David A. Craven, Adam Glesser
In this article we study several classes of `small' 2-groups: we complete the classification, started in [Stancu, 2006], of all saturated fusion systems on metacyclic p-groups for all primes p. We consider Suzuki 2-groups, and classify all center-free saturated fusion systems on 2-groups of 2-rank 2. We end by classifying all possible F-centric, F-radical subgroups in saturated fusion systems on 2-groups of 2-rank 2.
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We introduce the notion of a pro-fusion system on a pro-p group, which generalizes the notion of a fusion system on a finite p-group. We also prove a version of Alperin's Fusion Theorem for pro-fusion systems.
We classify all (saturated) fusion systems on bicyclic 2-groups. Here, a bicyclic group is a product of two cyclic subgroups. This extends previous work on fusion systems on metacyclic 2-groups (see [Craven-Glesser, 2012] and [Sambale, 2012]). As an application we prove Olsson's Conjecture for all blocks with bicyclic defect groups.
We describe a purely group-theoretic condition on an element g of a finite group G which implies that g has coefficient zero in every central idempotent element of the group ring RG, provided that R is a ring of prime characteristic. We use this condition to prove that the fusion system associated to a block of an alternating group is always isomorphic to the group fusion system of an alternating group.
We show that every (not necessarily saturated) fusion system can be realized as a full subcategory of the fusion system of a finite group. This result extends our previous work \cite{Park2010} and complements the related result \cite{LearyStancu2007} by Leary and Stancu.
For any prime $p$ and $S$ a $p$-group isomorphic to a Sylow $p$-subgroup of a rank $2$ simple group of Lie type in characteristic $p$, we determine all saturated fusion systems supported on $S$ up to isomorphism.
Block pro-fusion systems for profinite groups and blocks with infinite dihedral defect groups
本文引入profinite群块的块pro-fusion系统，证明Puig幂零块结构定理的profinite版本，进而得出以无限二面体pro-2群为亏群的块仅一个Morita等价类的结论。
Let G be a finite group. A complete system of pairwise orthogonal idempotents is constructed for the wreath product of G by the symmetric group by means of a fusion procedure, that is by consecutive evaluations of a rational function with values in the group ring. This complete system of idempotents is indexed by standard Young multi-tableaux. Associated to the wreath product of G by the symmetric group, a Baxterized form for the Artin generators of the symmetric group is defined and appears in the rational function used in the fusion procedure.
Continuing our previous work, we determine a minimal left characteristic biset $X$ for every exotic fusion system $\mathcal{F}$ on the extraspecial group $S$ of order $7^3$ and exponent $7$ discovered by Ruiz and Viruel, and analyze the finite group $G$ obtained from $X$ which realizes $\mathcal{F}$ as the full subcategory of the $7$-fusion system of $G$. In particular, we obtain an upper bound for the exoticity index for $\mathcal{F}$.
Let P be a finite metacyclic 2-group and F a fusion system on P. We prove that F is nilpotent unless P has maximal class or P is homocyclic, i.e. P is a direct product of two isomorphic cyclic groups. As a consequence we obtain the numerical invariants for 2-blocks with metacyclic defect groups. This paper is a part of the author's PhD thesis.
Proving a conjecture for fusion systems on a class of groups
本文证明了奇异与块奇异融合系统重合的猜想对于极大幂零类例外 p-群（p ≥ 5）成立，推广了作者先前结果并结合 Grazian-Parker 结论，同时验证了小素数情形。
Weight conjectures for fusion systems on an extraspecial group
本文验证了特殊群上融合系统的某些猜想，包括7素数的Ruiz-Viruel奇异融合系统，并验证了Robinson的普通重量猜想。
Exotic Fusion Systems Related to Sporadic Simple Groups (Version With Code)
本文借助 MAGMA 计算，对与散在单群 Co_1、F_3 和 M 在奇素数处相关的奇异及饱和融合系统进行分类。
Fusion invariant characters of p-groups
本文将Bárcenas--Cantarero定理推广到非饱和融合系，证明p-群的不可分解F-不变特征标数不小于F-共轭类数，并构造反例否定Cantarero--Combariza猜想。
For a prime $p$, we describe a protocol for handling a specific type of fusion system on a $p$-group by computer. These fusion systems contain all saturated fusion systems. This framework allows us to computationally determine whether or not two subgroups are conjugate in the fusion system for example. We describe a generation procedure for automizers of every subgroup of the $p$-group. This allows a computational check of saturation. These procedures have been implemented using MAGMA. We describe a program to search for saturated fusion systems $\mathcal{F}$ on $p$-groups with $O_p(\mathcal{F…
We complete the determination of saturated fusion systems on maximal class 3-groups of rank two.
Many of the conjectures of current interest in the representation theory of finite groups in characteristic $p$ are local-to-global statements, in that they predict consequences for the representations of a finite group $G$ given data about the representations of the $p$-local subgroups of $G$. The local structure of a block of a group algebra is encoded in the fusion system of the block together with a compatible family of Külshammer-Puig cohomology classes. Motivated by conjectures in block theory, we state and initiate investigation of a number of seemingly local conjectures for arbitrary t…
To any block idempotent $b$ of a group algebra $kG$ of a finite group $G$ over a field $k$ of characteristic $p>0$, Puig associated a fusion system and proved that it is saturated if the $k$-algebra $kC_G(P)e$ is split, where $(P,e)$ is a maximal $kGb$-Brauer pair. We investigate in the non-split case how far the fusion system is from being saturated by describing it in an explicit way as being generated by the fusion system of a related block idempotent over a larger field together with a single automorphism of the defect group.
We generalize the Reduction Theorem of Kessar-Stancu so it can be applicable to exotic fusion systems over the maximal nilpotency class of rank two $3$-groups. This is an essential step towards proving that these fusion systems are also block exotic.
A family of exotic fusion systems generalizing the group fusion systems on Sylow $p$-subgroups of $\mathrm{G}_2(p^a)$ and $\mathrm{Sp}_4(p^a)$ is constructed.
The aim of this paper is to generalise the notion of p-stability to fusion systems. We study the question how Qd(p) is involved in finite simple groups. We show that with a single exception a simple group involving Qd(p) has a subgroup isomorphic to either Qd(p) or a central extension of it by a cyclic group of order p. We define p-stability for fusion systems, characterise some of its properties and prove a fusion theoretic version of Thomson's maximal subgroup theorem. We introduce the notion of section p-stability both for groups and fusion systems and prove a version of Glauberman's theore…
We study saturated fusion systems on $p$-groups having sectional rank $3$ for all odd primes $p$. For $p\geq 5$, we obtain a complete classification of the ones that do not have any non-trivial normal $p$-subgroups.
Let $p$ be an odd prime, and let $S$ be a $p$-group with a unique elementary abelian subgroup $A$ of index $p$. We classify the simple fusion systems over all such groups $S$ in which $A$ is essential. The resulting list, which depends on the classification of finite simple groups, includes a large variety of new, exotic simple fusion systems.
Realizability of fusion systems by discrete groups
本文研究离散p-挠群上融合系统的可实现性，证明紧李群的融合系统可由线性挠群实现，并提出新工具证明某些融合系统不是序列可实现的，通过两大族例子加以说明。
We prove, when $S$ is a $2$-group of order at most $2^9$, that each reduced fusion system over $S$ is the fusion system of a finite simple group and is tame. It then follows that each saturated fusion system over a $2$-group of order at most $2^9$ is realizable. What is most interesting about this result is the method of proof: we show that among $2$-groups with order in this range, the ones which can be Sylow $2$-subgroups of finite simple groups are almost completely determined by criteria based on Bender's classification of groups with strongly $2$-embedded subgroups.
For a saturated fusion system $\mathcal F$ on a $p$-group $S$, we study the Burnside ring of the fusion system $B(\mathcal F)$, as defined by Matthew Gelvin and Sune Reeh, which is a subring of the Burnside ring $B(S)$. We give criteria for an element of $B(S)$ to be in $B(\mathcal F)$ determined by the $\mathcal F$-automorphism groups of essential subgroups of $S$. When $\mathcal F$ is the fusion system induced by a finite group $G$ with $S$ as a Sylow $p$-group, we show that the restriction of $B(G)$ to $B(S)$ has image equal to $B(\mathcal F)$. We also show that for $p=2,$ we can gain infor…
Fusion Systems on Sylow $3$-subgroups of Fischer and Monster sporadic groups -- I
本文研究菲舍尔和莫诺斯特奇奇异性群中3-子群的无核融合系统分类，证明3-群不支持奇异融合系统，为后续奇素数p的奇异性群Sylow子群融合系统分类奠定基础。
Representations of Coxeter groups over fusion rings and hyperplane complements
本文研究Coxeter群在融合环上的忠实实现及其Vinberg系统，证明其诱导超平面补集嵌入，从而为Artin–Tits群间的强可容许同态提供几何实现。
Fusion Systems and Simple Groups With Class Two Sylow $p$-subgroups
本文确定了支撑在幂零类二有限 $p$-群上的所有约化饱和融合系统，并由此给出了具有类二 Sylow $2$-子群的有限简单群分类的新证明。
We study group models for fusion systems and construct homology decompositions for the models of Robinson and Leary-Stancu type.
Let $q$ be a nontrivial odd prime power, and let $n \ge 2$ be a natural number with $(n,q) \ne (2,3)$. We characterize the groups $PSL_n(q)$ and $PSU_n(q)$ by their $2$-fusion systems. This contributes to a programme of Aschbacher aiming at a simplified proof of the classification of finite simple groups.
In this article, we compare two different notions of partially defined group strutures, namely partial groups and pregroups, as introduced by Chermak and Stallings respectively. In particular we prove that the category of pregroups can be seen as a full subcategory of the category of partial groups. We also bring out some conjugation properties about elements and subgroups of finite order in pregroups and their universal groups. We then use these to investigate the question of realisability of fusion systems in finite pregroups.
We study a certain family of simple fusion systems over finite $3$-groups, ones that involve Todd modules of the Mathieu groups $2M_{12}$, $M_{11}$, and $A_6=O^2(M_{10})$ over $\mathbb{F}_3$, and show that they are all isomorphic to the $3$-fusion systems of almost simple groups. As one consequence, we give new $3$-local characterizations of Conway's sporadic simple groups.
We determine for which known finite simple groups $G$ and which primes $p$ the $p$-fusion system of $G$ is simple. This means first collecting together the results that were already known (and correcting two errors made in an earlier study of this question), and then handling the remaining cases. At the same time, we develop some new tools to use when determining $O^{p'}(\mathcal{F})$ for arbitrary saturated fusion systems.
It is a long-standing open problem raised by Starostin to describe all finite groups with soluble centralizers of involutions. One can observe that if the centralizer fusion system of an involution is nilpotent, then the centralizer of that involution is soluble. In this paper, we classify the cases when the centralizer fusion system of a central involution in a finite group whose all involutions have soluble centralizers is a nilpotent fusion system. Indeed, we analyse the case when the solvable radical has odd order and the corresponding factor group is simple.
Given a saturated fusion system $\mathcal{F}$ over a $2$-group $S$, we prove that $S$ is abelian provided any element of $S$ is $\mathcal{F}$-conjugate to an element of $Z(S)$. This generalizes a Theorem of Camina--Herzog, leading to a significant simplification of its proof. More importantly, it follows that any $2$-block $B$ of a finite group has abelian defect groups if all $B$-subsections are major. Furthermore, every $2$-block with a symmetric stable center has abelian defect groups.
Fusion Systems on Sylow $3$-subgroups of Fischer and Monster sporadic groups -- II
该研究对散在群$\mathrm{Fi}_{24}'$和$\mathrm{M}$的西罗3-子群上的无核融合系统进行分类，完成了奇数$p$时散在群西罗$p$-子群无核融合系统分类项目，还扩展理论并描述算法，大幅加快找$p$-群无核融合系统的过程。
We prove here that with a very small number of exceptions, when $G$ is a sporadic simple group and $p$ is a prime such that the Sylow $p$-subgroups of $G$ are nonabelian, then $Out(G)$ is isomorphic to the outer automorphism groups of the fusion and linking systems of $G$. In particular, the $p$-fusion system of $G$ is tame in the sense of [AOV1], and is tamely realized by $G$ itself except when $G\cong M_{11}$ and $p=2$. From the point of view of homotopy theory, these results also imply that $Out(G)\cong Out(BG^\wedge_p)$ in many (but not all) cases.
We prove that the factorization of a saturated fusion system over a discrete $p$-toral group as a product of indecomposable subsystems is unique up to normal automorphisms of the fusion system and permutations of the factors. In particular, if the fusion system has trivial center, or if its focal subgroup is the entire Sylow group, then this factorization is unique (up to the ordering of the factors). This result was motivated by questions about automorphism groups of products of fusion systems.
For a prime number $p$, a finite $p$-group of order $p^n$ has maximal class if it has nilpotency class $n-1$. Here we examine saturated fusion systems on maximal class $p$-groups and, in particular, we describe all the reduFor a prime number $p$, a finite $p$-group of order $p^n$ has maximal class if and only if it has nilpotency class $n-1$. Here we examine saturated fusion systems $\mathcal F$ on maximal class $p$-groups $S$ of order at least $p^4$. The Alperin-Goldschmidt Theorem for saturated fusion systems yields that $\mathcal F$ is entirely determined by the $\mathcal F$-automorphisms o…
We finish the classification, begun in two earlier papers, of all simple fusion systems over finite nonabelian $p$-groups with an abelian subgroup of index $p$. In particular, this gives many new examples illustrating the enormous variety of exotic examples that can arise. In addition, we classify all simple fusion systems over infinite nonabelian discrete $p$-toral groups with an abelian subgroup of index $p$. In all of these cases (finite or infinite), we reduce the problem to one of listing all $\mathbb{F}_pG$-modules (for $G$ finite) satisfying certain conditions: a problem which was solve…
On Wielandt's Join Theorem for fusion systems and localities
本文研究融合系统和局部性中的威拉德定理，证明了其在融合系统和局部性中的版本，并提出一个独立的群论结果。
In this short note we study the cohomology algebra of saturated fusion systems using finite groups which realize saturated fusion systems and Hochschild cohomology of group algebras. A similar result to a theorem of Alperin is proved for varieties of cohomology algebras of fusions systems associated to block algebras of finite groups.
Let G be a finite group, p a fixed prime and P a Sylow p-subgroup of G. In this short note we prove that if p is odd, G is p-nilpotent if and only if P controls fusion of cyclic groups of order p. For the case p=2, we show that G is p-nilpotent if and only if P controls fusion of cyclic groups of order 2 and 4.
One way to understand the mod p homotopy theory of classifying spaces of finite groups is to compute their BZ/p-cellularization. In the easiest cases this is a classifying space of a finite group (always a finite p-group). If not, we show that it has infinitely many non-trivial homotopy groups. Moreover they are either p-torsion free or else infinitely many of them contain p-torsion. By means of techniques related to fusion systems we exhibit concrete examples where p-torsion appears.
We determine, for $p$ odd, all saturated fusion systems on a Sylow $p$-subgroup $S$ of the unitary group $SU_4(p)$ and we prove that they are all realizable by finite groups. In particular, we prove that $S$ does not support any exotic fusion systems.
This paper introduces the notion of fusion action system, an abstraction of the $p$-local data of a finite group acting on a finite set. Fusion action systems are closely connected with the theory of fusion systems; we detail the relationship of this new definition to the existing literature. We also develop a theory of classifying spaces of fusion action systems, analogous to the relationship between by $p$-local finite groups and fusion systems. This allows us to discuss the covering space theory arising from a group action in a fusion-theoretic context.
The transporter systems of Oliver and Ventura and the localities of Chermak are classes of algebraic structures that model the $p$-local structures of finite groups. Other than the transporter categories and localities of finite groups, important examples include centric, quasicentric, and subcentric linking systems for saturated fusion systems. These examples are however not defined in general on the full collection of subgroups of the Sylow group. We study here punctured groups, a short name for transporter systems or localities on the collection of nonidentity subgroups of a finite $p$-grou…
Suppose that all nontrivial subsections of a $p$-block $B$ are conjugate (where $p$ is a prime). By using the classification of the finite simple groups, we prove that the defect groups of $B$ are either extraspecial of order $p^3$ with $p \in \{3,5\}$ or elementary abelian.
For $S$ a Sylow $p$-subgroup of the group $\mathrm{G}_2(p)$ for $p$ odd, up to isomorphism of fusion systems, we determine all saturated fusion systems $\mathcal{F}$ on $S$ with $O_p(\mathcal{F})=1$. For $p \ne 7$, all such fusion systems are realized by finite groups whereas for $p=7$ there are $29$ saturated fusion systems of which $27$ are exotic.
Exotic Fusion System on a Subgroup of the Monster
本研究证明Grazian提出的怪兽群子群上的异常融合系统为块异常的，进而得出对所有截面秩3且p≥5的p-群，异常融合系统与块异常融合系统等价。
The fusion-stable tom Dieck homomorphism
本文证明了tom Dieck同态及其因子符号同态在限制到p-群的融合稳定子群时仍满射，进而解决了Mazza-Miller的问题，确定了Lefschetz同态满射当且仅当域为F₂。
Realizability of fusion systems by discrete groups: II
该研究对比离散p环面群上饱和融合系统的四种可实现性类型，针对特定局部有限群给出相关融合系统与链接系统的显式构造，证明了饱和性判定条件并推广了Cartan-Eilenberg稳定元素定理。
We show that every fusion system on a p-group S is equal to the fusion system associated to a discrete group G with the property that every p-subgroup of G is conjugate to a subgroup of S.
An $\mathcal{F}$-essential subgroup is called a pearl if it is either elementary abelian of order $p^2$ or non-abelian of order $p^3$. In this paper we start the investigation of fusion systems containing pearls: we determine a bound for the order of $p$-groups containing pearls and we classify the saturated fusion systems on $p$-groups containing pearls and having sectional rank at most $4$.
Large subgroups of fusion systems and localities
本文探讨了如何将Meierfrankenfeld等人关于有限简单群分类的程序结果转化为融合系统和局部性的概念，定义了大规模子群并分析了其性质。
For a finite group G of Lie type and a prime p, we compare the automorphism groups of the fusion and linking systems of G at p with the automorphism group of G itself. When p is the defining characteristic of G, they are all isomorphic, with a very short list of exceptions. When p is different from the defining characteristic, the situation is much more complex, but can always be reduced to a case where the natural map from Out(G) to outer automorphisms of the fusion or linking system is split surjective. This work is motivated in part by questions involving extending the local structure of a …
For a finite abelian $p$-group $A$ and a subgroup $Γ\le\text{Aut}(A)$, we say that the pair $(Γ,A)$ is fusion realizable if there is a saturated fusion system $\mathcal{F}$ over a finite $p$-group $S\ge A$ such that $C_S(A)=A$, $\textrm{Aut}_{\mathcal{F}}(A)=Γ$ as subgroups of $\text{Aut}(A)$, and $A$ is not normal in $\mathcal{F}$. In this paper, we develop tools to show that certain representations are not fusion realizable in this sense. For example, we show, for $p=2$ or $3$ and $Γ$ one of the Mathieu groups, that the only $\mathbb{F}_pΓ$-modules that are fusion realizable (up to extension…
Interpolated family of non group-like simple integral fusion rings of Lie type
本文利用类 Verlinde 公式计算 Rep(PSL(2,q)) 的通用融合规则并构造插值族 R_q，得到非群状单整融合环无限族，揭示其范畴化、Drinfeld 中心与弱群论性质的等价判据。
Transitive fusion systems over a class of finite p-groups
本文针对有限 p-群上传递融合系统的分类猜想，证明当非交换有限 p-群 S 的 p-秩为 2 时，S 必为阶 p³ 的超特殊群。
We give another proof of an observation of Thévenaz \cite{T1989} and present a fusion system version of it. Namely, for a saturated fusion system $\CF$ on a finite $p$-group $S$, we show that the number of the $\CF$-conjugacy classes of cyclic subgroups of $S$ is equal to the rank of certain square matrices of numbers of orbits, coming from characteristic bisets, the characteristic idempotent and finite groups realizing the fusion system $\CF$ as in our previous work \cite{P2010}.
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