Related papers: Reductions to simple fusion systems

We show that the automorphism group of a linking system associated to a saturated fusion system $\mathcal{F}$ depends only on $\mathcal{F}$ as long as the object set of the linking system is $\mathrm{Aut}(\mathcal{F})$-invariant. This was…

Lie group theory states that knowledge of a $m$-parameters solvable group of symmetries of a system of ordinary differential equations allows to reduce by $m$ the number of equations. We apply this principle by finding some \emph{affine…

Let $P$ be a closed convex cone in $\mathbb{R}^{n}$. Assume that $P$ is spanning i.e. $P-P=\mathbb{R}^{n}$ and pointed i.e. $P \cap -P=\{0\}$. Let $\alpha:=\{\alpha_{x}:x \in P\}$ be a $\sigma$-weakly continuous family of unital normal…

Let $G$ be a connected reductive complex algebraic group, and $E$ a complex elliptic curve. Let $G_E$ denote the connected component of the trivial bundle in the stack of semistable $G$-bundles on $E$. We introduce a complex analytic…

Let $\E/\Q$ be a fixed elliptic curve over $\Q$ which does not have complex multiplication. Assuming the Generalized Riemann Hypothesis, A. C. Cojocaru and W. Duke have obtained an asymptotic formula for the number of primes $p\le x$ such…

Glauberman's $Z^*$-theorem and analogous statements for odd primes show that, for any prime $p$ and any finite group $G$ with Sylow $p$-subgroup $S$, the centre of $G/O_{p^\prime}(G)$ is determined by the fusion system $\mathcal{F}_S(G)$.…

Drinfeld doubles of finite subgroups of SU(2) and SU(3) are investigated in detail. Their modular data - S, T and fusion matrices - are computed explicitly, and illustrated by means of fusion graphs. This allows us to reexamine certain…

This thesis is divided into two parts. In the first part we study completely integrable systems, and their underlying structures, in detail. We study their deformation theory and the different equivalence relations surrounding it. We…

A classification is given for factorizations of almost simple groups with at least one factor solvable, and it is then applied to characterize $s$-arc-transitive Cayley graphs of solvable groups, leading to a striking corollary: Except the…

The computation of the cohomology for finite groups of Lie type in the describing characteristic is a challenging and difficult problem. In earlier work, the authors constructed an induction functor which takes modules over the finite group…

Let $G$ be a finite group, $N(G)$ be the set of conjugacy classes of the group $G$. In the present paper it is proved $G\simeq L$ if $N(G)=N(L)$, where $G$ is a finite group with trivial center and $L$ is a finite simple group.

We classify finite groups in which the centralisers of certain non-central elements are soluble. This includes a full structural description of groups whose non-central element centralisers are all soluble, and a reduction theorem for the…

In this paper we show that all indecomposable nondegenerate set-theoretical solutions to the Quantum Yang-Baxter equation on a set of prime order are affine, which allows us to give a complete and very simple classification of such…

Let M be ternary, homogeneous and simple. We prove that if M is finitely constrained, then it is supersimple with finite SU-rank and dependence is $k$-trivial for some $k < \omega$ and for finite sets of real elements. Now suppose that, in…

We complete the determination of saturated fusion systems on maximal class 3-groups of rank two.

A key result in a 2004 paper by S. Arkhipov, R. Bezrukavnikov, and V. Ginzburg (ABG) gives an equivalence of the bounded derived category of finite dimensional modules for the principal block of a Lusztig quantum algebra at an $\ell^{th}$…

In this paper we study the cellularization of classifying spaces of saturated fusion systems with respect to classifying spaces of finite p-groups. We give explicit algebraic criteria to decide when a classifying space is cellular.…

We show that if $R$ is a, not necessarily unital, ring graded by a semigroup $G$ equipped with an idempotent $e$ such that $G$ is cancellative at $e$, the non-zero elements of $eGe$ form a hypercentral group and $R_e$ has a non-zero…

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…

We show that a weakly integral braided fusion category C such that every simple object of C has Frobenius-Perron dimension at most 2 is solvable. In addition, we prove that such a fusion category is group-theoretical in the extreme case…