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.     

Characters and Sylow 2-subgroups of maximal class revisited

We give two ways to distinguish from the character table of a finite group G if a Sylow 2-subgroup of G has maximal class. We also characterize finite groups with Sylow 3-subgroups of order 3 in terms of their principal 3-block.     

Finite groups with special Sylow 2-subgroups

Let T be a Sylow 2-subgroup of a simple group PSU (3, 2ⁿ), and Z a proper subgroup belonging to the center of T. We shall prove that a simple finite group whose Sylow 2-subgroup is isomorphic to T/Z coincides with PSU (3, 2ⁿ). As a consequence we list simple groups that can be represented in the form of a product of two Schmidt groups, i.e., of minimal nonnilpotent groups.Translated from Matematicheskie Zametki, Vol. 14, No. 2, pp. 217–222, August, 1973.     

Sylow 2-subgroups of rational solvable groups

A long-standing conjecture proposes that a Sylow 2-subgroup S of a finite rational group must be rational. In this paper we provide a counterexample to this conjecture, but we show that if G is solvable and S has nilpotence class 2, then S actually is rational.     

Elementary abelian 2-subgroups of compact Lie groups

We classify elementary abelian 2-subgroups of compact simple Lie groups of adjoint type. This finishes the classification of elementary abelian $p$ -subgroups of compact simple Lie groups (equivalently, complex linear algebraic simple groups) of adjoint type started in Griess (Geom Dedicata 39(3):253–305, 1991).     

Sylow 2-subgroups of the periodic groups saturated with finite simple groups

We prove that every 2-subgroup of a periodic group saturated with groups of Lie type over fields of odd characteristics whose Lie ranks are bounded as a whole is Chernikov. In particular, every such group is locally finite.     

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.     

On intersections of Sylow 2-subgroups in automorphism groups of finite simple groups of exceptional Lie type

We consider the problem of interpolation of finite sets of numerical data bounded in L _p -norms (1 ≤ p < ∞) by smooth functions that are defined in an n-dimensional Euclidean ball of radius R and vanish on the boundary of the ball. Under some constraints on the location of interpolation nodes, we obtain two-sided estimates with a correct dependence on R for the L _p -norms of the Laplace operators of the best interpolants.     

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?