In this article, we investigate the structure of a finite group G under the assumption that some subgroups of G are c-normal in G. The main theorem is as follows: Theorem A. Let E be a normal finite group of G. If all subgroups of Ep with order dp and 2dp (if p = 2 and Ep is not an abelian nor quaternion free 2-group) are c-normal in G, then E is p-hypercyclically embedded in G. We give some ap...

For any field K, we determine the quasi-simple subgroups G of the Chevalley group F4ðKÞ which are generated by a class S of so-called abstract transvection subgroups of G such that any member of S is contained in a long root subgroup of F4ðKÞ. First, we construct a polar space with point set S, which is embedded in the F4-geometry. This yields that a conjugate of G is contained in a classical s...

We classify all subgroups of SO(3) that are generated by two elements, each a rotation of finite order, about axes separated by an angle that is a rational multiple of π. In all cases we give a presentation of the subgroup. In most cases the subgroup is the free product, or the amalgamated free product, of cyclic groups or dihedral groups. The relations between the generators are all simple con...

We generalize the JSJ-splitting of Rips and Sela to give decompositions of nitely presented groups which capture splittings over certain classes of small subgroups. Such classes include the class of all 2-ended groups and the class of all virtually ZZ groups. The approach, called \track zipping", is relatively elementary, and diiers from the Rips-Sela approach in that it does not rely on the th...

We prove the convex combination theorem for hyperbolic n-manifolds. Many applications are given both in high dimensions and in 3 dimensions. One consequence is that given two geometrically finite subgroups of a discrete group of isometries of hyperbolic n-space, satisfying a natural condition on their parabolic subgroups, there are finite index subgroups which generated a subgroup that is an am...

Let us assume we are given an ultra-solvable set Q. S. Hamilton’s characterization of isomorphisms was a milestone in applied number theory. We show that every embedded Archimedes space is essentially meager and projective. Is it possible to study normal subgroups? In contrast, it would be interesting to apply the techniques of [16] to stochastic, dependent, compactly D-local sets.

We prove that abelian subgroups of the outer automorphism group of a free group are quasiisometrically embedded. Our proof uses recent developments in the theory of train track maps by FeighnHandel. As an application, we prove the rank conjecture for Out(Fn).

We prove the convex combination theorem for hyperbolic n-manifolds. Applications are given both in high dimensions and in 3 dimensions. One consequence is that given two geometrically finite subgroups of a discrete group of isometries of hyperbolic nspace, satisfying a natural condition on their parabolic subgroups, there are finite index subgroups which generate a subgroup that is an amalgamat...

We prove the convex combination theorem for hyperbolic n-manifolds. Applications are given both in high dimensions and in 3 dimensions. One consequence is that given two geometrically finite subgroups of a discrete group of isometries of hyperbolic n-space, satisfying a natural condition on their parabolic subgroups, and whose intersection is a separable subgroup, there are finite index subgrou...

We prove the convex combination theorem for hyperbolic n-manifolds. Applications are given both in high dimensions and in 3 dimensions. One consequence is that given two geometrically finite subgroups of a discrete group of isometries of hyperbolic nspace, satisfying a natural condition on their parabolic subgroups, there are finite index subgroups which generate a subgroup that is an amalgamat...