Volume 16, Issue 5

Abstract. For any torsion-free hyperbolic group Γ and any group G that is fully residually Γ, we construct algorithmically a finite collection of homomorphisms from  G to groups obtained from Γ by extensions of centralizers, at least one of which is injective. When G is residually Γ, this gives a effective embedding of  G into a direct product of such groups. We also give an algorithmic construction of a diagram encoding the set of homomorphisms from a given finitely presented group to Γ.

Abstract. We show that the rank does not decrease if one passes from a torsion-free locally quasi-convex hyperbolic group to the quotient by the normal closure of certain high powered elements. An argument provided by Ilya Kapovich further shows that the quasiconvexity assumption cannot be dropped without adding other assumptions.

Abstract. Kaplansky's zero divisor conjecture envisions that for a torsion-free group  G and an integral domain  R , the group ring does not contain non-trivial zero divisors. We define the length of an element  as the minimal non-negative integer  k for which there are ring elements  and group elements such that . We investigate the conjecture when  R is the field of rational numbers. By a reduction to the finite field with two elements, we show that if  for non-trivial elements in the group ring of a torsion-free group over the rationals, then the lengths of  and  cannot be among certain combinations. More precisely, we show for various pairs of integers ( i , j ) that if one of the lengths is at most  i , then the other length must exceed  j . Using combinatorial arguments we show this for the pairs ( 3 , 6 ) and ( 4 , 4 ). With a computer-assisted approach we strengthen this to show the statement holds for the pairs  and ( 4 , 7 ). As part of our method, we describe a combinatorial structure, which we call matched rectangles, and show that for these a canonical labeling can be computed in quadratic time. Each matched rectangle gives rise to a presentation of a group. These associated groups are universal in the sense that there is no counter-example to the conjecture among them if and only if the conjecture is true over the rationals.

Abstract. In this note we introduce the class of -groups (or Hall groups) related to the class of -groups defined by P. Hall in the 1950s. Establishing some basic properties of Hall groups we use them to obtain results concerning embeddings of abelian groups. In particular, we give an explicit classification of all abelian groups that can occur as subgroups in finitely generated metabelian groups. Hall groups allow us to give a negative answer to G. Baumslag's conjecture of 1990 on the cardinality of the set of isomorphism classes for abelian subgroups in finitely generated metabelian groups.

Abstract. The projective groups with and three related groups have normal subgroups that are sharply transitive on the corresponding projective line. This leads to five infinite nearfields which are not Dickson nearfields.

Abstract. Under some restrictions on the highest weight, the stabilizers of certain vectors in irreducible modules for the special linear groups with a rational action are determined. We consider infinitesimally irreducible modules whose highest weights have all coefficients at least 2 when expressed as a linear combination of the fundamental dominant weights and vectors whose nonzero weight components have weights that differ from the highest weight by a single simple root. For such vectors and modules a criterion for lying in the same orbit is obtained, and we prove that the stabilizers of vectors from different orbits are not conjugate. The orbit dimensions are also found. Furthermore, we show that these vectors do not lie in the orbit of a highest weight vector and their stabilizers are not conjugate to the stabilizer of such a vector.

Abstract. Let H be a subgroup of a finite group G . We say that H satisfies the partial Π-property in G if there exists a chief series of the group G such that for every G -chief factor () of the series , is a -number. In this paper, we investigate the structure of the finite groups G in which some primary subgroups satisfy the partial Π-property in G .

Abstract. Suppose that G is a finite group that admits a dihedral group of automorphisms generated by two involutions and such that . It is proved that if and satisfy a positive law of degree k , then G satisfies a positive law of degree that is bounded solely in terms of k and .

Abstract. Greither and Pareigis have established a connection between Hopf Galois structures on a Galois extension L / K with Galois group G , and the regular subgroups of the group of permutations on G , which are normalized by G . Byott has rephrased this connection in terms of certain equivalence classes of injective morphisms of G into the holomorphs of the groups N with the same cardinality of G . Childs and Corradino have used this theory to construct such Hopf Galois structures, starting from fixed-point-free endomorphisms of G that have abelian images. In this paper we show that a fixed-point-free endomorphism has an abelian image if and only if there is another endomorphism that is its inverse with respect to the circle operation in the near-ring of maps on G , and give a fairly explicit recipe for constructing all such endomorphisms.