THEOREM. Let F be the free profinite group on a set X, where \X\ > 2, and let n be a non-empty set of primes. Then F has a maximal abelian subgroup isomorphic to HpEn Zp. The idea of the proof is the following: we show that A — Ylpe7I1p is a free factor of Pa, i.e. fia ^ A *B for some profinite group B. To conclude from this that A is a maximal abelian subgroup of Fa (the general case then foll...

Theorem 1.1 (Kummer theory). Let m ∈ Z>0, and suppose that the subgroup μm(K) = {ζ ∈ K∗ : ζ = 1} of K∗ has order m. Write K∗1/m for the subgroup {x ∈ K̄∗ : x ∈ K∗} of K̄∗. Then K(K∗1/m) is the maximal abelian extension of exponent dividing m of K inside K̄, and there is an isomorphism Gal(K(K∗1/m)/K) ∼ −→ Hom(K∗, μm(K)) that sends σ to the map sending α to σ(β)/β, where β ∈ K∗1/m satisfies β = α.

Let G be a semisimple algebraic group defined over the rational numbers, K a maximal compact subgroup of G = G(R), and Γ ⊂ G(Q) a neat arithmetic subgroup. LetX = Γ\G/K be the locally symmetric space associated to Γ, and E the local system on X constructed out of a finite-dimensional, irreducible, algebraic representation E of G. Fix a maximally Q-split torus S in G; S is assumed to be nontrivi...

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. ...

Let (AS , S) be an Artin-Tits and X a subset of S ; denote by AX the subgroup of AS generated by X. When AS is of spherical type, we prove that the normalizer and the commensurator of AX in AS are equal and are the product of AX by the quasi-centralizer of AX in AS . Looking to the associated monoids A + S and A + X , we describe the quasi-centralizer of A+X in A + S thanks to results in Coxete...

The study of reductive group actions on a normal surface singularity X is facilitated by the fact that the group AutX of automorphisms of X has a maximal reductive algebraic subgroup G which contains every reductive algebraic subgroup of AutX up to conjugation. If X is not weighted homogeneous then this maximal group G is finite (Scheja, Wiebe). It has been determined for cusp singularities by ...

If G is a hyperbolic group, where G = H oφ Z, H is finitely presented, and φ is an automorphism of H, then H satisfies a polynomial isoperimetric inequality. Necessary and sufficient conditions of homological character are given for a finitely presented subgroup H of a hyperbolic group to be hyperbolic (resp. a quasi-convex subgroup). If Y is a connected subcomplex of the finite connected 2comp...

1: After identification of the algebra of exponential generating series with the shuffle algebra of ordinary formal power series, the exponential map exp! : XK[[X]] −→ 1 +XK[[X]] for the associated Lie group with multiplication given by the shuffle product is well-defined over an arbitrary field K by a result going back to Hurwitz. The main result of this paper states that exp! (and its recipro...

A method is proposed for the detection of item bias with respect to observed or unobserved subgroups. The method uses quasi-loglinear models for the incomplete subgroup x testscore x item 1 x x itek k contingency table. If subgroup membership is unknown the models are Haberman's incomplete-latent-class models. The (conditional) Rasch model is formulated as a quasi-loglinear model. The parameter...

The automorphism group of a curve is studied from the viewpoint canonical embedding and Petri’s theorem. A criterion for identifying as an algebraic subgroup general linear given. Furthermore, action extended to on generators minimal free resolution ring X.