相关论文: Burnside Rings of Fusions Systems and their Unit G…

We define the representation ring of a saturated fusion system $\mathcal F$ as the Grothendieck ring of the semiring of $\mathcal F$-stable representations, and study the dimension functions of $\mathcal F$-stable representations using the…

In general the endomorphisms of a non-abelian group do not form a ring under the operations of addition and composition of functions. Several papers have dealt with the ring of functions defined on a group which are endomorphisms when…

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…

The purpose of the present paper is to prove for finitely generated groups of type I the following conjecture of A.Fel'shtyn and R.Hill, which is a generalization of the classical Burnside theorem. Let G be a countable discrete group, f one…

Suppose that a finite $p$-group $P$ admits a Frobenius group of automorphisms $FH$ with kernel $F$ that is a cyclic $p$-group and with complement $H$. It is proved that if the fixed-point subgroup $C_P(H)$ of the complement is nilpotent of…

Let $G$ be a finite group and $\mathcal{U} (\mathbb{Z} G)$ the unit group of the integral group ring $\mathbb{Z} G$. We prove a unit theorem, namely a characterization of when $\mathcal{U}(\mathbb{Z}G)$ satisfies Kazhdan's property…

We explore the category of internal categories in the usual category of (right) group-sets, whose objects are referred to as categorified group-sets. More precisely, we develop a new Burnside theory, where the equivalence relation between…

Linking systems were introduced to provide algebraic models for $p$-completed classifying spaces of fusion systems. Every linking system over a saturated fusion system $\mathcal{F}$ corresponds to a group-like structure called a locality.…

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…

We relate the construction of groups which realize saturated fusion systems and signaliser functors with homology decompositions of p-local finite groups. We prove that the cohomology ring of Robinson's construction is in some precise sense…

Let B be a p-block of the finite group G. We observe that the p-fusion of G constrains the module structure of B: Any basis of B that is invariant under the left and right multiplications of a chosen Sylow p-subgroup S of G must in fact…

An $S$-ring (a Schur ring) is said to be separable with respect to a class of groups $\mathcal{K}$ if every algebraic isomorphism from the $S$-ring in question to an $S$-ring over a group from $\mathcal{K}$ is induced by a combinatorial…

This paper develops links between the Burnside ring of a finite group $G$ and the slice Burnside ring}. The goal is to gain a better understanding of ghost maps, idempotents, prime spectrum of these Burnside rings and connections between…

If G is a non-cyclic finite group, non-isomorphic G-sets X, Y may give rise to isomorphic permutation representations C[X] and C[Y]. Equivalently, the map from the Burnside ring to the representation ring of G has a kernel. Its elements are…

The famous Burnside-Schur theorem states that every primitive finite permutation group containing a regular cyclic subgroup is either 2-transitive or isomorphic to a subgroup of a 1-dimensional affine group of prime degree. It is known that…

Let $RG$ be the gruop ring of the group $G$ over ring $R$ and $\mathscr{U}(RG)$ be its unit group. Finding the structure of the unit group of a finite group ring is an old topic in ring theory. In, G. Tang et al: Unit Groups of Group…

For $p\in\{2,3\}$ it is known that a saturated $p$-fusion system is realizable if and only if each of its components is realizable by a finite simple group. For primes $p\geq 5$ this is false. Building on work of Broto, M{\o}ller, Oliver…

Saturated fusion systems are categories generalizing important aspects of conjugacy of $p$-subgroups in finite groups. It was shown by Chermak that there are group-like structures called regular localities associated to saturated fusion…

For a finite group $G$, we define a ghost ring and a mark homomorphism for the double Burnside ring of left-free $(G,G)$-bisets. In analogy to the case of the Burnside ring $B(G)$, the ghost ring has a much simpler ring structure, and after…

The proper subgroup $B$ of the group $G$ is called {\it strongly embedded}, if $2\in\pi(B)$ and $2\notin\pi(B \cap B^g)$ for any element $g \in G \setminus B $ and, therefore, $ N_G(X) \leq B$ for any 2-subgroup $ X \leq B $. An element $a$…