On Tate-Shafarevich groups over galois extensions

LetA be an abelian variety defined over a number fieldK. LetL be a finite Galois extension ofK with Galois groupG and let III(A/K) and III(A/L) denote, respectively, the Tate-Shafarevich groups ofA overK and ofA overL. Assuming these groups are finite, we compute [III(A/L) ^G ]/[III(A/K)] and [III(A/K)]/[N(III(A/L))], where [X] is the order of a finite abelian groupX. Especially, whenL is a quadratic extension ofK, we derive a simple formula relating [III(A/L)], [III(A/K)], and [III(A ^x/K)] whereA x is the twist ofA by the non-trivial characterχ ofG.     

On galois groups and their maximal 2-subgroups

LetG be a finite group of even order, having a central element of order 2 which we denote by −1. IfG is a 2-group, letG be a maximal subgroup ofG containing −1, otherwise letG be a 2-Sylow subgroup ofG. LetH=G/{±1} andH=G/{±1}. Suppose there exists a regular extensionL ₁ of ℚ(T) with Galois groupG. LetL be the subfield ofL ₁ fixed byH. We make the hypothesis thatL ₁ admits a quadratic extensionL ₂ which is Galois overL of Galois groupG. IfG is not a 2-group we show thatL ₁ then admits a quadratic extension which is Galois over ℚ(T) of Galois groupG and which can be given explicitly in terms ofL ₂. IfG is a 2-group, we show that there exists an element α ε ℚ(T) such thatL ₁ admits a quadratic extension which is Galois over ℚ(T) of Galois groupG if and only if the cyclic algebra (L/ℚ(T).a) splits. As an application of these results we explicitly construct several 2-groups as Galois groups of regular extensions of ℚ(T).     

On the galois number and the minimal degree of doubly transitive groups

Consider the action of a finite group G on a set M. Then the Galois number is defined to be 1 + f, where fis the maximal number of fixed points of an element in G, which does not act as the identity on M. We determine the Galois number and the minimal degree of all doubly transitive permutation groups.     

Demuškin groups of rank ℵ0 as absolute galois groupsas absolute galois groups

We produce a complete classification of the groups mentioned in the title. This result complements earlier work of Serre and Labute [Lab2] and is further motivated by the basic question: which profinite groups occur as groups of automorphisms of algebraically closed fields?     

Polynomials with frobenius galois groups

An effective characterization of polynomials of degree n whose Galois groups are Frobenius groups with kernel of order n is given. Some examples of such polynomials are listed.     

A galois theory for the banach algebra of continuous symmetric functions on absolute galois groups

In this paper we extend the techniques and the basic results of the classical Galois theory of the fields extension is an algebraic closure of Q, to the algebras extension ? ? ?_sym(G), where this last is the ?— algebra of all the continuous symmetric functions ? defined on the absolute Galois group with values in ?.     

Pro-p galois groups of rank ≤4

Letp be a prime >2, letF be a field of characteristic ≠p containing a primitivep-th root of unity and letG _F (p) be the Galois group of the maximal Galois-p-extension ofF. Ifrk G _F (p)≤4 thenG _F (p) is a free pro-p product of metabelian groups orG _F (p) is a Demuškin group of rank 4.     

On automorphism groups of some finite groups

We show that if n > 1 is odd and has no divisor p4 for any prime p, then there is no finite group G such that│Aut(G)│ = n.     

Pro-P galois groups of function fields over local fields

For non-archimedean local field K and a prime number p we compute the finitely generated pro-p (closed) subgroups of the absolute Galois group of K(t). In addition, we characterize the finitely generated pro-p groups which occur as the maximal pro-p Galois group of algebraic extensions of K(t) containing a primitive pth root of unity.     

On the simplicity of some kac-moody groups

Recently, Marcuson extended the classical construction of Tits systems in Steinberg groups to include the Kac-Moody Steinberg groups associated with the infinite dimensional versions of the great Lie algebras. If these Lie algebras and their Kac-Moody groups are viewed as limits of their finite dimensional counterparts, more direct methods may be employed. In fact, the Kac-Moody Chevalley groups of these Lie algebras are seen to be simple.     

On the derived series of some groups

We solve Problems 17.82 and 17.86(b) posed by Mikhailov in the Kourovka Notebook [1]. Namely, we construct: (1) an example of a finitely presented group H in which the intersection H ^(ω) of all terms of the derived series is distinct from its commutant; (2) an example of a balanced presentation 〈x ₁, x ₂, x ₃|r ₁, r ₂, r ₃〉 of the trivial group for which F(x ₁, x ₂, x ₃)/[R ₁,R ₂] is not a residually soluble group (here R _i (i = 1, 2) denotes the normal closure of r _i in F(x ₁, x ₂, x ₃)). The construction of the second example is related to some approach to the Whitehead asphericity conjecture.     

On the simplicity of some kac-moody groups

Recently, Marcuson extended the classical construction of Tits systems in Steinberg groups to include the Kac-Moody Steinberg groups associated with the infinite dimensional versions of the great Lie algebras. If these Lie algebras and their Kac-Moody groups are viewed as limits of their finite dimensional counterparts, more direct methods may be employed. In fact, the Kac-Moody Chevalley groups of these Lie algebras are seen to be simple.     

On the normalizer of finitely generated subgroups of absolute galois groups of uncountable hilbertian fields of characteristic 0

For a fieldK and a positive integere let N _e (K) be the set of alle-tuplesσ = (σ ₁, …,σ _e)εG(K) ^e that generate a selfnormalizer closed subgroup ofG(K). Chatzidakis proved, that ifK is Hilbertian and countable, then N _e (K) has Haar measure 1. IfK is Hilbertian and uncountable, this need not be the case. Indeed, we prove that ifK ₀ is a field of characteristic 0 that contains all roots of unity,T is a set of cardinality ℵ₁ which is algebraically independent overK ₀ andK =K ₀(T), then neither N _e (K) nor its complement contain a set of positive measure. In particular N _e (K) is a nonmeasurable set. This work was partially supported by an NSF grant #DMS-H603187, while the second author enjoyed the hospitality of Rutgers University.     

On the IYB-property in some solvable groups

A finite group G is called Involutive Yang-Baxter (IYB) if there exists a bijective 1-cocycle ${\chi: G \longrightarrow M}$ for some ${\mathbb{Z}G}$ -module M. It is known that every IYB-group is solvable, but it is still an open question whether the converse holds. A characterization of the IYB property by the existence of an ideal I in the augmentation ideal ${\omega\mathbb{Z}G}$ complementing the set 1?G leads to some speculation that there might be a connection with the isomorphism problem for ${\mathbb{Z}G}$ . In this paper we show that if N is a nilpotent group of class two and H is an IYB-group of order coprime to that of N, then ${N \rtimes H}$ is IYB. The class of groups that can be obtained in that way (and hence are IYB) contains in particular Hertweck’s famous counterexample to the isomorphism conjecture as well as all of its subgroups. We then investigate what an IYB structure on Hertweck’s counterexample looks like concretely.