 2.2, we show that if two groups G and H have isomorphic Frobenius p-fusion systems, then they are isomorphic in B∆p . Note that in 1.36 we always assumed that biset categories were small by restricting the object class to a transversal for the isomorphism classes of finite groups, which is a set. This was so we could later consider the free Q-module generated over the set of all morphisms. Suppose that C is defined in the same way as B but has as objects the class of finite groups. Using the Mackey formula (1.20), it is easy to see that

if ϕ: H → G is a group isomorphism, then [G × H/∆ϕ(H)] is a C-isomorphism between H and G. From this we can conclude that B and C have isomorphic skeletons, hence are equivalent categories. To avoid having to reference the particular choice of representatives for group isomorphism classes, we assume in this chapter that all biset categories and related categories have as objects the class of ﬁnite groups, which we denote by G.

2.1 Notation. By ¡ and ∆ we mean the classes of left-free subgroups and bifree subgroups. That is,

¡ := {¡G×H | G, H ∈ G} ,

∆ := {∆G×H | G, H ∈ G} .

For a prime number p and pair G, H ∈ G, let ¡p,G×H and ∆p,G×H denote the subsets

27 of ¡G×H and ∆G,H consisting of p-subgroups. Then ¡p and ∆p are deﬁned to be the classes

¡p := {¡p,G×H | G, H ∈ G} ,

∆p := {∆p,G×H | G, H ∈ G} .

¡p ¡p ∆p ∆p The pairs (G, ¡p) and (G, ∆p) give rise to categories B := B , B := B , and Z(p) Z(p) ∆p ∆p B := B as per 1.36, though some care is needed since ¡p,G×G and ∆p,G×G do not e Z(p) contain ∆(G) when G is not a p-group. That B¡p and B∆p actually possess identity morphisms follows from Dress’s result in A.33. In particular, for G ∈ G, the element p := P λ U in where λ = 0 if U is not a p-group and λ = 1 if U is a p-group U∈SG U ZSG U U is in the image of the classical mark homomorphism Φ: Z(p)B(G) → Z(p)SG. Setting −1 ep := Φ (p), it turns out that the diagonal embedding ∆: Z(p)B(G) → Z(p)B(G, G)

¡p sends ep to the multiplicative identity element of Z(p)B (G, G).

In this chapter, ρG,H denotes the mark homomorphism as deﬁned in 1.34.

2.1 A Special Class of Groups

2.2 Lemma. If ﬁnite groups G and H are isomorphic in B¡, then they are isomorphic as groups.

∼ ¡ ¡ Proof. Suppose that G = H in B . Then there is some a ∈ Z(p)B (G, H) and ¡ ¡ b ∈ Z(p)B (H,G) such that a ·H b = 1G in B . As ∆(G) appears as a summand of ρG,G(1G) = ρG,G(a ·H b) = ρG,H (a) ·H ρH,G(b), there must be some left-free subgroups L ≤ G × H and M ≤ H × G such that L ∗ M = ∆(G). This only occurs when G is isomorphic to a subgroup of H. It follows by symmetry that H is isomorphic to a subgroup of G, hence G =∼ H as groups.

2.3 Lemma. If p-groups P and Q are isomorphic in B¡p , then they are isomorphic as groups.

¡p ¡ Proof. In this case, Z(p)B (P,Q) = Z(p)B (P,Q) so P and Q are in fact isomorphic in B¡, and the result follows from 2.2.

28 2.4 Proposition. Suppose G, H ∈ G are p-nilpotent and have normal Sylow p- subgroups P and Q, respectively. Then the following are equivalent:

1. G =∼ H in B¡p .

2. P =∼ Q in B¡p .

3. P =∼ Q as groups.

4. G =∼ H in B∆p .

Proof. Given any ﬁnite group G with Sylow p-subgroup P , let |G|p0 := |G|/|P | denote the p0 part of |G|. A group G is p-nilpotent with a normal Sylow p-subgroup precisely when G can be written as a direct product of its Sylow p-subgroup and a Hall p0-subgroup.

Suppose that G = Gp × Gp0 , H = Hp × Hp0 , and K = Kp × Kp0 are p-nilpotent groups 0 with respective Sylow p-subgroups Gp, Hp, Kp, and respective Hall p -subgroups Gp0 ,

Hp0 , Kp0 . Deﬁne

¡p ¡p 1 iG,H : Z(p)B (Gp,Hp) → Z(p)B (G, H), [Gp × Hp/L] 7→ [G × H/L], |G|p0

Since the Sylow p-subgroups of G and H are normal, if ¡(U, α, V ) is a p-subgroup of

G × H, then ¡(U, α, V ) ≤ Gp × Hp so the map is surjective. As Gp0 ≤ CG(Gp) and

Hp0 ≤ CH (Hp), the G×H-conjugacy class of ¡(U, α, V ) is equal to the Gp×Hp-conjugacy class of ¡(U, α, V ), so this map is injective.

If A and B are subgroups of Hp, then

A\H/B = A(hp, hp0 )B | hp ∈ Hp, hp0 ∈ Hp0 = AhpB × hp0 | hp ∈ Hp, hp0 ∈ Hp0

= A\Hp/B × Hp0 .

In other words, each (A, B)-double coset of Hp gives rise to |H|p0 (A, B)-double cosets of H. Since Hp ≤ CH (Hp0 ) if T is a transversal for the (A, B)-double cosets of H and P h P r T is a transversal for the (A, B)-double cosets of H , then U = |H| 0 U p p h∈T p r∈Tp for any U ≤ Hp. In particular, suppose L ∈ ¡G×H and M ∈ ¡H×K are p-groups and let T be a transversal for the (p2(L), p1(M))-double cosets of H and Tp a transversal for

29 the (p2(L), p1(M)-double cosets of Hp. Then 1 iG,H ([Gp × Hp/L]) ·H iH,K ([Hp × Kp/M]) = [G × H/L] ·H [H × K/M] |G|p0 |H|p0 1 X = [G × K/(L ∗ (h,1)M)] |G|p0 |H|p0 h∈T |H|p0 X = [G × K/(L ∗ (q,1)M)] |G|p0 |H|p0 q∈Tp X (q,1) = iG,K ([Gp × Kp/(L ∗ M)])

q∈Tp

= iG,K ([Gp × Hp/L] ·Hp [Hp × Kp/M]).

This shows that the collection of maps {iG,H | G, H ∈ G} respect the tensor product of

¡p bisets. In particular, if γ : Hp → Gp is a B -isomorphism, then iG,H (γ): H → G is a

¡p B -isomorphism. The same statements hold when ¡p is replaced with ∆p. (1 =⇒ 3). In this case, an argument similar to 2.2 shows that P =∼ Q as groups. (3 =⇒ 4). As in the statement of the problem, let P and Q denote the Sylow p-subgroups of G and H, respectively. If ϕ: Q → P is a group isomorphism, then

∆p [P × Q/∆ϕ(Q)] is a B -isomorphism between Q and P , so iG,H ([P × Q/∆(P, ϕ, Q)]) is a B∆p -isomorphism between G and H, as desired. (4 =⇒ 1). This follows from the deﬁnitions of the relevant categories. (2 ⇐⇒ 3). The claim is an immediate consequence of 2.3.

¡p 2.5 Notation. Recall that Z(p)Be (G, H) has a N-grading respecting the tensor prod-

¡p uct of subgroups (1.35). We may decompose any a ∈ Z(p)Be (G, H) via a = a0 + a+,

∆p L ¡p where a0 ∈ Z(p)Be (G, H) and a+ ∈ n>0 Z(p)Ben (G, H).

2.6 Proposition. If two ﬁnite groups G and H are isomorphic in B¡p , then they are isomorphic in Be∆p and have isomorphic Sylow p-subgroups.

∼ ¡p Proof. Let P ∈ Sylp(G) and Q ∈ Sylp(H) and suppose that G = H in B . This means

¡p 0 ¡p 0 there exists γ ∈ Z(p)B (G, H) and γ ∈ Z(p)B (H,G) such that γ ·H γ = 1G and 0 ¡p 0 ¡p 0 γ ·G γ = 1H . Set γe := ρG,H (γ) and γe := ρH,G(γ ). By an argument essentially identical to the proof of 2.2, we can conclude that P =∼ Q as groups. Of course, G =∼ H in Be¡p

30 via γe, so

0 0 0 0 0 0 e1G = (γe0 + γe+) ·H (γe0 + γe+) = (γe0 · γe0) + (γe0 · γe+ + γe+ · γe0 + γe+ · γe+).

0 ∆p 0 0 0 L ¡p But e1G, γe0 · γe0 ∈ Z(p)Be (G, G) while γe0 · γe+ + γe+ · γe0 + γe+ · γe+ ∈ n>0 Z(p)Ben (G, G), 0 0 ∼ which implies that γe0 ·H γe0 = e1G. Similarly, we can deduce that γe0 ·G γe0 = e1H , so G = H in Be∆p .

2.2 Fusion Preserving Isomorphisms

Our goal in this section is to show that if G and H are ﬁnite groups with respective Sylow p-subgroups P and Q, and α: Q → P is an isomorphism between the

Frobenius p-fusion systems FQ(H) and FP (G), then G and H are isomorphic in the category B∆p .

2.7 Deﬁnition (Fusion preserving isomorphism). Suppose F is a fusion system on a p-group S and E is a fusion system on a p-group T .A fusion preserving isomorphism from S to T or an isomorphism between F and E is a group isomorphism α: S → T such that whenever ϕ ∈ HomF (P,Q), the map

αϕ: α(P ) → α(Q), α(x) 7→ α(ϕ(x))

is in HomE (α(P ), α(Q)). This deﬁnition generalizes to the case where α is not injective so long as each F-morphism preserves ker α, which is to say that the map αϕ: α(P ) → α(Q) is well-deﬁned provided ϕ(P ∩ ker α) ≤ ker α. We won’t need to make use of this more general concept of a morphism between F and E.

2.8 Notation. Given a ﬁnite group G and P ∈ Sylp(G), let SfG,P denote a transversal

for the G-conjugacy classes of p-subgroups of G such that each element of SfG,P is a

∆p subgroup of P . The identity morphism e1G ∈ Z(p)Be (G, G) is the element

X + e1G := [∆(U)]G×G.

U∈SeG,P

2.9 Lemma. Let G and H be ﬁnite groups with respective Sylow p-subgroups P and

Q, and suppose α: Q → P is an isomorphism of the fusion systems FQ(H) and FP (G).

31 ∆p The element γeG,H,α ∈ Z(p)Be (G, H) deﬁned by

X + γeG,H,α := [∆α(V )]G×H V ∈SeH,Q

∆p is an isomorphism in Z(p)Be (G, H) between H and G with inverse

X + γeH,G,α−1 := [∆α−1 (α(V ))]H×G. V ∈SeH,Q

−1 Proof. Since α is a fusion preserving isomorphism, AutH (V ) = α AutG(α(V ))α for any V ≤ Q. From this we have

Nα−1 = p1(NG×H (∆α(V ))) −1 = g ∈ NG(α(V )) | α cgα ∈ AutH (V )

= NG(α(V ))

and

p2(NH×G(∆α−1 (α(V )))) = p1(NG×H (∆α(V )))

= NG(α(V ))

for any V ≤ Q. Given a section (V,U) of H let CH (V,U) denote the collection of elements

h ∈ NH (U) ∩ NH (V ) such that ch induces the identity morphism on V/U. Since α is

an isomorphism, CH (V, ker α) = CH (V ) for any V ≤ Q. Let A be a transversal for

G/NG(α(V )). From [BD12, Lemma 4.5], for any V ≤ Q we have that

+ + a := [∆α(V )]G×H ·H [∆α−1 (α(V ))]H×G = X (g,k) −1 −1 ∆(α(V ), αch α , α(V )) = (g,ch,k)∈A×AutH (V )×A X (g,k) ∆(α(V ), cx, α(V ))

(g,cx,k)∈A×AutG(α(V ))×A + is an integral multiple of [∆(α(V ))]G×G. Set L := NG×G(∆(α(V ))). The G × G-orbit of ∆(α(V )) contains |G|2 |G|2 [G × G : L] = = |k1(L)| · |p2(L)| |CG(α(V ))| · |NG(α(V ))| |G| |N (α(V ))| |G| = · G · |NG(α(V ))| |CG(α(V ))| |NG(α(V ))|

= |A × AutG(α(V )) × A|

32 + −1 elements, forcing a = [∆(α(V ))]G×G. Symmetrically, for any U ≤ P with V := α (U) we can deduce

−1 + + + [∆(V, α |U ,U)]H×G ·G [∆(U, α|V ,V )]G×H = [∆(V )]H×H . Then     X + X + γG,H,α ·H γH,G,α−1 =  [∆α(V )]  ·  [∆α−1 (α(V ))]  e e  G×H  H  H×G V ∈SeH,Q V ∈SeH,Q X + + = [∆α(V )]G×H ·H [∆α−1 (α(V ))]H×G

V ∈SeH,Q X + = [∆(α(V ))]G×G

V ∈SeH,Q X + = [∆(U)]G×G

U∈SeG,P

= e1G.

Similarly,

γeH,G,α−1 ·G γeG,H,α = e1H .

2.10 Lemma. Let G and H be ﬁnite groups with respective Sylow p-subgroups P

and Q, and suppose α: Q → P is an isomorphism of the fusion systems FQ(H) and 0 FP (G). For any subgroups V and V of Q, the following numbers are equal:

0 0 1. The number nH (V,V ) of H-conjugates of V contained in V .

0 0 2. The number nG(α(V ), α(V )) of G-conjugates of α(V ) contained in α(V ).

0 3. The number nG×H (∆α(V ), ∆α(V )) of G × H-conjugates of ∆α(V ) contained in 0 ∆α(V ).

Proof. Set F := FQ(H) and E := FP (G). Since α is an isomorphism between F and E, it follows that | HomF (R,S)| = | HomE (α(R), α(S))| for any R,S ≤ Q, so in particular

| AutH (V )| = | AutG(α(V ))| and

33 0 0 This establishes nH (V,V ) = nG(α(V ), α(V )).

Now suppose that {V1,...,Vn} is the collection of H-conjugates of V contained 0 0 in V . Let X denote the collection of G × H-conjugates of ∆α(V ) contained in ∆α(V ). We claim that

X = Y := {∆α(V1),..., ∆α(Vn)} .

Since α is an isomorphism of F and E, it follows that {α(V1), . . . , α(Vn)} is the collection 0 h of G-conjugates of α(V ) in α(V ). For each Vk there is some h ∈ H such that V = Vk −1 g and some g ∈ G such that cg = αchα |α(V ). Note in particular that α(V ) = α(Vk). −1 Moreover, cgαch |Vk = α|Vk so

(g,h) 0 ∆α(V ) = ∆α(Vk) ≤ ∆α(V ).

(g,h) 0 h This shows Y ⊆ X. On the other hand, if ∆α(V ) ≤ ∆α(V ) then V = Vk for some −1 (g,h) k ∈ {1, . . . , n} and cg = αchα , so cg(α(V )) = α(Vk), whence ∆α(V ) = ∆α(Vk). 0 0 Therefore X ⊆ Y . This establishes that nG×H (∆α(V ), ∆α(V )) = |X| = nH (V,V ), as desired.

2.11 Lemma. Let G and H be ﬁnite groups with respective Sylow p-subgroups P

and Q, and suppose α: Q → P is an isomorphism of the fusion systems FQ(H) and

FP (G). Let {V1,...,Vn} denote a transversal of the FQ(H)-isomorphism classes. Then

the matrix representation for the Z(p)-linear map

∆p ∆p ∆p ρG,H : Z(p)B (G, H) → Z(p)Be (G, H)

with respect to the standard bases {[G × H/∆α(Vi)] | i = 1, . . . , n} and

{∆α(Vi) | i = 1, . . . , n} is the corresponding subtable of marks for H.

Proof. Set F := FQ(H) and E := FP (G). We must show that

∆p (ρG,H )ij = Mij, i, j ∈ {1, . . . , n} ,

where M is the subtable of marks for H corresponding to {V1,...,Vn}. Recall that V Mij = |(H/Vj) i | = |NH (Vi,Vj)|/|VJ |. Since α is an isomorphism of F and E, for every −1 V ≤ Q it follows that α AutH (V )α = AutG(α(V )) and moreover p2(NG×H (∆α(V ))) =

NH (V ). Fix some i, j ∈ {1, . . . , n}. By invoking 2.10 to justify the third equality below

34 we obtain:

∆p |NG×H (∆α(Vi), ∆α(Vj))| (ρG,H )ij = |CG(α(Vi))| · |Vj| n (∆ (V ), ∆ (V )) · |N (∆ (V ))| = G×H α i α j G×H α i |CG(α(Vi))| · |Vj| n (V ,V ) · |C (α(V ))| · |N (V )| = H i j G i H i |CG(α(Vi))| · |Vj| |N (V ,V )| = H i j |Vj|

= Mij, as desired.

2.12 Theorem. If G and H have isomorphic Frobenius p-fusion systems, then G and H are isomorphic in the category B∆p .

Proof. Suppose P ∈ Sylp(G) and Q ∈ Sylp(H). If α: Q → P is a fusion preserv-

ing isomorphism between FQ(H) and FP (G), then in light of 2.11 and Dress’s result concerning idempotents of single Burnside rings recorded in A.33, both of the

∆p ∆p elements γeG,H,α ∈ Be (G, H) and γeH,G,α−1 ∈ Be (H,G) as deﬁned in 2.9 are in ∆p ∆p ∆p the images of the mark homomorphisms ρG,H : Z(p)B (G, H) → Z(p)Be (G, H), and ∆p ∆p ∆p ∆p −1 ρH,G : Z(p)B (H,G) → Z(p)Be (H,G), respectively. Thus (ρG,H ) (γeG,H,α) is an isomorphism between H and G in B∆p .

35 Chapter 3

Characteristic Idempotents

By using Möbius inversion, we can recover an explicit inverse for the mark S homomorphism ρG,H defined in 1.34. In this chapter we utilize this inverse to find various expressions for the characteristic idempotent ωF of a fusion system F, and use these expressions to show that the characteristic idempotents of unsaturated fusion systems don’t necessarily respect bideflation. In particular, we construct an unsaturated fusion system F on the elementary abelian 3-group of order 27 and quotient system F/T such S that BidefT (ωF ) 6= ωF/T .

3.1 Calculating the Characteristic Idempotent

3.1 Notation. Recall that when G and H are ﬁnite groups, ¡G×H denotes the set of left-free subgroups of G × H and ∆G×H denotes the set of bifree subgroups of G × H. + + Also, for L ∈ SG×H , [L]G×H or simply [L] denotes the G × H-orbit sum of L. Given a fusion system F on a ﬁnite p-group S, recall that S(F) denotes the collection of twisted diagonal subgroups ∆(U, ϕ, V ) ∈ ∆S×S such that ϕ is an F- isomorphism. We let S](F) denote a transversal for the S × S-conjugacy classes of S(F).

3.2 Deﬁnition (The idempotent ωF ). Let F be an arbitrary fusion system on a ﬁnite p-group S and set S := S(F). The characteristic idempotent ωF of F is the element in

In [BD12, Section 7], Boltje and Danz construct a commuting triangle of bijections

f Fus(S) Idem(S)

h g Sys(S)

where Fus(S) is the set of fusion systems on S, Idem(S) is a set of idempotents in ∆ QB (S, S) satisfying certain technical conditions, and Sys(S) is the set of subsets of

∆S×S satisfying conditions I and II in 1.32. The map f takes a fusion system F to its characteristic idempotent ωF , the map h takes a fusion system F to the system S(F), and the map g takes an idempotent to the corresponding ﬁxed point system. These bijections extend bijections obtained by Ragnarsson and Stancu in [RS13] between saturated fusion ∆ systems and certain idempotents in Z(p)B (S, S). Consequently, F is saturated if and ∆ only if ωF ∈ Z(p)B (S, S).

3.3 Deﬁnition. Let F be a fusion system on a p-group S. For L := ∆(β(W ), β, W ) F in S(F), deﬁne the rational number aL by

nV µSS F |W | · |S| X β W,V aL := , |Nβ| · |CS(β(W ))| | HomF (V,S)| W ≤V ≤S

V where nβ is the number of F-morphisms in HomF (V,S) that extend β, i.e., the cardinality of

{α ∈ HomF (V,S) | α|W = β} .

37 3.4 Lemma (4.8, [BD12]). Let G and H be ﬁnite groups and L = ¡(U, ϕ, V ) be an

element of ¡G×H . Then ¡ −1 + |CG(U)| X SH G × H (ρG,H ) ([L] ) = |W | · µW,V . |NG×H (L)| ¡(ϕ(W ), ϕ|W ,W ) W ≤V

The next proposition establishes that the coefficients of ωF in terms of the ∆ F standard basis for QB (S, S) are precisely the numbers aL as defined in 3.3. Using this fact we can recapitulate a result already in the literature that the S × S/1-coefficient of

ωF is 0.

3.5 Proposition. Let F be a fusion system on a p-group S. Let S˜ be a transversal of the S × S-orbits of the system S := S(F). Then

X F ωF = aL · [S × S/L]. L∈S˜

F In particular, when S 6= {1} we have a1 = 0, where 1 is the trivial subgroup of S × S.

∼ Proof. For any L ≤ G × H, since L/(k1(L) × k2(L)) = pi(L)/ki(L) for i ∈ {1, 2} it follows that |L| = |k1(L)|·|p2(L)|. In the case that L = NS×S(∆(β(W ), β, W )) for some subgroup W ≤ S and isomorphism β : W → β(W ), we have k1(L) = CS(β(W )) and

−1 p2(L) = h ∈ NS(W ) | ∃g ∈ NS(β(W )) : ch = β cgβ = Nβ.

From the deﬁnitions of ρS,S and ωF ,

X |S| ρS,S(ωF ) = ∆(U, α, V ). |CS(U)| · | HomF (U, S)| ∆(U,α,V )∈S

In light of the inversion formula for ρS,S (3.4), we have

−1 X |S| · [S × S : NS×S(L)] −1 + ωF = · ρS,S([L]S×S) |CS(p1(L))| · | HomF (p1(L),S)| L∈S

X |S| 1 X SS = · |M|µp (M),p (L)[S × S/M], | HomF (p1(L),S)| |S × S| 1 1 L∈S M≤L

38 and by changing notation slightly, X 1 X SS S × S = |W |µW,V , |S| · | HomF (U, S)| ∆(α(W ), α|W ,W ) ∆(U,α,V )∈S W ≤V

so after changing the order of summation,

SS X |W | X X µW,V S × S = |S| | HomF (V,S)| ∆(α(W ), α|W ,W ) W ≤S W ≤V ≤S α∈HomF (V,S) SS V X |W | X X µW,V nβ S × S = |S| | HomF (V,S)| ∆(β(W ), β, W ) W ≤S β∈HomF (W,S) W ≤V ≤S SS V X |W | X µW,V nβ |S × S| S × S = · |S| | HomF (V,S)| |NS×S(L)| L L∈S˜, W ≤V ≤S L:=∆(β(W ),β,W ) V SS X |W | · |S| X nβ µW,V S × S = |Nβ| · |CS(β(W ))| | HomF (V,S)| L L∈S˜ W ≤V ≤S X F = aL · [S × S/L], L∈S˜

F as desired. Now assume that S 6= {1}. To see a1 = 0 it suﬃces to recall that

X SS µ1,V = 0. V ∈SS

3.6 Corollary. Let F be a fusion system on a p-group S such that every F-morphism α extends to S. Let S](F) be a transversal of the S × S-orbits of the system S(F). Then X |S| S × S ωF = . |Z(S)| · | AutF (S)| ∆(S, β, S) ∆(S,β,S)∈S^(F) In particular, if S is abelian and F is saturated, then

X 1 S × S ωF = . | AutF (S)| ∆(S, β, S) ∆(S,β,S)∈S(F)

Proof. For any subgroup V of S deﬁne n(V ) := | {α ∈ AutF (S) | α|V = IdV } | and S note that n(V ) = nβ for any β ∈ HomF (V,S). By assumption, HomF (V,S) = | AutF (S)| {α|V | α ∈ AutF (S)} and so | HomF (V,S)| = n(V ) . Now suppose that W ≤ V ≤ S

39 V and let β ∈ HomF (W, S). Then nβ = n(W )/n(V ) and so

V SS SS X nβ µW,V X n(W )n(V )µW,V = | HomF (V,S)| n(V ) · | AutF (S)| W ≤V ≤S W ≤V ≤S

n(W ) X SS = µW,V | AutF (S)| W ≤V ≤S  0 if W 6= S, =  1 if W = S.  | AutF (S)|

Thus X |S| S × S ωF = , |Z(S)| · | AutF (S)| ∆(S, β, S) ∆(S,β,S)∈S^(F) as desired. Now suppose that S is abelian and that F is saturated. Then S×S acts trivially on S(F) hence S](F) = S(F). Moreover, if W is a subgroup of S and β ∈ HomF (W, S) then β(W ) is normal in S and so is fully F-normalized. Hence by the Extension Axiom for saturated fusion systems, β extends to Nβ = S. The claim follows from the previous arguments.

As a consequence of 3.6 we can deduce that ωFS (S) is the multiplicative identity of B(S, S).

3.7 Corollary. Let S be a p-group and set F := FS(S). Then S × S ω = . F ∆(S)

Proof. By assumption, given some α ∈ HomF (S, S) there is an s ∈ S such that (s,1) ∆(S, α, S) = ∆(S, cs,S) = ∆(S). Thus we can take S](F) = {∆(S)}. Clearly every morphism in F extends to S, so after recalling that AutF (S) = Inn(S), the result is evident from 3.6.

3.8 Example. We presently compute ωF when F is a saturated fusion system on

S := D8, the dihedral group with 8 elements. There are three saturated fusion systems

on S [Lin07]. Namely, the saturated fusion systems FS(S), FS(S4) where S4 is the

symmetric group on four letters, and FS(SL(3, F2)).

40 ∼ Recall that Aut(S) = S so AutF (S) is a 2-group. If F is a saturated fusion system on S, then AutS(S) is a Sylow 2-subgroup of AutF (S) and so it must follow that

AutF (S) = AutS(S).

Case 1: The case where F = FS(S) was computed already in 3.7.

Case 2: Next we consider when F = FS4 (S). Given

S = r, s | r4 = s2 = 1, srs = r−1 we can embed S into S4 via the identiﬁcation r = (1234) and s = (12)(34). The subgroup Q := h(12)(34), (13)(24)i is maximal in S and normal in S4 since it contains all elements of cycle type (2, 2). Since S4 acts transitively by conjugation on the non-identity ∼ ∼ elements of Q we know that AutF (Q) = Aut(Q) = Aut(V4) = S3. The subgroups

R := h(13)(24), (13)i and T := h(1234)i both have three conjugates in S4 so by an

order argument NS4 (R) = S = NS4 (T ), which implies that AutF (R) = AutS(R) and

AutF (T ) = AutS(T ). We calculate the coeﬃcients of ωF and obtain −1 S × S 1 S × S S × S ωF = + + . 3 ∆(Q) 3 ∆(Q, c(12),Q) ∆(S)

Case 3: Let G := GL(3, F2) = SL(3, F2) and set F := FS(G). As before present S = r, s | r4 = s2 = 1, srs = r−1 . Write Q := s, r2 , R := rs, r2 and T := hri. Recall that the two copies of Klein’s four group in S, namely Q and R, are not conjugate in G whence HomF (Q, S) = AutF (Q) and HomF (R,S) = AutF (R). Also, G acts transitively by conjugation on the subgroups of order 2 in S, from which we can ∼ ∼ conclude that AutF (Q) = S3 = AutF (R). Moreover, if W is a subgroup of S such that

|W | = 2, then | HomF (W, S)| = 5. Let α be the F-automorphism of Q which switches hsi with r2 and ﬁxes sr2 . Let γ be the F-automorphism of R which switches hrsi with r2 and ﬁxes hsri.

Let ζ be the element of HomF (hsi ,S) with codomain hrsi. We can compute ωF from the table below.

41 nV µ ∆(β(W ), β, W ) |W |·|S| P β W,V aF |Nβ |·|CS (β(W ))| W ≤V ≤S | HomF (V,S)| ∆(β(W ),β,W ) ∆( r2 ) (2 · 8)/(8 · 8) = 1/4 8/15 2/15 ∆(hsi) (2 · 8)/(4 · 4) = 1 −2/15 −2/15 ∆(hrsi) see above see above −2/15 2 ∆( r , α|hsi, hsi) (2 · 8)/(4 · 8) = 1/2 −2/15 −1/15 2 ◦ ∆( r , α|hsi, hsi) (2 · 8)/(8 · 4) = 1/2 −2/15 −1/15 2 ∆( r , γ|hrsi, hrsi) (2 · 8)/(4 · 8) = 1/2 −2/15 −1/15 2 ◦ ∆( r , γ|hrsi, hrsi) (2 · 8)/(8 · 4) = 1/2 −2/15 −1/15 ∆(hrsi , ζ, hsi) (2 · 8)/(4 · 4) = 1 1/5 1/5 ∆(hrsi , ζ, hsi)◦ see above see above 1/5 ∆(Q) (4 · 8)/(8 · 4) = 1 −1/3 −1/3 ∆(R) see above see above −1/3 ∆(T ) (4 · 8)/(8 · 4) = 1 0 0 ∆(Q, α, Q) (4 · 8)/(4 · 4) = 2 1/6 1/3 ∆(R, γ, R) see above see above 1/3 ∆(S) (8 · 8)/(8 · 2) = 4 1/4 1

3.2 Bideﬂation of Characteristic Idempotents

Ragnarasson and Stancu [RS13, Section 8] show that bideﬂation preserves characteristic idempotents in the case when F is a saturated fusion system on S. This means S that given a strongly F-closed subgroup T of S, BidefT (ωF ) = ωF/T , where F/T is the quotient fusion system F/T . We now show that this result does not extend to unsaturated fusion systems. Our counterexample relies on constructing a quotient system F/T for an unsaturated fusion system F that strictly contains the normalizer subsystem

NF (T ) of T in F. In other words, we need some F-morphism ϕ: R → S that doesn’t extend to any F-morphism ψ : RT → S.

3.9 Deﬁnition. Let ψ : G → H be a group homomorphism and suppose ϕ: A → B is a group homomorphism between subgroups A and B of G such that ϕ(A∩ker ψ) ≤ ker ψ. We can transfer ϕ along ψ to obtain a well-deﬁned group homomorphism

ψϕ: ψ(A) → ψ(B), ψ(a) 7→ ψ(ϕ(a)).

42 In the case that H is a normal subgroup of G and ψ : G → G/H is the canonical projection morphism and ϕ(A ∩ T ) ≤ T , we denote ψϕ by ϕ.

3.10 Deﬁnition (Strongly F-closed). Suppose that F is a fusion system on a p-group

S. A subgroup T of S is strongly F-closed provided that every ϕ ∈ HomF (R,S) satisﬁes ϕ(R∩T ) ≤ T . Note in particular that T is normal in S, since F contains all maps induced from conjugation in S.

3.11 Deﬁnition (Quotient System). Suppose that F is a fusion system on a p-group S and that T is strongly F-closed. The quotient fusion system F/T is the category with objects all the subgroups of S/T and morphisms deﬁned by

HomF/T (R/T, Q/T ) := {ϕ | ϕ ∈ HomF (R,Q)} , where R and Q are subgroups of S containing T and ϕ is the transfer of ϕ along the canonical projection morphism S → S/T as deﬁned in 3.9. Note that ϕ makes sense in this context since ϕ(T ) = T . One can easily verify that F/T is a fusion system on S/T .

It is important to note that when ϕ ∈ HomF (R,S) for some arbitrary subgroup R of S that doesn’t necessarily contain T , the homomorphism ϕ: RT/T → S/T is not generally a F/T -morphism, as ϕ: R → S might not extend to any F-morphism ψ : RT → S. When

3.12 Definition (Bideflation). Given a finite group S and normal subgroup T of S, bideflation from S to T is the Z-linear map

S BidefT : B(S, S) → B(S/T, S/T ), [X] 7→ [T \X/T ] where T \X/T is the set of (T,T )-orbits. In other words, viewing X as a S × S-set, T \X/T = X/(T × T ). When F is a fusion system on S and T is strongly F-closed [RS13, Lemma 8.4] shows S × S S/T × S/T ) BidefS = T ∆(ϕ(P ), ϕ, P ) ∆(ϕ ¯(P T/T ), ϕ,¯ P T/T ) for any F-morphism ϕ.

With the next proposition we explicitly construct a fusion system F and quotient system F/T such that the bideﬂation of ωF is not equal to the characteristic idempotent of F/T .

43 3.13 Proposition. There exists an unsaturated fusion system F on a p-group S and a quotient system F/T such that

S BidefT (ωF ) 6= ωF/T .

Proof. First suppose that F is some arbitrary fusion system on S and that T is strongly F-closed. For R ≤ S, let R¯ denote the image of R under the canonical projection π : S → S/T , i.e., R¯ = RT/T . From the proof in 3.5,

On the other hand,

SS X |W | X X µW,V S × S ω = . F/T |S| | Hom (V, S)| ∆(β(W ), β, W T ≤W ≤S W ≤V ≤S F/T β∈HomF/T (V,S)

S Thus, to show that BidefT (ωF ) 6= ωF/T it suﬃces to show that there is a subgroup W of S with T ≤ W ≤ S and a σ ∈ HomF/T (W, V ) such that

X X µW,V 6= (*) Hom (V, S) W ≤V ≤S F/T γ∈HomF/T (V,S), γ|W =σ X |Q ∩ T | X X µQ,P . |T | |HomF (P,S)| Q≤W, Q≤P ≤S α∈HomF (P,S), QT =W α|W =σ

Here, and for the rest of the proof, µ denotes the Möbius function of SS.

For each V ≥ T let πV denote the projection map AutF (V ) → AutF/T V . −1 Recall that for each α ∈ HomF (V,S) we have πV (α) = α · ker(πV ) where AutF (V ) acts on HomF (V,S) from the right via the usual function composition. In particular,

44 of (*) the summand for Q = W is

It is easy to verify that S satisﬁes conditions I and II of 1.32, so S induces a fusion system F on S with the property that HomF (P,S) = {ι: P → S} for all P 6= R and

HomF (R,S) = {ι: R → S, γ}, where ι denotes the inclusion map.

Since S is abelian, Nγ = S and all subgroups are fully F-normalized, but by construction γ does not extend to S and so F is not saturated. Also, T is strongly F-closed and so we may consider the quotient fusion system F/T . Recall that the morphisms of F/T are the projections of morphisms in F whose domain contains T , so by construction HomF/T (W, S) = ι: W → S .

45 Set σ := ι: W → S. Since γ 6= σ we deduce that for all P ≤ S, there exists precisely one morphism α ∈ HomF (P,S) such that α restricts to σ. There are

33 − 3 = 4 32 − 3

2-dimensional subspaces P which contain R. There are

32 − 3 = 3 3 − 1 subspaces Q < W such that Q + T = W . It is clear that the poset

{P ≤ S : Q < W, Q + T = W, Q ≤ P } ordered by subgroup inclusion is, up to isomorphism, independent of the choice of complement Q. Let U 6= R be a ﬁxed complement of T . Putting this all together, we consider (**) and see

X |Q ∩ T | X X µQ,P = |T | |HomF (P,S)| Q

46 Chapter 4

A Generalized Burnside Functor

Bouc shows [Bou10, Chapter 5] that the subfunctor lattice of the Q-Burnside functor is isomorphic to a certain collection of subsets of a transversal B for the isomorphism classes of B-groups. We generalize this result to the case of the functor QS , the Yoneda functor Hom (1, −): A → -Mod, by demonstrating that the subfunctor AQ Q Q lattice of QS is also isomorphic to same poset as the Q-Burnside functor. Additionally, we prove that QS is the decondensation of the Q-Burnside functor. Our need to make sense of idempotent condensation and decondensation over non-unital rings motivates our present study of pseudo-rings.

4.1 Pseudo-rings

Here we recall the concept of a pseudo-ring, also known as a non-unital ring or simply a rng. We develop some basic theory regarding the class of admissible modules over a pseudo-ring, with a mind to apply this to the study of the A˜-module associated to the functor QS . Most of the material here had to be developed as needed, so we include our proofs, but it is likely well known to experts. A possible source for further study is [Wis91].

4.1 Deﬁnition (Pseudo-ring). A pseudo-ring R is a triple (R, +, ·) where R is a set equipped with two binary operations + and · (called addition and multiplication, respectively) such that satisfy all the usual ring actions except that there might not be a multiplicative identity element in R. Let (R, +, ·)(S, +0, ∗) be pseudo-rings. A function f : R → S is a pseudo-ring

47 homomorphism provided that f is a group homomorphism and f(a · b) = f(a) ∗ f(b) for all a, b ∈ R. Note that even if R and S are rings, a pseudo-ring morphism f : R → S does not necessarily satisfy f(1R) = 1S, where 1R and 1S are the multiplicative identity elements in R and S, respectively. We can deﬁne R-modules in the usual way, except of course we do not have an axiom specifying that the multiplicative identity acts trivially.

4.2 Deﬁnition (Enough idempotents). A pseudo-ring R has enough idempotents provided that there is a subset E ⊆ R of pairwise orthogonal idempotents that give a L L Z-module decomposition R = e∈E eR = e∈E Re. The subset E is called a complete set of idempotents of R. It is easy to verify that if R has enough idempotents, then it

also has local units, which means that for any ﬁnite subset {r1, . . . , rn} of R there is there

is an idempotent e ∈ R such that eri = ri = rie for each i = 1, . . . , n. A pseudo-ring R is idempotent provided that R2 = R. It’s clear that if R has enough idempotents, then it is idempotent.

4.3 Deﬁnition (Admissible module). Let R and S be pseudo-rings. A left R-module M is admissible provided that RM = M and a right R-module M is admissible provided MR = M. We denote by R-Mod∗ the full subcategory of R-Mod that has the class of admissible R-modules as objects. If an (R,S)-bimodule is admissible as a left R-module and as a right S-module we say it is an admissible (R,S)-bimodule.

4.4 Lemma (Characterizations of admissible modules). Let R be a pseudo-ring with enough idempotents and let M be a left R-module. The following are equivalent:

1. M is admissible.

L 2. If E is a complete set of idempotents of R, then M = e∈E eM.

3. For every x ∈ M there is an idempotent e ∈ R such that x = ex.

Proof. (1 ⇐⇒ 2) and (2 =⇒ 3). Suppose that E is any complete set of idempotents Pn Pm for R. If M is admissible then any x ∈ M may be written as x = i=1( j=1 ei,jri,j)xi where ei,j ∈ E, ri,j ∈ R, and xi ∈ M. We thus obtain the Z-module decomposition L L M = e∈E eM. Conversely, if M = e∈E eM then every x ∈ M may be written as

48 Pn i=1 eixi for ei ∈ E and xi ∈ M, yielding M ⊆ RM. This establishes (1 ⇐⇒ 2). Now Pn Pn suppose (2) holds. Given that x = i=1 eixi, take e := i=1 ei and notice ex = x. This establishes (2 =⇒ 3). (3 =⇒ 1). This is immediate from the deﬁnition of RM.

4.5 Lemma. Let R be a pseudo-ring with enough idempotents and M be an admissible R-module. Any subquotient of M is also admissible.

Proof. Let L ≤ N ≤ M be R-submodules of M and suppose m + L ∈ N/L. By Lemma 4.4, there is some e ∈ R such that em = m, whence e(m + L) = (em) + L = m + L. By invoking Lemma 4.4 again it follows that the subquotient N/L is an admissible R-module.

4.6 Lemma. Suppose R has enough idempotents and M is an R-module. Then the canonical R-module homomorphism

R ⊗R M → RM, r ⊗ m 7→ rm for r ∈ R, m ∈ M

is an R-module isomorphism.

Pn Proof. The map is clearly surjective. To see it is injective, suppose that i=1 ri ⊗ mi Pn is sent to 0, which just means i=1 rimi = 0. Since R has enough idempotents it has local units, so there is some idempotent e ∈ R such that eri = ri for each i = 1, . . . , n. Then n n n X X X ri ⊗ mi = eri ⊗ mi = e ⊗ rimi = e ⊗ 0 = 0, i=1 i=1 i=1 whence the canonical homomorphism is injective.

Condensation and Decondensation

4.7 Notation. We now assume throughout that R is an idempotent pseudo-ring, E is a set of pairwise orthogonal idempotents for R, and

M S := eRf. e,f∈E

49 R 4.8 Deﬁnition. The condensation functor CS : R- Mod → S- Mod and decondensa- R tion functor DS : S- Mod → R- Mod are deﬁned on objects by ! R M R M CS (M) := eM and DS (N) := Re ⊗S N, e∈E e∈E where N is a S-module and M is an R-module. When R and S are understood, we shall write C and D instead.

4.9 Lemma. Condensation and decondensation produce admissible modules. That is, if M is an R-module and N is a S-module, then C(M) is an admissible S-module and D(N) is an admissible R-module.

Proof. Let M be an R-module. Then C(M) = ⊕e∈EeM is an admissible S-module, Pn since any element of C(M) is of the form m := i=1 eimi (for ei ∈ E, mi ∈ M), and Pn i=1 eim = m. Now suppose that N is any S-module. Since R2 = R by assumption,

2 RD(N) = R(⊕e∈ERe) ⊗S N = (⊕e∈ER eG) ⊗S N = (⊕e∈ERe) ⊗S N = D(N).

Thus D(N) is an admissible R-module.

4.10 Remark. In general DC(M) =6∼ M and CD(N) =6∼ N for an R-module M and a S-module N, since DC(M) and CD(N) are always admissible. However, if N is an admissible S-module, then M CD(N) = C Re ⊗S N e∈E M M = f Re ⊗S N f∈E e∈E ∼ M = fRe ⊗S N f,e∈E

= S ⊗S N =∼ N,

where the last isomorphism follows from 4.6.

Our next lemma will allow us to conclude later that the A˜-module associated to QS is the decondensation of the B-module associated to the Q-Burnside functor.

50 4.11 Lemma. Let R be a pseudo-ring and E ⊆ R a set of orthogonal idempotents and deﬁne M S := eRf. e,f∈E For any e ∈ E, 0 ! ! M M ∼ Re ⊗ eRe0 = Re0 S e∈E e∈E as R-modules.

Proof. First, note S is a sub-pseudo-ring of R and E is a complete set of idempotents L L for S. Set M := e∈E Re ⊗S e∈E eRe0 . If e 6= f are elements of E and a, b ∈ R, then ae ⊗S fbe0 = 0 since e = eee ∈ eRe ⊆ S and ef = 0. Thus M is the Z-span of

{ae ⊗ ere0 | a, r ∈ R}. The map ! ! M M ϕ: Re × eRe0 → Re0, (x, y) 7→ xy e∈E e∈E is clearly S-balanced and so induces an R-module morphism ϕ: M → Re0. Also, we can deﬁne an R-module homomorphism

ψ : Re0 → M, ae0 7→ ae0 ⊗ e0.

For any a, r ∈ R and e ∈ E we have ψϕ(ae ⊗ ere0) = ψ(aeere0e0) = ae(ere0) ⊗ e0 =

ae ⊗ ere0, which shows ϕ is injective. On the other hand, for any a ∈ R we have

ϕψ(ae0) = ϕ(ae0 ⊗ e0) = ae0, whence ϕ is surjective.

4.2 Decondensation of the Burnside Functor

Throughout, let G denote a transversal for the isomorphism classes of ﬁnite groups.

51 4.12 Remark. Recall (1.38) that we have Q-vector space decompositions

˜ M A = QSG×H , G,H∈G M A = QSG×H , G,H∈G M B = QB(G, H), G,H∈G

˜ κ where A has a Q-algebra structure induced from the Q-bilinear maps − ∗˜H −. P For a G ∈ G, recall that the elements eG := g∈G ∆(G, cg,G) and 1G :=

|G|∆(G) are idempotents in A. In particular, 1G is the multiplicative identity of κ G×G κ (QSG×G, ∗G) and eG is the multiplicative identity of ((QSG×G) , ∗G). Their images ˜ ˜ ˜ in A via the Q-algebra isomorphism ζ : A → A are denoted by e˜G and 1G, respectively. The collections {e˜G | G ∈ G} and also 1˜G | G ∈ G are sets of orthogonal idempotents of A˜. Note

˜ κ ˜ M 1G ∗˜ A = QSG×H , H∈G ˜ κ ˜ M A ∗˜ 1G = QSH×G. H∈G

˜ κ ˜ κ ˜ ˜ It follows that 1G ∗˜ A ∗˜ 1H = QSG×H and that 1G | G ∈ G is a complete set of idempotents for A˜. From [BD13, Proposition 3.9] we have a Q-algebra isomorphism

∼ M κ κ M κ ˜ κ B = e˜G ∗˜ QSG×H ∗˜ e˜H = e˜G ∗˜ A ∗˜ e˜H . G,H∈G G,H∈G

After identifying B with this sub-algebra of A˜, we see that {e˜G | G ∈ G} is a complete set of idempotents for B. Note in particular that for the trivial group 1 we have   κ M κ ˜ κ κ M B ∗˜ e˜1 =  e˜G ∗˜ A ∗˜ e˜H  ∗˜ e˜1 = QB(G, 1) G,H∈G G∈G

This is the admissible B-module associated to the usual Q-Burnside functor, which we recall is the Yoneda functor Hom (1, −): B → -Mod. Recall that denotes the BQ Q Q QS Yoneda functor Hom (1, −): A → -Mod, AQ Q Q

52 which assigns to a group G ∈ G the Q-vector space spanned by the subgroup lattice of G × 1, which can be canonically identiﬁed with QSG, hence the notation. As QS is a Q-linear functor, it’s associated admissible A˜-module is

M ˜ κ ˜ ˜ κ QSG×1 = A ∗˜ 11 = A ∗˜ e˜1. G∈G

4.13 Notation. Given G ∈ G and a subgroup H ≤ G, we let (G, H) denote H × 1 viewed as a subgroup of G × 1, which we can canonically identify with H ∈ QSG.

As an application of 4.11 and our observations in 4.12, we obtain:

A˜ κ 4.14 Theorem. The decondensation DB(B ∗˜ e˜1) of the admissible B-module associated to the Q-Burnside functor is isomorphic as an A˜-module to the admissible ˜ ˜ κ ˜ A-module A ∗˜ 11 associated to the functor QS .

Proof. In 4.11, replace R with A˜, S with B, and e0 with e˜1 = 1˜1. To wit, ! ! A˜ κ M ˜ κ M κ ˜ κ ∼ ˜ κ ˜ DB(B ∗˜ e˜1) = A ∗˜ e˜G ⊗B e˜G ∗˜ A ∗˜ e˜1 = A ∗˜ 11. G∈G G∈G

4.15 Notation. To avoid a proliferation of symbols, we will begin to use concatenation to denote multiplication in A˜. However, to avoid mistaking the action of operators, we will occasionally continue to write A˜ ∗˜κ (G, H), where (G, H) is the subgroup H × 1 ≤ G × 1.

4.3 The Action of Elementary Subgroups

To understand the subfunctor lattice of QS we need to study the A˜-module ˜ structure of its associated A-module ⊕H∈GQS (H, 1), via examining expressions of the κ form L ∗˜H (V × 1) for L ≤ G × H and V ≤ H. Recall from 1.30 that for L ≤ G × H and V ≤ H,  0 if V 6= p (L), κ  2 L ∗˜H (V × 1) = L, p2(L)×1 a · (p1(L) × 1) if V = p2(L).  p1(L)×1

53 This motivates us to study the coeﬃcients aL,p2(L)×1 in more detail. p1(L)×1

4.16 Definition. Let G and H be finite groups and L a subgroup of G × H. Define

˜L 0 0 0 Y := L ∈ SG×H | L ≤ L, p1(L ) = p1(L) .

4.17 Lemma. Let G and H be ﬁnite groups and L ≤ G × H. Then

L,p2(L)×1 1 X Y˜L 0 a = µ 0 |k2(L )| p1(L)×1 |H| L ,L L0∈Y˜L

L,p2(L)×1 0 0 Proof. From the deﬁnitions, consists of those (L ,V × 1) ∈ G×H × H×1 Yp1(L)×1 S S such that

0 0 1. L ≤ L and V ≤ p2(L),

0 2. p1(L ) = p1(L),

0 0 3. V = p2(L ),

0 0 4. p1(L) × 1 ≤ L ∗ (V × 1).

0 0 L,p2(L)×1 The condition V = p2(L ) implies that the projection of onto its ﬁrst com- Yp1(L)×1 0 0 0 0 0 ponent is injective. Together, p1(L ) = p1(L) and V = p2(L ) imply that L ∗ (V × 1) =

p1(L) × 1. It’s an easy consequence that the map

L,V ×1 ˜L 0 0 0 YU×1 → Y , (L ,V ) 7→ L

0 0 L,p2(L)×1 0 0 is a poset isomorphism. For any (L ,V × 1) ∈ we have k1(V × 1) = V and Yp1(L)×1 0 0 0 0 0 consequently k2(L ) ∩ k1(V × 1) = k2(L ) from which it follows that κ(L ,V × 1) = −1 0 |H| |k2(L )|. In light of 1.28, the result follows.

4.18 Definition. Let G be a finite group and N a normal subgroup of G. The rational number mG,N is defined by

1 X m := µSG |X|. G,N |G| X,G X≤G,XN=G

Note in particular that mG,1 = 1.

54 The coeﬃcients mG,N play an important role in determining the subfunctor lattice of the Burnside functor, and appear when considering how deﬂation acts on the idempotents of QB(G).

4.19 Lemma (5.2.2, [Bou10]). Let G and H be ﬁnite groups and L ∈ SG×H a left-free subgroup of the form L = ¡(ϕ(H), ϕ, H), where ϕ: H → G is a group homomorphism. Then as an element of Q,

L,p1(L)×1 | ker ϕ| a = mH,ker ϕ. p2(L)×1 |H|

Proof. The number aL,p1(L)×1 does not depend on whether L is viewed as a subgroup of p2(L) G × H or ϕ(H) × H, so we can without loss of generality assume that ϕ is surjective.

Let K denote the kernel of ϕ and suppose X ≤ G. As the ﬁbers of ϕ|X are simply the left cosets xK such that x ∈ X, it follows that ϕ(X) = G if and only if XK = H. This demonstrates that Y˜L is parameterized by the X ≤ G such that XK = H, ˜L 0 0 i.e., Y = {L ≤ L | L = ¡(G, ϕ|X ,X)}. In particular, the convex subposet X := ˜L ˜L 0 0 {X ∈ SH | XK = H} of SH is isomorphic to Y via the map Y → X ,L 7→ p2(L ). 0 ˜L ∼ 0 For any L = ¡(G, ϕ|X ,X) ∈ Y , as X/(X ∩ K) = XK/K = H/K, we have |k2(L )| = |X ∩ K| = |K| · |X|/|H|. From Lemma 4.17 we obtain

L,p2(L)×1 1 X Y˜L 0 a = µ 0 |k2(L )| p1(L)×1 |H| L ,L L0∈Y˜L 1 X = µX |X ∩ K| |H| X,H X∈X 1 X = µSH |X ∩ K| |H| X,H X≤H, XK=H |K| X = µSH |X| |H|2 X,H X≤H, XK=H |K| = m . |H| H,K

The results in 4.19 and 1.29 allow us to determine how the ﬁve elementary subgroups act upon ⊕G∈GQSG×1. We collect these facts together for later reference. We follow with a lemma establishing that the ∗˜κ-product decomposition of elementary

55 subgroups spans QSG×H . We shall leverage this lemma repeatedly when investigating the subfunctor lattice of QS . Recall that (G, H) denotes H × 1 viewed as a subgroup of G × 1.

4.20 Corollary. Let G and H be ﬁnite groups. Then

G κ 1 G κ 1 1. If H ≤ G, then IndH ∗˜ (H,H) = |H| (G, H) and ResH ∗˜ (G, H) = |G| (H,H).

κ 1 2. If ϕ: H → G is a group isomorphism, then Isoϕ ∗˜ (H,H) = |H| (G, G).

G κ −1 3. If N ¢ G, then DefG/N ∗˜ (G, G) = [G : N] · mG,N · (G/N, G/N) and G κ −1 InfG/N ∗˜ (G/N, G/N) = [G : N] (G, G).

G κ In particular, whether DefG/N ∗˜ (G, G) is zero or not depends only on the number

mG,N .

4.21 Notation. Let G and H be ﬁnite groups and suppose X ∈ SG×H corresponds to

G P1 the quintuple (P1,K1, η, P2,K2). Let R(X) := Ind ∗ Inf ∗ Isoη denote the right- P1 P1/K1 free part of X and and L(X) := DefP1 ∗ ResH the left-free part of X. Note that P2/K2 P2 p2(L(X)) = p2(X) and p1(R(X)) = p1(X).

4.22 Lemma (A spanning set for QSG×H ). Let G and H be ﬁnite groups and suppose X ≤ G × H. Then

X R(X) ∗˜κ L(X) = |q(X)|−1 · N, P2/K2 N≤X, p1(N)=p1(X), p2(N)=p2(X)

where q(X) is as in 1.9. As a consequence, QSG×H is the Q-span of

κ {R(X) ∗˜ L(X) | X ∈ SG×H } .

Proof. At the risk of confusion but in order to maintain consistency with the rest of this document, set L := R(X) and M := L(X). Recall from 1.28 that

κ X L,M L ∗˜H M = aN N N≤L∗M, p1(N)=p1(L), p2(N)=p2(M)

56 where,

L,M X YN 0 0 aN = µ(L0,M 0),(L,M) · κ(L ,M ). 0 0 L,M (L ,M )∈YN L,M 0 0 0 But YN consists of those (L ,M ) ≤ (L, M) such that p1(L ) = p1(N) = p1(L), 0 0 0 0 p2(L ) = p1(M ), p2(M ) = p2(N) = p2(M). As L is right-free, p1(L ) = p1(L) implies that L0 = L since the subgroup lattice of a right-free subgroup is isomorphic to the 0 subgroup lattice of its left projection. Similarly, since M is left-free, p2(M ) = p2(M) 0 L,M implies that M = M. This establishes that YN = {(L, M)}, whence

L,M −1 aN = κ(L, M) = |q(X)| .

κ Now we prove that QSG×H is spanned by elements of the form R(X) ∗˜ L(X) by showing, via strong induction on the order of subgroups of G × H, that X ≤ G × H κ is in the Q-span of {R(Y ) ∗˜ L(Y ) | |Y | ≤ |X|}. If X is the trivial subgroup of G × H, then R(X) is the trivial subgroup of G × 1 and L(X) is the trivial subgroup of 1 × H. κ κ Thus R(X) ∗˜1 L(X) = X, so Q(R(X) ∗˜1 L(X)) = QX. Now let 1 < X ≤ G × H be arbitrary. Since X R(X) ∗˜κ L(X) = |q(X)|−1X + |q(X)|−1 N, N

{R(X) ∗˜κ L(X)} ∪ {R(Y ) ∗˜κ L(Y ) | |Y | < |X|} .

This establishes the result.

4.4 Subfunctors

Now we are in a position to show that QS has the same subfunctor lattice as the Q-Burnside functor. While most of the arguments in [Bou10, Chapter 5] carry through without modiﬁcation, one hurdle to overcome is that a subgroup L ≤ G × H doesn’t decompose in A˜ into its right and left-free parts in as simple a manner as it does in A. The remedy is to liberally apply 4.22. As usual, G is some transversal for the isomorphism classes of ﬁnite groups. Recall also that if U ≤ G, then (G, U) denotes the subgroup U × 1 ≤ G × 1. Whenever X and Y are modules over some ring, we let X ≤ Y signify that X is a submodule of Y .

57 4.23 Deﬁnition (Subfunctors). Let F,E : C → D be functors between categories C and D, where the objects of D have a set structure and the morphisms of D are set maps with possibly richer structure. We say F is a subfunctor of E, denoted F ≤ E, provided that F (X) ⊆ E(X) for every object X in C and F (f) is the restriction of E(f) to F (X) for any C-morphism f : X → Y .

4.24 Deﬁnition (G). For a ﬁnite group G ∈ G, let G denote the subfunctor of QS associated to the cyclic A˜-module generated by (G, G). For an arbitrary H ∈ G, it follows that κ κ G(H) = 1˜H ∗˜ A˜ ∗˜ (G, G).

4.25 Proposition. Let G ∈ G. The following are equivalent:

1. If H ∈ G such that |H| < |G|, then G(H) = 0.

2. If H ∈ G such that G(H) 6= 0, then G is isomorphic to a subquotient of H.

3. If 1 < N ¢ G, then mG,N = 0.

G κ 4. If 1 < N ¢ G, then DefG/N ∗˜ (G, G) = 0 in QSG/N .

˜ κ G G Proof. (1 =⇒ 4) Suppose that 1 < N ¢G, and then note 1G/N ∗˜ DefG/N = DefG/N and G κ ˜ ˜ κ G κ DefG/N ∗˜ (G, G) ∈ 1G/N A∗˜ (G, G) = G(G/N) = 0, so in particular DefG/N ∗˜ (G, G) = 0. (3 ⇐⇒ 4). This is immediate from 4.20. κ (3 =⇒ 2). Suppose that 1˜H A˜ ∗˜ (G, G) 6= 0. Then from 4.22 there is some subgroup X ≤ H × G, determined by the quintuple (P1,K1, η, P2,K2), such that

R(X) ∗˜κ L(X) ∗˜κ (G, G) 6= 0.

This forces P = G, so L(X) = DefG and 2 G/K2

DefG ∗˜κ(G, G) = [G : K ]−1m (G/K , G/K ) 6= 0. G/K2 2 G,K2 2 2 ∼ Our initial assumption forces K2 to be the trivial group and so P1/K1 = G, as desired. (2 =⇒ 1). If |H| < |G|, then G cannot be isomorphic to a subquotient of H.

58 Any ﬁnite group G satisfying the equivalent conditions in 4.25 is called a B- group.

4.26 Deﬁnition (B-group). A ﬁnite group G is a B-group (over Q) provided that G κ mG,N = 0 (or equivalently DefG/N ∗˜ (G, G) = 0) for every non-trivial normal subgroup N of G. G/N κ G −1 G ˜ As Def(G/N)/(G/M) ∗˜ DefG/N = [G : N] DefG/M in A, G is a B-group if and

only if mG,N = 0 for all minimal normal subgroups.

4.27 Proposition. Let G and H be in G. If H is isomorphic to a quotient of G, then

G ≤ H . In the case that H is a B-group, the converse also holds.

Proof. Suppose that N ¢G and ϕ: H → G/N is a group isomorphism. Then A˜∗˜κ (H,H) G κ contains InfG/N ∗˜ Isoϕ(H,H), which is a non-zero multiple of (G, G). This establishes κ κ A˜ ∗˜ (G, G) ≤ A˜ ∗˜ (H,H), which is equivalent to G ≤ H . Now suppose that H is a B-group and A˜∗˜κ (G, G) ≤ A˜∗˜κ (H,H). In particular, κ κ (G, G) ∈ 1˜GA˜ ∗˜ (H,H). But as |G|∆(G) ∗˜ (G, G) = (G, G), we actually have that (G, G) ∈ |G|∆(G) ∗˜κ A˜ ∗˜κ (H,H). By 4.22, there is some X ≤ G × H corresponding κ κ to the quintuple (P1,K1, η, P2,K2) such that R(X) ∗˜ L(X) ∗˜ (H,H) 6= 0 and also κ |G|∆(G) ∗˜ R(X) 6= 0. From the second equality we immediate have that P1 = G. Since ResH ∗˜κ(H,H) = 0 when P < H, we must have P = H. From our assumption that P2 2 2 H ˜κ H is a B-group, DefK2 ∗ (H,H) = 0 whenever K2 6= 1, so K2 = 1. Recalling that η is ∼ ∼ ∼ an isomorphism between P2/K2 and P1/K1, we have G/K1 = P1/K1 = P2/K2 = H, precisely as desired.

4.28 Deﬁnition. Let F : AQ → Q-Mod a functor. A ﬁnite group H is minimal for F provided that F (H) 6= 0 and F (K) = 0 whenever |K| < |H|.

4.29 Proposition. Suppose F is a subfunctor of QS . If H is a minimal group for F , then H is a B-group. Also, F (H) is the 1-dimensional subspace Q(H,H) of QSH×1 and H ≤ F . In particular, if H is a B-group, then H (H) = Q(H,H).

Proof. Suppose that H is a minimal group for H. By deﬁnition of QS , we have that ˜ F (H) ⊆ QSH×1. First note that if M is the A-module associated to the subfunctor F ,

59 then H κ H κ˜ ˜ H κ˜ ˜ ResK ∗˜ F (H) = ResK ∗˜ 1H M = 1K (ResK ∗˜ 1H M) ⊆ 1K M = F (K).

We may write any x ∈ F (H) as

X x = λU (H,U) U≤H

H κ −1 for λU ∈ Q. For K < H, we have ResK ∗˜ x = |H| · λK (K,K), which is an element of F (K) = 0, so λK = 0. This establishes that F (H) is the one dimensional subspace κ Q(H,H). In particular, (H,H) ∈ M, so of course A˜ ∗˜ (H,H) ≤ M, from which it follows that H ≤ F . When |K| < |H|, H (K) ≤ F (K) = 0, so H is a B-group by 4.25.

Given that H is a B-group, 4.25 shows that H is a minimal group for H , hence

H (H) = Q(H,H).

4.30 Proposition. Let G ∈ G. Then there exists a group β(G) such that:

1. Every minimal group for G is isomorphic to β(G).

2. G = β(G).

3. β(G) is isomorphic to a quotient of G.

∼ 4. mG,N 6= 0 for any N ¢ G such that G/N = β(G).

Proof. Let H be a minimal group for G. We prove that G = H and that H is isomorphic to a quotient of G. Because H is a minimal group for a subfunctor of QS , it is κ a B-group and H ≤ G by 4.29. As G(H) = 1˜H A˜∗˜ (G, G) 6= 0, there is some X ≤ H×G κ κ corresponding to the quintuple (P1,K1, η, P2,K2) such that R(X)∗˜ L(X)∗˜ (G, G) 6= 0. In particular, P = G and 0 6= Iso ∗˜κ DefP2 ∗˜κ ResG , but as this product is in (P /K ) 2 η K2 P2 G 1 1 and H is minimal for G, we must have that mG,K2 6= 0, P1 = H, and K1 = 1, so H is isomorphic to the quotient G/K2 of G, whence G ≤ H from 4.27. This establishes that G = H , proving (2) and (3) after we set β(G) := H. 0 Given another minimal group H for G, the above arguments yield H = H0 , so H is a quotient of H0 and vice versa since both H and H0 are B-groups (4.27). Hence the set of minimal groups for G consists of the isomorphism class of a single ﬁnite group β(G), which is isomorphic to some quotient G/N where mG,N 6= 0, proving (1)

60 ∼ in particular. If G/M = G/N, then 0 6= mG,N = mG,M , the equality following from [Bou10, Proposition 5.3.4]. This shows (4) and completes the proof.

4.31 Theorem. Let G ∈ G.

1. If H is a B-group and also is isomorphic to a quotient of G, then H is isomorphic to a quotient of β(G).

2. Let N ¢ G. Then the following are equivalent:

(a) mG,N 6= 0. (b) β(G) is isomorphic to a quotient of G/N.

3. In particular, if N ¢ G, then G/N =∼ β(G) if and only if G/N is a B-group and

mG,N 6= 0.

Proof. 1. Since H is isomorphic to a quotient of G, β(G) = G ≤ H (4.30, 4.27), and since H is a B-group, H is isomorphic to a quotient of β(G) (4.27).

G κ ˜ κ 2. (a =⇒ b). If mG,N 6= 0, then DefG/N ∗˜ (G, G) 6= 0, so (G/N, G/N) ∈ A∗˜ (G, G),

hence G/N ≤ G = β(G). As β(G) is a B-group it is thus isomorphic to a quotient of G/N (4.27).

(b =⇒ c). In this case, β(G/N) = G/N ≤ β(G), so since β(G) is a B-group it is isomorphic to a quotient of β(G/N). On the other hand, β(G/N) is isomorphic to a quotient of G/N and hence of G, so since β(G/N) is a B-group, (1) implies that β(G/N) is isomorphic to a quotient of β(G).

(c =⇒ a). Assume β(G) =∼ β(G/N). From 4.30 there is a subgroup M/N ¢ ∼ ∼ G/N such that β(G/N) = (G/N)/(M/N) and mG/N,M/N 6= 0. Then G/M = β(G/N) =∼ β(G) and

0 6= mG,M = mG,N · mG/N,G/N ,

where the equality is proved in [Bou10, Proposition 5.3.1]. It follows that mG,N 6= 0.

∼ 3. If G/N = β(G), then mG,N 6= 0 (4.30) and G/N is a B-group. Conversely, if G/N ∼ is a B-group and mG,N 6= 0, then (2) gives β(G) = β(G/N) = G/N.

61 4.32 Deﬁnition. Let B denote a transversal for the set of isomorphism classes of B-groups. Deﬁne the relation on B by G H provided that H is isomorphic to a quotient of G. Then (B, ) is a poset. A subset X of B is closed provided it contains all ascending chains that begin in X, i.e., G is an element of X whenever G H for some H ∈ X.

4.33 Lemma. If F is a subfunctor of QS and G is a ﬁnite group, then F (G) is the sum of the one dimensional subspaces Q(G, H) such that (G, H) ∈ F (G).

Proof. Let M denote the A˜-module associated to F and suppose that x ∈ F (G). Then x ∈ 1˜GM and X x = λU (G, U), U≤G where λU ∈ Q. Suppose that H ≤ G and λH 6= 0. Then viewing |G|∆(H) as an κ κ element of QSG×G, we have |G|∆(H) ∗˜ x = λH (G, H). As |G|∆(H) ∗˜ x ∈ M and κ κ 1˜G ∗˜ |G|∆(H) = |G|∆(H), we actually have λH (G, H) = |G|∆(H)∗˜ x ∈ 1˜GM = F (G). This proves the claim.

4.34 Theorem. Let S denote the subfunctor lattice of QS and T the collection of closed subsets of B, ordered by subset inclusion. The map

Θ: S → T,F 7→ {H ∈ B | H ≤ F }

is an isomorphism of posets with inverse

X Ψ: T → S, X 7→ H . H∈X

Proof. First we establish that Θ actually takes values in T . Suppose that F ∈ S. If

H ∈ Θ(F ) and G ∈ B such that G H, then G ≤ H ≤ F , so G ∈ Θ(F ). This establishes that Θ(F ) is closed. Moreover, Θ and Ψ are obviously poset morphisms. Let M denote the A˜-module associated to F and suppose G ∈ B is arbitrary.

62 Our next aim is to show that

X F (G) = H (G). H≤G, H ≤F From 4.33 we have X F (G) = Q(G, H). H≤G, (G,H)∈M G κ −1 ˜ κ If (G, H) ∈ M, then ResH ∗˜ (G, H) = |G| (H,H) ∈ M, so A ∗˜ (H,H) is contained G κ in M, i.e., H ≤ F . Conversely, if H ≤ M, then (H,H) ∈ M and IndH ∗˜ (H,H) = −1 |H| (G, H) ∈ M, so Q(G, H) ≤ F (G). This establishes that X F (G) = Q(G, H). H≤G, H ≤F ˜ ˜ κ G ˜ ˜ But when H ≤ G, H (G) = 1GA ∗˜ (H,H) contains (G, H) since IndH ∈ 1GA, so X X F (G) ⊇ H (G) ⊇ Q(G, H) = F (G), H≤G, H≤G, H ≤F H ≤F which shows that X F (G) = H (G), H≤G, H ≤F as claimed. For an arbitrary G ∈ G, we now have

X X F (G) ⊇ K (G) ⊇ K (G) = F (G), K∈G, K≤G, K ≤F K ≤F so X F (G) = K (G). K∈G, K ≤F This means X F = H . H∈G, H ≤F

Recall that H = β(H) and β(H) ∈ B, so X F = H . H∈B, H ≤F

63 We have shown ΨΘ(F ) = F . Conversely, suppose that X ∈ T . Then ! ( ) X X ΘΨ(X) = Θ K = H ∈ B | H ≤ K K∈X K∈X clearly contains X. For the other containment, note that

X X H ≤ K =⇒ Q(H,H) = H (H) ⊆ K (H) K∈X K∈X X κ =⇒ (H,H) ∈ 1˜H A˜ ∗˜ (K,K) K∈X κ X κ =⇒ |H|∆(H) ∗˜ (H,H) ∈ |H|∆(H)1˜H A˜ ∗˜ (K,K) K∈X =⇒ ∃K ∈ X : ∆(H)A˜ ∗˜κ (K,K) 6= 0, which implies that there is some L ≤ H × K with p1(L) = H and p2(K) such that κ κ R(L)∗˜ L(L)∗˜ (K,K) 6= 0. As K ∈ B, we must have k2(L) = 1 and so K is isomorphic to a quotient of H, i.e., H K. But since X is closed H ∈ X, so X ⊇ ΘΨ(X), proving ΘΨ(X) = X, which completes the proof.

4.35 Corollary. Let G be a B-group.

1. The functor G has a unique maximal subfunctor

X νG := H . H∈B, HG, H=6∼G

0 0 2. If F < F are subfunctors of QS such that F /F is simple, then there is a 0 0 unique G ∈ B such that G ≤ F and G 6≤ F . In particular, G + F = F and

G ∩ F = νG.

Proof. (1) By 4.34 the subfunctor lattice of G is isomorphic to the poset of closed subsets of

Θ(G) = {H ∈ B | H ≤ G} .

However, H ≤ G if and only if H G since G is a B-group (4.27). Thus

Θ(G) = {H ∈ B | H G} .

64 If X is a closed subset of Θ(G), then either G ∈ X, in which case X = Θ(G), or

X ⊆ {H ∈ B | H G, H =6∼ G} , which is itself a closed subset of Θ(G). This establishes that νG is the unique maximal subfunctor of G. 0 0 (2) Suppose that G and G are elements of Θ(F ) \ Θ(F ), so G and G0 are subfunctors of F 0 but not F . As F 0/F is simple, if X is a closed subset of Θ(F 0) satisfying Θ(F ) ⊆ X and G ∈ X, then X = Θ(F 0), hence G0 ∈ X. Therefore,

G0 ∈ Θ(F ) ∪ {H ∈ B | H G} , but as G0 ∈/ Θ(F ) it must be that G0 ∈ {H ∈ B | H G}, so G0 G. By symmetry, 0 0 0 0 G G , hence G = G . As G ≤ F but G 6≤ F , it follows from the simplicity of F /F 0 ∼ that G +F = F . Moreover, F /F = G/(G ∩F ) is simple so G ∩F = νG, from (1).

4.36 Remark. Bouc [Bou10, Theorem 5.4.14] demonstrates that the subfunctor lattice of the Q-Burnside functor is also isomorphic as a poset to the set of closed subsets of

B, by the map sending a subfunctor F to the set {H ∈ B | eH ≤ F }, where eH is the functor associated to the cyclic B-module generated by (G, G). Consequently, we have shown that QS and the Q-Burnside functor have isomorphic subfunctor lattices. This is somewhat surprising, since the exact condensation functor can map simple modules to 0, hence one would initially expect QS to have a larger subfunctor lattice than the Q-Burnside functor. Our next result shows that this lattice isomorphism actually arises from condensation.

4.37 Proposition. Set x := (G, G), the element G × 1 ∈ SG×1. Let Bx denote

the cyclic B-submodule of Be˜1 generated by x and Ax˜ the cyclic A˜-submodule of A˜e˜1 A˜ ˜ generated by x. Then CB(Ax) = Bx as B-modules. In other words, the condensation of the subfunctor G of QS is the subfunctor eG of the Q-Burnside functor.

G×1 ˜ ˜ Proof. Since x ∈ (QSG×1) =e ˜GAe˜1, we see e˜Gx = x. Recall that e˜H Ae˜G =e ˜H Be˜G.

65 We have

A˜ ˜ M ˜ CB(Ax) = e˜H Ax H∈G M = e˜H A˜e˜Gx H∈G M = e˜H B˜e˜Gx H∈G

= Be˜Gx = Bx,

precisely as desired.

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69 Appendix A

Single Burnside Rings

A.1 Sets with a Group Action

Throughout this section G denotes a group.

A.1 Deﬁnition (Opposite group). Let (G, •) be a group. The opposite group of G is the group (G◦,?) where G◦ = G as sets and where g ?h := h•g for elements g, h ∈ G.

A.2 Deﬁnition (G-sets). Given a set X, a left G-action on X or a left group action of G on X is a map G × X → X, (g, x) 7→ g · x satisfying

1. (gh) · x = g · (h · x) for all g, h ∈ G, x ∈ X, and

2. 1G · x = x for all x ∈ X.

When there is little possibility for confusion, we will write gx instead of g·x for each g ∈ G and x ∈ X. The map g · −: X → X, x 7→ g · x is a bijection as it is invertible, and it’s easy to see that ϕ: G → Sym(X), g 7→ g·− is a group homomorphism. Conversely, given a group homomorphism ψ : G → Sym(X) it’s clear that G × X → X, (g, x) 7→ [ψ(g)](x) is a left G-action on X, hence the two notions are equivalent. A left G-set is a pair (X, ·) such that X is a set and · is a left group action of G on X. As is frequently done, we will identify the underlying set X with (X, ·) when the group action is understood. In a similar fashion, we can deﬁne a right G-action on a set X as a left action of G◦ on X. As it’s convenient to envisage G acting on the right in a literal sense, we note that a right G action on X is equivalent to a map X × G → X, (x, g) 7→ x · g

70 satisfying x·(gh) = (x·g)·h for all g, h ∈ G, x ∈ X and x·1G = x for all x ∈ X.A right G-set is a set X equipped with a right G-action. When we say without qualiﬁcation that X is a G-set we mean that X is a left G-set. As the inversion map G → G◦, g 7→ g−1 is a group isomorphism, we see that every left action of G on X induces a right action of G on X by x · g := g−1 · x and vice versa. By a set with a group action we mean a G-set X for some not necessarily speciﬁed group G.

A.3 Example. A group G acts on itself by left and right multiplication and conjugation. It acts on its subgroup lattice by conjugation. If H ≤ G, then G acts on G/H via u · (gH) := (ug)H for all u, g ∈ G. Unless stated otherwise, the left coset space G/H is understood to have this left G-action. The disjoint union and Cartesian product of two G-sets are again G-sets in the obvious way.

A.4 Deﬁnition (G-set morphism). Suppose that X and Y are G-sets for a group G. A morphism of G-sets is a map f : X → Y such that f(g · x) = g · f(x) for all g ∈ G and x ∈ X. Sometimes G-set morphisms are called G-equivariant maps. If f is invertible, we call it a G-set isomorphism and we say that X and Y are isomorphic as G-sets, written ∼ X = Y . We denote the set of G-equivariant maps from X to Y by HomG(X,Y ).

A.5 Deﬁnition (Orbits, stabilizers, ﬁxed points). Suppose G is a group and X is a G-set. A subset Y ⊆ X is G-invariant provided g · y ∈ Y for each g ∈ G and y ∈ Y . Let x ∈ X. The G-orbit of x or the orbit of x under G is the set Gx := {g · x | g ∈ G}. The collection X/G of G-orbits in X forms a partition of X. Sometimes

X/G is denoted by XG and is called the set of coinvariants of X. In the case that X consists precisely of one orbit we say that X is transitive.

The stabilizer subgroup of x is the set stabG(x) := {g ∈ G | g · x = x}. Note −1 that for g, h ∈ G we have h(gx) = gx if and only if (g hg)x = x, whence stabG(gx) = g stabG(x). Given a subset U of G, the set of U-ﬁxed points of X is the set XU consisting of the points in X on which U acts trivially, i.e., XU = {x ∈ X | ∀u ∈ U, u · x = x}. For a ﬁxed g ∈ G, note that x ∈ XU if and only if gx ∈ XgUg−1 , from which it follows

71 that the map XU 7→ XgUg−1 sending x 7→ gx is bijective. In particular, |XH | = |XK | whenever H and K are conjugate in G. We will write Xg in the place of X{g}. Note that taking ﬁxed points commutes with the product and coproduct. That is,

(X q Y )H = XH q Y H , and (X × Y )H = XH × Y H for any subset H of G and any two G-sets X and Y .

A.6 Definition (Faithful and free actions). Let G be a group and X a G-set and ϕ: G → Sym(X) the group homomorphism induced by the action of G on X. The kernel of ϕ is the kernel of the action of G on X. From this definition, g ∈ G is in the kernel of the group action if and only if g · x = x for all x ∈ X. We say that G acts faithfully, or that the action of G on X is faithful or that X is a faithful G-set provided that ker ϕ is trivial. From this definition, if g ∈ ker ϕ, then Xg = X.

A G-set X is free provided that its point stabilizers are trivial, i.e., stabG(x) =

1G for all x ∈ X. When X is a free G-set we say that G acts freely on X. Note that g X = ∅ for each non-trivial element g ∈ G, so in particular any free action is also faithful.

A.7 Theorem (Orbit-stabilizer theorem). Let X be a transitive G-set. Then there exists some subgroup H of G such that X is isomorphic to G/H. In particular, when

G and X are ﬁnite and x ∈ X, we have |Gx| = [G : stabG(x)].

Proof. Let x ∈ X and set H := stabG(x). The map f : G/H → X, gH 7→ gx is clearly a well-deﬁned bijection. It is G-equivariant since f(u · gH) = f((ug)H) = (ug)x = u · (gx) = u · f(gH).

A.8 Proposition. Let H and K be subgroups of G, and let X be a G-set.

1. There is a bijection H HomG(G/H, X) ↔ X .

2. In particular, there is a bijection

H HomG(G/H, G/K) ↔ (G/K)

72 H −1 H and (G/K) = gK : g ∈ G, g Hg ≤ K . It follows that (G/K) = ∅ if and

only if H 6≤G K.

3. G/H =∼ G/K as G-sets if and only if H and K are conjugate in G. Moreover, the

transversal T (SG) of conjugacy classes of subgroups of G is also a complete set of representatives for the isomorphism classes of transitive G-sets.

Proof. (1) Let f : G/H → X be a G-set morphism such that f(H) = xf . Observe that H xf = f(H) = f(hH) = h · f(H) = h · xf for all h ∈ H, so xf ∈ X . As f is completely determined by the value f(H), the map

H HomG(G/H, X) → X , f 7→ f(H) is injective. It is surjective since for a fixed x ∈ XH , the map f : G/H → X defined by f(gH) = g · x is a G-set morphism. (2) The bijection is immediate from (1). Now, gK ∈ (G/K)H ⇐⇒ H · gK = gK ⇐⇒ (g−1Hg) · K = K ⇐⇒ g−1Hg ≤ K, as desired. −1 (3) Suppose that H =G K and write K = xHx for some x ∈ G. We claim that the map ϕ: G/K → G/H defined by ϕ(gK) = (gx)H is a G-set isomorphism. To see ϕ is well-defined note that if gK = uK for g, u ∈ G, then gxHx−1 = uxHx−1, whence gxH = uxH, i.e., ϕ(gK) = ϕ(uK). That ϕ is a G-set morphism is obvious. It is surjective because {gx | g ∈ G} = G and G/H is transitive. To establish injectivity, observe that for g, y ∈ G

ϕ(gK) = ϕ(yK) ⇐⇒ gxH = yxH ⇐⇒ y−1gxH = xH ⇐⇒ y−1gxHx−1 = xHx−1 ⇐⇒ y−1gK = K ⇐⇒ gK = yK.

∼ Conversely, suppose that G/H = G/K. Then HomG(G/H, G/K) 6= ∅ and By (2) H we know that (G/K) 6= ∅, so H ≤G K. By symmetry, K ≤G H, thus K =G H, as desired. As every transitive G set is of the form G/H for some subgroup H, this establishes that T (SG) is also a transversal for the isomorphism classes of transitive G-sets.

73 A.9 Corollary (Counting ﬁxed points). If G is a ﬁnite group with subgroups H and H −1 K, then |(G/K) | = |NG(H,K)|/|K|, where NG(H,K) := g ∈ G | gHg ≤ K is the transporter in G of H into K.

Proof. This is immediate from A.8.

A.10 Corollary. Every G-set is isomorphic to a disjoint union of transitive G-sets of the form G/H, where H is a subgroup of G.

Proof. Suppose X is a G-set. As the G-orbits of X form a partition, there is an indexing ` set Λ and a subset {xλ | λ ∈ Λ} of X such that X = λ∈Λ Gxλ. By deﬁnition of orbit and A.7, there exist G-set isomorphisms fλ : Gxλ → G/ stabG(xλ) for each λ ∈ Λ. The ` ` coproduct map λ∈Λ fλ : X → λ∈Λ G/ stabG(xλ) the desired G-set isomorphism.

A.11 Corollary (Orbit counting theorem). Let G be a ﬁnite group and X a ﬁnite G-set. Then 1 X |X/G| = |Xg|. |G| g∈G

Proof. Observe that

X |Xg| = |{(g, x) ∈ G × X | g · x = x}| g∈G X = | stabG(x)| x∈X X |G| = (by A.7) |Gx| x∈X X 1 = |G| |Gx| x∈X X X 1 = |G| |A| A∈X/G x∈A X = |G| 1 A∈X/G = |G| · |X/G|, as desired.

74 A.12 Corollary. Let G be a ﬁnite group and X a ﬁnite G-set. Then there exist

unique natural numbers nH (X) for each H ∈ T (SG), independent of the particular

transversal T (SG), such that

∼ a X = nH (X) · G/H,

H∈T (SG)

where nH (X) · G/H denotes the nH (X)-fold disjoint union of G/H with itself.

Proof. As in the proof of A.10, there is a ﬁnite subset {x1, . . . , xn} of X such that `n ∼ `n X = i=1 Gxi, whence X = i=1 G/ stabG(xi). From A.8, T (SG) is a complete set of representatives for the isomorphism classes of transitive G-sets, hence after grouping the

G-sets in the coproduct according to their isolcass, we obtain natural numbers nH (X) for each H ∈ T ( ) such that X =∼ ` n (X)·G/H. Their uniqueness follows from SG H∈T (SG) H the fact that the stabilizer subgroups of two points in the same orbit are conjugate.

A.13 Definition (Table of marks). Let G be a finite group. The matrix M defined H by MH,K := |(G/K) | = |NG(H,K)|/|K| (by A.9) for H,K ∈ T (SG) is called the

table of marks of G. If T (SG) is given a total ordering satisfying H K only if |H| ≤ |K|, then by A.8 we know M is upper triangular with non-zero diagonal entries

MH,H = [NG(H): H]. In particular, M is not singular.

We present a theorem due to Burnside that in the sequel allows us to embed the Burnside ring B(G) into its ghost ring via a mark homomorphism.

A.14 Theorem. Let G be a ﬁnite group, and X and Y be ﬁnite G-sets. Then X =∼ Y if and only if |XH | = |Y H | for any subgroup H of G.

Proof. First suppose that f : X → Y is a G-set isomorphism. For any subgroup H of G and x ∈ XH we have f(x) = f(hx) = hf(x) for all h ∈ H, hence f(XH ) ⊆ Y H . By symmetry, f −1(Y H ) ⊆ XH , which proves that |XH | = |Y H |. Conversely, suppose that |XH | = |Y H | for every subgroup H of G. By A.12, there exist natural natural numbers nK (X) and nK (Y ) for each K ∈ T (SG) such that X =∼ ` n (X) · G/K and Y =∼ ` n (Y ) · G/K. As taking ﬁxed points K∈T (SG) K K∈T (SG) K

75 commutes with the disjoint union, for each H ∈ T (SG) we obtain the equation

X H (nK (X) − nK (Y ))|(G/K) | = 0.

K∈T (SG)

In other words, if M denotes the table of marks of G and v is the |T (SG)|-dimensional vector whose Kth coordinate vK is given by vK := nK (X) − nK (Y ), then Mv = 0. But

as M is non-singular, it must follow that v = 0 and thus nK (X) − nK (Y ) = 0 for each ∼ K ∈ T (SG). It follows that X = Y , as desired.

A.2 Operations on Sets with a Group Action

Many of the elementary operations on group representations can be abstracted to G-sets. First, we make clear the connection between permutation representations and G-sets.

A.15 Deﬁnition (The category of G-sets). The category G-Set has as objets the G-sets and as morphisms the G-equivariant maps together with the usual function composition. The subcategory consisting of ﬁnite G-sets is denoted by G-set.

A.16 Definition (Permutation representations). Suppose R is a commutative ring. Let G be a finite group, X a finite G-set and RX the free R-module generated by X. The action of G on X extends R-linearly to an action on RX, and this induces an RG- module structure on RX. In this context, we call RX a permutation RG-module and the representation afforded by RX is a permutation representation.

A.17 Deﬁnition (Restriction along homomorphisms). Let ϕ: H → G be a group homomorphism and suppose X is a G-set. The morphism ϕ induces an H-set structure on X in the obvious way: h · x := ϕ(h) · x for all h ∈ H and x ∈ X. We say that this

H-set structure on X is obtained by restriction along ϕ and write Resϕ X to denote said H-set. Clearly any G-set morphism f : X → Y is also an H-set morphism between

Resϕ X and Resϕ Y . When we view f as an H-set morphism in this way we write

Resϕ f : Resϕ X → Resϕ Y . From these constructions we obtain the restriction functor along ϕ

Resϕ(−): G-set → H-set sending a G-set X to Resϕ X and a G-set morphism f : X → Y to Resϕ f.

76 A.18 Definition (Elementary operations). Let G be a group, H a subgroup of G, and G X a G-set. The restriction functor along the inclusion H,→ G is denoted by ResH . Likewise, if N is a normal subgroup of G and π : G → G/N is the canonical projection G homomorphism, the restriction functor along π is denoted InfG/N and is called the inflation functor from G/N to G. Given a group isomorphism ϕ: G → H, the restriction functor along ϕ−1 is called the isogation functor from G to H and is denoted by Isoϕ. In particular, Isoϕ takes a G-set to an H-set. We conclude with two more functors that are not interpreted so readily as restrictions along homomorphisms. Suppose now that H ≤ G and that X is an H-set. The direct product G×X carries the structure of a left H-set via h·(g, x) := (gh−1, hx·) G for all g ∈ G, x ∈ X, and h ∈ H. Let IndH X denote the fiber product G×X/H and note that it has a left G-set structure induced from left multiplication, i.e., g · (g0, x)H := (gg0, x)H for any g, g0 ∈ G and x ∈ X. Given an H-set morphism f : X → Y we G G G obtain a G-set morphism IndH f : IndH X → IndH Y , (g, x)H 7→ (g, f(x))H, which is well-defined since f is H-equivariant. This yields an induction from H to G functor G IndH : H-set → G-set. Now suppose that N is a normal subgroup of G and X a left G-set. As above, G/N × X carries a left G-set structure via g · (aN, x) := (ag−1N, g · x) for g, a ∈ G, x ∈ X. The set of G-orbits (G/N × X)/G has a natural left G/N-action, and hence we obtain a deflation from G to G/N functor

G DefG/N : G-set → G/N-set,X 7→ (G/N × X)/G.

It turns out that all of the ﬁve operations above can be realized by tensoring with the appropriate elementary biset appearing in 1.13.

There are some basic relations amongst the elementary operations in A.18 that mirror the usual relations between induction and restriction of representations. For a more comprehensive list, confer [Bou10, p. 2].

A.19 Theorem (Identities). Let G be a ﬁnite group and suppose that H and K are subgroups of G.

1. (Mackey formula for G-sets). If Y is an H-set, then the following are isomorphic

77 K-sets: G G ∼ a K H ResK IndH Y = IndK∩gH Isocg ResKg∩H Y, g∈[K\G/H] where [K\G/K] is a transversal for the set of (K,H)-double cosets of G.

2. (Frobenius reciprocity). If X is a G-set and Y is an H-set, then

G ∼ G G X × IndH Y = IndH ((ResH X)) × Y )

as G-sets. In the special case where Y = H/H, we obtain the G-set isomorphism

∼ G G X × (G/H) = IndH ResH X.

Proof. For sketches of the proofs, confer [Haz00, p. 743].

A.3 The Single Burnside Ring

Consult [Sol67] and [Haz00, pp. 741-802] for a more extensive treatment of the single Burnside ring. Unless otherwise indicated, G always denotes a finite group. In order to define the Burnside ring B(G), we first need the notion of the Grothendieck group of a commutative monoid.

A.20 Deﬁnition (Grothendieck group of a monoid). Suppose that (M,?) is commutative monoid and let (F, +) be the free abelian group generated by M. Further, let

F0 be the subgroup of F generated by {m + n − (m ? n) | m, n ∈ M}. Note that in F , m + n is not generally equal to m ? n. The Grothendieck group K0(M) of M is deﬁned to be the quotient F/F0. The Grothendieck group functor K0(−) from the category of commutative monoids to the category of abelian groups is left adjoint to the forgetful functor in the opposite direction.

A.21 Definition (Single Burnside ring). Let G be a finite group. The set of isomorphism classes of finite G-sets forms a commutative monoid (M,?) under (X) ? (Y ) := (X ` Y ), where X and Y are finite G-sets and (X) denotes the isomorphism class of

X. The Grothendieck group K0(M) of this commutative monoid is the single Burnside group B(G). Given an isomorphism class (X), we denote its image in B(G) by [X]. ∼ As is the case with the coproduct, it is easy to verify that if X1 = Y1 and ∼ ∼ X2 = Y2 as G-sets, then X1 × X2 = Y1 × Y2. The direct product of G-sets induces a

78 semiring structure on M, which extends linearly to the free abelian group F generated by

M, endowing F with a ring structure. The subgroup F0 of F is an ideal, hence B(G) has a commutative ring structure given by [X][Y ] := [X × Y ]. The multiplicative identity of B(G) is [•], where • is a singleton set with trivial G-action. When B(G) is endowed with this multiplicative structure it is called the single Burnside ring. The standard basis of B(G) is the set [G/K] as K runs through a transversal for the G-conjugacy classes of subgroups of G.

It doesn’t follow a priori that [X] − [Y ] = 0 if and only if X =∼ Y , that is, two non-isomorphic G-sets could potentially have the same image in B(G). Our ﬁrst result concerning the single Burnside ring B(G) is to establish that this is not the case.

A.22 Proposition. Let X and Y be ﬁnite G-sets for a ﬁnite group G. Then [X] = [Y ] in B(G) if and only if X and Y are isomorphic as G-sets.

Proof. If X and Y are isomorphic as G-sets, then they are in the same isomorphism class, hence their image in B(G) is the same. Conversely, [X] = [Y ] if and only if there

are natural numbers n and m and ﬁnite G-sets Ai, Bi, Cj, Dj where 1 ≤ i ≤ n and 1 ≤ j ≤ m such that n X (X) + [(Ai) + (Bi) − (Ai q Bi)] = i=1 m X (Y ) + [(Cj) + (Dj) − (Cj q Dj)] , j=1 which is equivalent to

n !  m  a a ∼ X q (Ai q Bi) q  (Cj q Dj) = i=1 j=1

 m  n ! a a Y q  (Cj q Dj) q (Ai q Bi) . j=1 i=1 For any subgroup H of G, if we take ﬁxed points of both sides we obtain

n !  n  H X H H X H H |X | + |Ai | + |Bi | +  |Cj | + |Dj | = i=1 j=1

n !  n  H X H H X H H |Y | + |Ai | + |Bi | +  |Cj | + |Dj | , i=1 j=1

79 which implies that |XH | = |Y H | for any subgroup H of G. From A.14 we can conclude that X and Y are isomorphic, as desired.

The multiplication of two standard basis elements within B(G) is described by the following proposition. As an initial lemma we explicitly determine how induction behaves on sets with a transitive group action.

A.23 Lemma. Let G be a group with subgroups K0 ≤ K ≤ G. Then as G-sets, G 0 ∼ 0 IndK (K/K ) = G/K

G 0 0 Proof. Recall that IndK (K/K ) = G ×K (K/K ). Consider the map

0 0 f : G/K → G ×K (K/K ) sending the left coset gK to the K-orbit (g, K0)K. This map is well-deﬁned because if g0 = gk0 for some k0 ∈ K0, then f(g0K0) = (gk0,K0)K = (gk0, k0−1K0)K = (g, K0)K = 0 0 0 0 f(gK ). It is G-equivariant since f(gK) = (g, K )K = g(1G,K )K = gf(K ). Any 0 0 element in G ×K (K/K ) can be written as (g, K )K for some g ∈ G, so f is certainly surjective. In general, K acts freely on G × X whenever X is a K-set, so each orbit in G × (K/K0) has cardinality |K|. Since there are |G|[K : K0] elements in G × (K/K0), 0 0 0 we see that |G/K | = |G|/|K | = |G ×K (K/K ), so f is injective as well.

A.24 Proposition (Multiplication of standard basis elements). Suppose G be a ﬁnite group with subgroups H and K, and let [K\G/H] be a set of representatives for the (K,H)-double cosets of G. Then

X [G/H] · [G/K] = G/(K ∩ gH ) g∈[K\G/H]

in B(G).

∼ G Proof. First note that G/H = IndH H/H as G-sets. In light of the identities in A.19,

80 we have G-set isomorphisms

∼ G G G/H × G/K = IndK ResK G/H (Frobenius reciprocity) ∼ G G G = IndK (ResK IndH H/H) ! ∼ G a K H = IndK IndK∩gH Isocg ResKg∩H H/H (Mackey formula) g∈[K\G/H] ! ∼ G a K g g = IndK IndK∩gH (K ∩ H /K ∩ H ) g∈[K\G/H] ! ∼ G a g = IndK K/K ∩ H (A.23) g∈[K\G/H] ∼ a G g = IndK K/K ∩ H g∈[K\G/H] a =∼ G/K ∩ gH (A.23). g∈[K\G/H]

This establishes the result.

A.25 Deﬁnition (Classical mark homomorphism). Let G be a ﬁnite group. Together, A.12 and A.22 demonstrate that the Burnside ring B(G) is a free Z-module with basis elements [G/H], where H ranges through a transversal for the conjugacy classes of subgroups of G.

Let ZSG denote the free Z-module whose canonical basis is the set SG of subgroups of G. A typical element a ∈ will be written P a · H, where ZSG H∈SG H aH ∈ Z. We endow ZSG with the ring structure deﬁned on the standard basis SG by  0 if U 6= V , U · V := U if U = V ,

where U, V ∈ SG. The conjugation action of G on SG extends Z-linearly to an action G on ZSG, and the set of G-ﬁxed points (SG) is spanned precisely by set of orbit sums [H]+ ∈ | H ∈ , where [H]+ := P 1 · K is an element of for any G ZSG S G K∈[H]G ZSG

H ∈ SG.

For subgroup H of G, deﬁne ΦH : B(G) → Z on the standard basis of B(G) by

H ΦH : B(G) → Z, [G/K] 7→ |(G/K) |.

81 As taking ﬁxed points commutes with taking disjoint unions and direct products, the map X ρ: B(G) → ZSG, a 7→ ΦH (a) · H H∈SG is a ring homomorphism, which we call the classical mark homomorphism of B(G) into

the classical ghost ring ZSG. Occasionally, especially when considering multiple groups, we might denote the classical mark homomorphism from B(G) to ZSG by ρG. As two H H G-sets X and Y are isomorphic if and only if |X | = |Y | for each H ∈ SG (A.14), the classical mark homomorphism is injective. Consequently, we may view B(G) as a H gHg−1 subring of ZSG if we wish. Moreover, since |X | = |X | for any H ∈ SG, g ∈ G, G we actually have that ρ(B(G)) ⊆ (ZSG) . Some authors refer to the ﬁxed point set G (ZSG) as the classical ghost ring.

Dress describes the cokernel of the classical mark homomorphism in [Dre69].

A.26 Theorem (Cokernel of the classical mark homomorphism). Let G

be a ﬁnite group with subgroups H, K in SG and set n(K,H) := | {x ∈ N (K)/K | hx, Ki = H} |. The element P α · H, where α ∈ , of G G H∈SG H H Z the ghost ring ZSG is in the image of the classical mark homomorphism ρ if and only

if for any K ∈ SG,

X n(K,H)αH ≡ 0 mod |NG(K)/K|.

H∈SG

Proof. Consult [Dre69] or [Haz00, Theorem 3.2.1].

A.27 Deﬁnition. Let R be a commutative ring and G a ﬁnite group. After tensoring the Z-module B(G) with R we obtain an R-algebra RB(G) := R ⊗Z B(G), which we

call the single Burnside R-algebra. Similarly, deﬁne RSG := R ⊗Z ZSG and Rρ :=

R ⊗Z ρ: RB(G) → RSG, and note that G-conjugation action on ZSG extends R- linearly to RSG. We identify 1 ⊗ B(G) ⊆ QB(G) with B(G) and 1 ⊗ ZSG ⊆ QSG with ZSG. G As the classical mark homomorphism ρ: B(G) → (ZSG) ⊆ ZSG is an injective Z-module morphism between two modules with the same rank over Z, the map

G Qρ := Q ⊗Z ρ: QB(G) → (QZSG) ⊆ QSG

82 H is a Q-algebra isomorphism. Generally, given a ∈ RB(G) and H ∈ SG we write |a | :=

RΦH (a).

A.28 Definition (Primitive idempotents in QB(G)). Let G be a finite group and + H ∈ SG. The inverse image of the orbit sum [H]G ∈ ZSG ⊆ QSG under Qρ is denoted G by eH . From the definition, for any K ∈ SG  1 if K is G-conjugate to H, G  ρK (eH ) = 0 otherwise.

G The set eH | H ∈ T (SG) is the set of primitive idempotents of QB(G). Independently, Gluck [Glu81] and Yoshida [Yos83] explicitly determined these idempotents.

A.29 Lemma. Let G be a ﬁnite group.

1. Let H ∈ SG. For any a ∈ QB(G),

G H G a · eH = |a | · eH .

H Conversely, if b ∈ QB(G) such that a · b = |a | · b for all a ∈ QB(G), then G b ∈ QeH .

G G 2. If H is a proper subgroup of G, then ResH eG = 0. Conversely, if b ∈ QB(G) G G satisﬁes ResH b = 0 for any proper subgroup H of G, then b ∈ QeG.

3. Let H ∈ SG and g ∈ G. Then

G G H G 1 G H g H gH ResH eH = eH , eH = IndH eH , (eH ) = egH . [NG(H): H]

Proof. Confer [Haz00, Lemma 3.3.1].

A.30 Definition (Möbius function). A poset (A, ≤) is called locally finite provided that for every pair a, b ∈ A the interval [a, b] := {x ∈ A | a ≤ x ≤ b} is a finite subset of A. Suppose that (A, ≤) is a locally finite poset. Let I denote the set of intervals of A and suppose R is a commutative ring. The incidence R-algebra A of (A, ≤) is the set of functions from I to R. If f ∈ A and a, b ∈ A we suppress notation a bit and write f(a, b) instead of f([a, b]). We can endow A with an R-algebra structure via the

83 convolution of functions. Given f, g ∈ A, deﬁne the convolution f ∗ g by (f ∗ g)(a, b) := P x∈[a,b] f(a, x) · g(x, b).

The multiplicative identity of A is Kronecker delta function δ(a, b) := δa,b. Let ξ ∈ A denote the function that constantly evaluates to 1. One can show that ξ is a central unit and that multiplication by ξ is the discrete analog to integration. The inverse of ξ is the Möbius function µ, which we will denote by µA if we wish to emphasize the underlying poset. For aesthetic typesetting reasons, we might occasionally write A A µa,b := µ (a, b), especially when values of µ appear as coeﬃcients. We can recursively deﬁne µ by   1 if a = b,  !  µ(a, b) := − P µ(a, x) if a < b,  a≤x

Later, we will need to consider the Möbius function µ of the subgroup lattice of a p-group. In this case, Hall [Hal33] gives a closed form for µ.

A.31 Lemma. Let p be a prime and P a p-group with subgroups H and K. Further,

let µ denote the Möbius function of the subgroup lattice SP . Then  k (k) ∼ k (−1) p 2 if H ¢ K and H/K = (Z/pZ) , µ(H,K) = 0 otherwise.

Proof. For an elegant proof, confer [HIO89, Corollary 3.5].

A.32 Theorem. Let G be a ﬁnite group and H ∈ SG. Then

G 1 X eH = |K|µ(K,H)[G/K], |NG(H)| K≤H

where µ is the Möbius function of SG.

Proof. Confer [Haz00, Lemma 3.3.2].

84 G A natural task is to determine under what conditions eH is actually in B(G). G More generally, given a set π of primes, we would like to know when eH ∈ Z(π)B(G). The following theorem of Dress [Dre69] gives an answer.

A.33 Theorem. Let G be a ﬁnite group and S a collection of subgroups of G that

is closed under G-conjugation. Set T (S) := S ∩ T (SG). For a set π of prime numbers, the following are equivalent:

P G 1. The idempotent H∈T (S) eH ∈ Z(π)B(G).

2. If H and K are subgroups of G such that H ¢ K and K/H is cyclic of prime order p ∈ π, then H ∈ S if and only if K ∈ S.

P G In particular, when π = {p} and S is the set of p-subgroups of G, then e := P ∈T (S) eP P is in Z(p)B(G). Note that e is the pre-image of the element := P ∈S P ∈ ZSG under the usual mark homomorphism.

85 Notation

Notation Description Page List S BR The R-biset category over a commutative ring R and 24 a class S of consisting of subsets SG,H ⊆ SG×H for each pair of ﬁnite groups G and H.

¡G×H and ∆G×H The collection of left-free and bifree subgroups of 13 product group G × H

Func(C, D) Functors from C to D 6

HomC(A, B) Morphisms between A and B in the category C 6 H\G/K The set of (H,K)-double cosets in G 5 [H\G/K] A transversal for the collection of (H,K)-double 5 cosets in G [H]G The set of G-conjugates of a subgroup H of G 5 [H]+ The orbit sum P K in 81 G K∈[H]G ZSG H =G K H is conjugate to K in G 5 HomR(M,N) R-module morphisms from M to N 6 HomC(X, −) Yoneda functor of X 6 HomG(X,Y ) Set of G-set morphisms from X to Y 71 H ≤G K H is subconjugate to K in G 5

 −1 Nϕ Nϕ := y ∈ NP (Q): ϕ ◦ (cy|Q) ◦ ϕ ∈ AutP (ϕ(Q)) 9

RSG The extension of scalars to R of the integral ghost 82 ring ZSG. RB(G) The extension of scalars to R of the integral Burnside 82 ring B(G).

SG Subgroup lattice of G 5 T (SG) Transversal for the conjugacy classes of subgroups of 5 G stabG(x) Stabilizer subgroup of x 71 Sym(X) Symmetric group of a set X 5

86 Notation Description Page List

X/G The set of G-orbits of the G-set X, also known as the 71 set of coinvariants and occasionally denoted by XG

L,M YN The poset summed over to determine the coeﬃcient 20 L,M aN

Z(π) Localization of Z at with respect to Z \ ∪p∈πpZ 5

87 Index

admissible module, 48 fully normalized,9 morphisms, 31 B-group, 59 quotient system, 43 bifree, 12 saturation,9 bifree Burnside group, 21 strongly F-closed, 43 biset, 11 bifree, 12 G-set, 70 elementary, 13 ghost ring, 81 left-free, 12 Grothendieck group, 78 morphism, 11 group action, 70 opposite, 11 coinvariants, 71 right-free, 12 faithful, 72 tensor product, 15 free, 72 transitive, 11 G-equivariant, 71 biset category, 24 invariant subset, 71 biset functor, 25 orbit, 71 Burnside R-algebra, 82 orbit counting, 74 Burnside ring, 78 restriction along a morphism, 76 marks, 75 stabilizer, 71 multiplication, 80 transitive, 71 single, 78 isomorphism class,6 double Burnside algebra, 15 double Burnside group, 15 left-free Burnside group, 21 double Burnside ring, 15 Möbius, 83 elementary biset, 13 mark homomorphism, 81 elementary subgroup, 13 cokernel, 82 enough idempotents, 48 marks, 75 free G-set, 72 opposite group, 70 Frobenius p-fusion system,8 orbit, 71 fusion system,8 orbit sum, 81 Nϕ,9 characteristic idempotent, 36 permutation module, 76 fully centralized,9 permutation representation, 76

88 pseudo-ring, 47 subfunctor, 58 section, 12 table of marks, 75 sets with a group action, 70 tensor product of bisets, 15 single Burnside ring, 78 twisted category algebra, 24 ghost ring, 81 twisted diagonal, 12 mark homomorphism, 81 stabilizer, 71 Yoneda functor,6