The symmetric spaces the title refers to are the spaces X = G=K where G is a connected semisimple Lie group with nite center and K is a maximal compact subgroup. The Fourier transform on X is deened by means of the Iwasawa decomposition G = NAK of G where N is nilpotent and A abelian. Let g; n; a; k denote the corresponding Lie algebras. We also need the group M = K A ; the centralizer of A in ...

Write Θ for the stable character associated to a finite dimensional representation E of a connected real reductive group G. Let M be the centralizer of a maximal torus T , and denote by ΦM (γ,Θ ) Arthur’s extension of |D M (γ)| 1 2Θ(γ) to T (R). In this paper we give a simple explicit expression for ΦM (γ,Θ ), when γ is elliptic in G.

We study the properties of certain center-like subsets of groups, which are obtained by localizing setwise commutativity conditions. 1. Introduction. Let G denote a group with center Z = Z(G); and for each subset A of G, denote by C(A) the centralizer of A in G. In this paper, we define several " center-like " subsets of G, investigate their properties and establish conditions for them to coinc...

A X g. We exhibit natural embeddings of M(A X) in M(A) that give rise to monomorphisms from 1 (M(A X)) to 1 (M(A)). We call the images of these monomorphisms intersection subgroups of type X and prove that they form a conjugacy class of subgroups of 1 (M(A)). Recall that X in L(A) is modular if X+Y is an element of L(A) for all Y in L(A). We call X in L(A) supersolvable if there exists a chain ...

‎given an integer $n$‎, ‎we denote by $mathfrak b_n$ and $mathfrak c_n$ the classes of all groups $g$ for which the map $phi_{n}:gmapsto g^n$ is a monomorphism and an epimorphism of $g$‎, ‎respectively‎. ‎in this paper we give a characterization for groups in $mathfrak b_n$ and for groups in $mathfrak c_n$‎. ‎we also obtain an arithmetic description of the set of all integers $n$ such that a gr...

here we consider all finite non-abelian 2-generator $p$-groups ($p$ an odd prime) of nilpotency class two and study the probability of having $n^{th}$-roots of them. also we find integers $n$ for which, these groups are $n$-central.

A group G is said to have restricted centralizers if for each g in the centralizer $$C_G(g)$$ either finite or has index G. Shalev showed that a profinite with virtually abelian. Given set of primes $$\pi $$ , we take interest groups -elements. It shown such an open subgroup form $$P\times Q$$ where P abelian pro- and Q '$$ subgroup. This significantly strengthens result from our earlier paper.

A depth two extension A | B is shown to be weak depth two over its double centralizer V A (V A (B)) if this is separable over B. We introduce a notion of codepth two coalgebra homomorphism g : C → D, dual to a depth two algebra homomorphism. It is shown that the endomorphism ring of bicomodule endomorphisms End D C D forms a right bialgebroid over the centralizer subalgebra g * : D * → C * of t...

Python implementation of permutations is presented. Three classes are introduced: Perm for permutations, Group for permutation groups, and PermError to report any errors for both classes. The class Perm is based on Python dictionaries and utilize cycle notation. The methods of calculation for the perm order, parity, ranking and unranking are given. A random permutation generation is also shown....