The normaliser decomposition for p-local finite groups

Decomposition of locally compact coset spaces

In a previous article by the author and P. Wesolek, it was shown that a compactly generated locally compact group $G$ admits a finite normal series $(G_i)$ in which the factors are compact, discrete or irreducible in the sense that no closed normal subgroup of $G$ lies properly between $G_{i-1}$ and $G_{i}$. In the present article, we generalize this series to an analogous decomposition of the coset space $G/H$ with respect to closed subgroups, where $G$ is locally compact and $H$ is compactly generated. This time, the irreducible factors are coset spaces $G_{i}/G_{i-1}$ where $G_{i}$ is compa…

On Mackey Decomposition for locally profinite groups

To study induced representation of some class of groups, Mackey's theory is very useful. In this paper, we consider some generalization of Mackey's theory for locally profinite groups. In particular, we give conditions on groups under which we have the Mackey decomposition and some examples such that we do not have the Mackey decomposition in some sense.

通过覆盖实现的规范图分解

Canonical graph decompositions via coverings

本文提出一种基于整数参数的规范方法，将有限图分解为高度连通的局部部分，并通过更粗的模型描述其全局结构。该方法利用覆盖的缠结树结构投影，将图子式中的缠结理论应用于任意图，并扩展至局部有限Cayley图，为有限生成群提供规范分解。

Decompositions of the automorphism group of a locally compact abelian group

It is well known that every locally compact abelian group L can be decomposed as L_1 \oplus R^n, where L_1 contains a compact-open subgroup. In this paper, we use this decomposition to study the topological group Aut(L) of automorphisms of L, equipped with the g-topology. We show that Aut(L) is topologically isomorphic to a matrix group with entries from Aut(L_1), Hom(L_1, R^n), Hom(R^n, L_1), and GL_n(R), respectively. It is also shown that the algebraic portion of the decomposition is not specific to locally compact abelian groups, but is also true for objects with a well-behaved decompositi…

Decompositions of locally compact contraction groups, series and extensions

A locally compact contraction group is a pair (G,f) where G is a locally compact group and f an automorphism of G which is contractive in the sense that the forward orbit under f of each g in G converges to the neutral element e, as n tends to infinity. We show that every surjective, continuous, equivariant homomorphism between locally compact contraction groups admits an equivariant continuous global section. As a consequence, extensions of locally compact contraction groups with abelian kernel can be described by continuous equivariant cohomology. For each prime number p, we use 2-cocycles t…

A note on Bruhat decomposition of GL(n) over local principal ideal rings

Let A be a local commutative principal ideal ring. We study the double coset space of GL(n,A) with respect to the subgroup of upper triangular matrices. Geometrically, these cosets describe the relative position of two full flags of free primitive submodules of A^n. If k is the length of the ring, we determine for which of the pairs (n,k) the double coset space depend on the ring in question. For n=3, we give a complete parametrisation of the double coset space and provide estimates on the rate of growth of the number of double cosets.