On the Distribution of Galois Groups

We propose a conjecture on the distribution of number fields with given Galois group and bounded norm of the discriminant. This conjecture is known to hold for abelian groups. We give some evidence relating the general case to the composition formula for discriminants, give a heuristic argument in favor of the conjecture, and present some computational data.     

Normal integral bases and strict ray class groups modulo 4

Let F be a number field. We construct three tamely ramified quadratic extensions which are ramified at most at some given set of finite primes, such that K₃⊂K₁K₂, both K₁/F and K₂/F have normal integral bases, but K₃/F has no normal integral basis. Since Hilbert-Speiser's theorem yields that every finite and tamely ramified abelian extension over the field of rational numbers has a normal integral basis, it seems that this example is interesting (cf. [5] J. Number Theory 79 (1999) 164; Theorem 2). As we shall explain below, the previous papers (Acta Arith. 106 (2) (2003) 171-181; Abh. Math. Sem. Univ. Hamburg 72 (2002) 217-233) motivated the construction. We prove that if the class number of F is bigger than 1, or the strict ray class group of F modulo 4 has an element of order ?3, then there exist infinitely many triplets (K₁,K₂,K₃) of such fields.     

Reflection Theorems and the Tame Kernel of a Number Field

Let L be a number field containing the pth primitive root of unity ζ _p . We investigate the p-rank of the ideal class groups of some subfields of L by using reflection theorems and establish relations between the p-rank of the ideal class groups and that of groups of units of some subfields of L.

Let F be a number field and 𝒪 _F the ring of integers in F. We also study the p-rank of tame kernels of F and establish relations between the p-rank of K ₂𝒪 _F and that of some direct summands of the ideal class group of F(ζ _p ).     

A New Characterization of the Mathieu Groups and Alternating Groups with Degrees Primes

In the past thirty years, several kinds of quantitative characterizations of finite groups especially finite simple groups have been investigated by many mathematicians. Such as quantitative characterizations by group order and element orders, by element orders alone, by the set of sizes of conjugacy classes, by dimensions of irreducible characters, by the set of orders of maximal abelian subgroups and so on. Here the authors continue this topic in a new area tending to characterize finite simple groups with given orders by some special conjugacy class sizes, such as largest conjugacy class sizes, smallest conjugacy class sizes greater than 1 and so on.     

Automorphisms of symmetric groups are toy examples for mapping class groups

We give a new proof establishing the automorphism groups of the symmetric groups inspired by the analogous result of Ivanov for the extended mapping class group. As a key tool, we consider the actions on the Kneser graphs.     

Quadratic fields with noncyclic 5- or 7-class groups

We shall show that the number of quadratic fields with absolute discriminant ≤x and noncyclic 5- or 7-class group is ≫x ^1/4 improving the existing known bound for g=5 and for g=7 in Byeon (Ramanujan J. 11:159–163, 2006). This work was supported by KRF-2005-070-C00004.     

Automorphisms of Hyperbolic Groups and Graphs of Groups

Using the canonical JSJ splitting, we describe the outer automorphism group Out(G) of a one-ended word hyperbolic group G. In particular, we discuss to what extent Out(G) is virtually a direct product of mapping class groups and a free abelian group, and we determine for which groups Out(G) is infinite. We also show that there are only finitely many conjugacy classes of torsion elements in Out(G), for G any torsion-free hyperbolic group. More generally, let Γ be a finite graph of groups decomposition of an arbitrary group G such that edge groups G_e are rigid (i.e. Out(G_e) is finite). We describe the group of automorphisms of G preserving Γ, by comparing it to direct products of suitably defined mapping class groups of vertex groups.     

Countable Primitive Groups

We give a complete characterization of countable primitive groups in several settings including linear groups, subgroups of mapping class groups, groups acting minimally on trees and convergence groups. The latter category includes as a special case Kleinian groups as well as subgroups of word hyperbolic groups. As an application we calculate the Frattini subgroup in many of these settings, often generalizing results that were only known for finitely generated groups. In particular, we answer a question of G. Higman and B.H. Neumann on the Frattini group of an amalgamated product. Received: January 2006, Revision: May 2006, Accepted: May 2006     

Logarithmic Frobenius structures and Coxeter discriminants

We consider a class of solutions of the WDVV equation related to the special systems of covectors (called ∨-systems) and show that the corresponding logarithmic Frobenius structures can be naturally restricted to any intersection of the corresponding hyperplanes. For the Coxeter arrangements the corresponding structures are shown to be almost dual in Dubrovin's sense to the Frobenius structures on the strata in the discriminants discussed by Strachan. For the classical Coxeter root systems this leads to the families of ∨-systems from the earlier work by Chalykh and Veselov. For the exceptional Coxeter root systems we give the complete list of the corresponding ∨-systems. We present also some new families of ∨-systems, which cannot be obtained in such a way from the Coxeter root systems.     

Projective character degree patterns of 2-groups

Abstract

We discuss the prospects for finding a “core class,” i.e., a well-behaved class of non-free abelian groups of cardinality ?₁ such that every non-free abelian group of cardinality ?₁ has a subgroup in the core class.