We study some geometric properties of actions on nonpositively curved spaces related to complete reducibility and semisimplicity, focusing on representations of a finitely generated group Γ in the group G of rational points of a reductive group over a local field, acting on the associated space (symmetric space or affine building). We prove that the space of completely reducible classes is the ...

In this paper we study the normalizer decomposition of a compact Lie group G using information fusion system F on maximal discrete p-toral subgroup. We prove that there is an injective map from set conjugacy classes chains F-centric, F-radical subgroups to p-centric, p-stubborn continuous subgroups. The bijection when π0(G) finite p-group. also classifying space chain mod p equivalent correspon...

The Möbius function of the subgroup lattice a finite group G has been introduced by Hall and applied to investigate several different questions. We propose following generalization. Let A be automorphism Aut(G) denote CA(G) set A-conjugacy classes subgroups G. For H ≤ let [H]A = { Ha ∣ a∈ A} element containing H. may define an ordering in way: [K]A if K for some ∈ A. consider μA corresponding p...

Abstract We prove that for a Banach algebra A having bounded $\mathcal {Z}(A)$ -approximate identity and every $\mathbf {[IN]}$ group G with weight w which is either constant on conjugacy classes or satisfies $w \geq 1$ , {Z}(L^{1}_{w}(G) \otimes ^{\gamma } A) \cong \mathcal {Z}(L^{1}_{w}(G)) . As an application, we discuss the conditions under {Z}(L^{1}_{\omega }(G,A))$ enjoys certain algebrai...

in this paper we give some general results on the non-splitextension group $overline{g}_{n} = 2^{2n}{^{cdot}}sp(2n,2), ngeq2.$ we then focus on the group $overline{g}_{4} =2^{8}{^{cdot}}sp(8,2).$ we construct $overline{g}_{4}$ as apermutation group acting on 512 points. the conjugacy classes aredetermined using the coset analysis technique. then we determine theinertia factor groups and fischer...

Given a group $G$, we write $x^G$ for the conjugacy class of $G$ containing element $x$. A famous theorem B. H. Neumann states that if is in which all classes are finite with bounded size, then derived $G'$ finite. We establish following result. Let $n$ be positive integer and $K$ subgroup such $|x^G|\leq n$ each $x\in K$. $H=\langle K^G\rangle$ normal closure $K$. Then order $H'$ $n$-bounded. ...

We consider the pseudo-Anosov elements of the mapping class group of a surface of genus g that fix a rank k subgroup of the first homology of the surface. We show that the smallest entropy among these is comparable to (k + 1)/g. This interpolates between results of Penner and of Farb and the second and third authors, who treated the cases of k = 0 and k = 2g, respectively, and answers a questio...

We classify the spherical birational sheets in a complex simple simply-connected algebraic group. use classification to show that, when $G$ is connected reductive group with derived subgroup, two conjugacy classes $\mathcal{O}_1$, $\mathcal{O}_2$ of lie same sheet, up shift by central element $G$, if and only coordinate rings $\mathcal{O}_1$ are isomorphic as $G$-modules. As consequence, we pro...

As preparation for a duality theory for saddle programs, a partial conjugacy correspondence is developed among equivalence classes of saddle functions. Three known conjugacy correspondences, including Fenchel's correspondence among convex functions and Rockafellar's extension of it to equivalence classes of saddle functions, are shown to be degenerate special cases. Additionally, two new corres...

By algebraic group theory, there is a map from the semisimple conjugacy classes of a finite group of Lie type to the conjugacy classes of the Weyl group. Picking a semisimple class uniformly at random yields a probability measure on conjugacy classes of the Weyl group. We conjecture that this measure agrees with a second measure on conjugacy classes of the Weyl group induced by a construction o...