Related papers: Saturated fusion systems on $p$-groups of maximal …

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,…

Tarnauceanu [Archiv der Mathematik, 102 (1), (2014), 11--14] gave a characterisation of elementary abelian $2$-groups in terms of their maximal sum-free sets. His theorem states that a finite group $G$ is an elementary abelian $2$-group if…

We consider the Burnside ring $A(\mathcal{F})$ of $\mathcal{F}$-stable $S$-sets for a saturated fusion system $\mathcal{F}$ defined on a $p$-group $S$. It is shown by S. P. Reeh that the monoid of $\mathcal{F}$-stable sets is a free…

In this paper we prove characterizations of $p$-nilpotency for fusion systems and $p$-local finite groups that are inspired by results in the literature for finite groups. In particular, we generalize criteria by Atiyah, Brunetti,…

We show that every saturated fusion system $\mathcal{F}$ has a unique minimal $\mathcal{F}$-characteristic biset $\Lambda_\mathcal{F}$. We examine the relationship of $\Lambda_\mathcal{F}$ with other concepts in $p$-local finite group…

We classify fusion systems $\mathcal{F}$ in which $O_p(\mathcal{F})=\{1\}$, and there are two $\mathrm{Aut}_{\mathcal{F}}(S)$-invariant essential subgroups whose normalizer systems generate $\mathcal{F}$. We employ the amalgam method and,…

In a 1969 article, A. Dress described the prime ideals of the Burnside ring of a finite group G and the inclusion relations between them. One may ask whether similar results exist for the Burnside ring of a saturated fusion system F on a…

Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms. For any fixed prime divisor $p$ of $|G|$, we provide a complete characterization of the structure of a group $G$ in which every maximal $A$-invariant…

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…

Let $G$ be the generalized free product of two groups with an amalgamated subgroup. We propose an approach that allows one to use results on the residual $p$-finiteness of $G$ for proving that this generalized free product is residually a…

We define here two new classes of saturated fusion systems, reduced fusion systems and tame fusion systems. These are motivated by our attempts to better understand and search for exotic fusion systems: fusion systems which are not the…

Let $q$ be a power of a fixed prime $p$. We classify up to isomorphism all simple saturated fusion systems on a certain class of $p$-groups constructed from the polynomial representations of $\mathrm{SL}_2(q)$, which includes the Sylow…

Transfer systems are combinatorial objects which classify $N_\infty$ operads up to homotopy. By results of A. Blumberg and M. Hill, every transfer system associated to a linear isometries operad is also saturated (closed under a particular…

Let $G$ be a finite group and let $p$ be a prime. In this paper, we study the structure of finite groups with a large number of $p$-regular conjugacy classes or, equivalently, a large number of irreducible $p$-modular representations. We…

Let $\mathfrak{J}$ be a class of non-abelian simple groups and $\mathfrak{X}$ be a class of groups. A chief factor $H/K$ of a group $G$ is called $\mathfrak{X}$-central in $G$ provided $(H/K)\rtimes G/C_G(H/K)\in\mathfrak{X}$. We say that…

In this paper we describe an algorithm for finding the nilpotency class, and the upper central series of the maximal normal p-subgroup N(G) of the automorphism group, Aut(G) of a bounded (or finite) abelian p-group G. This is the first part…

Let $G$ be a finite group and $S$ a subset of $G$. Then $S$ is product-free if $S \cap SS = \emptyset$, and complete if $G^{\ast} \subseteq S \cup SS$. A product-free set is locally maximal if it is not contained in a strictly larger…

The structure of finite and locally finite groups in which every element has prime power order (CP-groups) is well known. In this paper we note that the combination of our earlier results with the available information on the structure of…

A character table $X$ for a saturated fusion system $\mathcal{F}$ on a finite $p$-group $S$ is the square matrix of values associated to a basis of the lattice of virtual $\mathcal{F}$-stable ordinary characters of $S$. We investigate a…

In this paper we give a classification of the rank two p-local finite groups for odd p. This study requires the analisis of the possible saturated fusion systems in terms of the outer automorphism group ant the proper F-radical subgroups.…