Abelian Quotients Research Articles

Pro- pGalois Groups of Algebraic Extensions of [formula omitted

For a prime number pwe characterize the finitely generated maximal pro- pGalois groups of algebraic extensions of Q . This generalizes a characterization by Jensen and Prestel of the maximal abelian quotients of these Galois groups. As an application we show that the Witt rings of the algebraic extensions of Q with finitely many square classes have elementary type.

Finiteness properties and abelian quotients of graph groups

Finiteness properties and abelian quotients of graph groups

The Bieri-Neumann-Strebel Invariants for Graph Groups

Given a finite simplicial graph ${\cal G}$, the graph group $G{\cal is the group with generators in one-to-one correspondence with the vertices of ${\cal G}$ and with relations stating two generators commute if their associated vertices are adjacent in ${\cal G}$. The Bieri-Neumann-Strebel invariant can be explicitly described in terms of the original graph ${\cal G}$ and hence there is an explicit description of the distribution of finitely generated normal subgroups of $G{\cal G}$ with abelian quotient. We construct Eilenberg-MacLane spaces for graph groups and find partial extensions of this work to the higher dimensional invariants.

Abelian quotients of subgroups of the mapping class group of surfaces

Abelian quotients of subgroups of the mapping class group of surfaces

Some finite varieties of Lie algebras

in the class whenever all n-generated subalgebras of L are in the class. Thus the classes JV of nilpotent Lie algebras and d of abelian Lie algebras are 2-recognizable. The problems are to find if a given class is n-recognizable and to find the smallest n. We consider the classes &‘JV of Lie algebras which have an abelian ideal with nilpotent quotient, JV&, of Lie algebras which have a nilpotent ideal with abelian quotient, % of supersolvable Lie algebras, and Y of solvable Lie algebras. The same notation is used for the corresponding classes of finite groups. Most notation is standard except where indicated.

On Abelian Quotients of Primitive Groups

On Abelian Quotients of Primitive Groups

Finite permutation groups with large abelian quotients

Finite permutation groups with large abelian quotients

On abelian quotients of primitive groups

It is shown that if G G is a primitive permutation group on a set of size n n , then any abelian quotient of G G has order at most n n . This was motivated by a question in Galois theory. The field theoretic interpretation of the result is that if M / K M/K is a minimal extension and L / K L/K is an abelian extension contained in the normal closure of M M , then the degree of L / K L/K is at most the degree of M / K M/K .

An abelian quotient of the mapping class groupI g

An abelian quotient of the mapping class groupI g

Capitulation in Class Field Extensions of Type (p, p)

Let K be a number field, K(1) its Hilbert class field, that is, the maximal abelian unramified extension of K, let K(2) be the Hilbert class field of K(1), and let G = Gal(K(2)/K) (alternatively, for p a prime the first and second p class fields enjoy properties analogous to those of the respective class fields discussed in this introduction; the particulars may be found surrounding Lemma 2). Since G/G’ is the largest abelian quotient of G, G/G′ = Gal (K(1))/K) and so G’ is the abelian group Gal(K(2)K(1)); moreover, class field theory provides (Artin) maps φK, φK(1) which are isomorphisms of the class groups Ck, Ck(1) onto G/G′, G′ respectively. In the remarkable paper [1] E. Artin computed the composition VG′where e is the homomorphism induced on the class groups by extending ideals of K to ideals of K(1), and he gave a formula for computing VG′, the now familiar transfer (Verlagerung) homomorphism, in terms of the group G alone (see Lemma 1).

A new class of counterexamples to the integrability problem

For many years, it was believed that the solvability of the integrability problem for a transitive Lie pseudogroup depends only on the local solvability of linear differential operators arising from the abelian quotients in Guillemin's Jordan-Hölder decomposition for the transitive Lie algebra associated to the pseudogroup. We provide an example of a transitive pseudogroup for which the integrability problem is not solvable and whose corresponding Jordan-Hölder sequence has only non-abelian quotients.

Abelian quotients of the mapping class group of a 2-manifold

Abelian quotients of the mapping class group of a 2-manifold

Abelian quotients of the Teichmüller modular group

Abelian quotients of the Teichmüller modular group