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 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....

For each d, we construct CAT(0) cube complexes on which Cremona groups of rank d act by isometries. From these actions deduce new and old group-theoretical dynamical results about groups. In particular, study the behavior irreducible components exceptional loci. This leads to proofs regularization theorems, such as with property FW. We also find constraints degree growth for non-pseudo-regulari...

Write E for the stable discrete series character associated to an irreducible nite-dimensional representation E of a connected real reductive group G. Let M be the centralizer of the split component of a maximal torus T , and denote by M ( ; E) Arthur's extension of jDG M ( )j 12 E( ) to T (R). In this paper we give a simple explicit expression for M ( ; E), when is elliptic in G. We do not ass...

Let p be a prime,G a finite group which has a normal p-subgroup containing its own centralizer inG, and R a commutative local ring with residue class field of characteristic p. In this paper, it is shown that if α is an augmented automorphism of RG which fixes a Sylow p-subgroup P of G, there is ρ ∈ Aut(G) such that xαρ = x for all x ∈ P , and αρ is an inner automorphism of RG.

We assume that the “table head” of the ordinary character table of the sporadic simple Janko group J4 is known. That is, the conjugacy classes and their centralizer orders are known as well as all p-th power maps, for primes p up to the maximal element order in J4, which is 66. For the sake of simplicity, we take the table head of the character table contained in the GAP Character Table Library...