In this note we prove that every finite collection of connected algebraic subgroups the group triangular automorphisms affine space generates a solvable subgroup.

If G is a finite p-solvable group for some prime p, A a solvable subgroup of the automorphism group of G of order prime to |G| such that A stabilises a p-block b of G and acts trivially on a defect group P of b, then there is a Morita equivalence between the block b and its Watanabe correspondent w(b) of CG(A), given by a bimodule M with vertex ∆P and an endo-permutation module as source, which...

We present a general rectilinear Steiner tree problem in the plane and prove that it is solvable on the Hanan grid of the input points. This result is then used to show that several variants of the ordinary rectilinear Steiner tree problem are solvable on the Hanan grid, including | but not limited to | Steiner trees for rectilinear (or iso-thetic) polygons, obstacle-avoiding Steiner trees, gro...

A graph product is the fundamental group of a graph of groups Amongst the simplest examples are HNN groups and free products with amalgamation. The conjugacy problem is solvable for recursively presented graph products with cyclic edge groups over finite graphs if the vertex groups have solvable conjugacy problem and the sets of cyclic generators in them are semicritical. For graph products ove...

we describe under various conditions abelian subgroups of the automorphism‎ ‎group $mathrm{aut}(t_{n})$ of the regular $n$-ary tree $t_{n}$‎, ‎which are‎ ‎normalized by the $n$-ary adding machine $tau =(e‎, ‎dots‎, ‎e,tau )sigma _{tau‎ ‎}$ where $sigma _{tau }$ is the $n$-cycle $left( 0,1‎, ‎dots‎, ‎n-1right) $‎. ‎as‎ ‎an application‎, ‎for $n=p$ a prime number‎, ‎and for $n=4$‎, ‎we prove that...

By exploiting the hidden algebraic structure of the Schrödinger Hamiltonian, namely the sl(2), we propose a unified approach of generating both exactly solvable and quasi-exactly solvable potentials. We obtain, in this way, two new classes of quasi-exactly solvable systems one of which is of periodic type while the other hyperbolic. Tracking down solvable quantum potentials has always aroused i...

A classification is given for groups which can occur as the fundamental group of some compact 3-manifold. In most cases we are able to determine the topological structure of a compact 3-manifold whose fundamental group is known to be solvable. Using the results obtained for solvable groups, we are able to extend some known results concerning nilpotent groups of closed 3-manifolds to the more ge...

Abstract The study of the complexity equation satisfiability problem in finite groups had been initiated by Goldmann and Russell (Inf. Comput. 178 (1), 253–262, 2002) where they showed that this is for nilpotent while it -complete non-solvable groups. Since then, several results have appeared showing can be solved polynomial time certain solvable G having a normal subgroup H with factor / . Thi...

The main result in this paper is as follows: Theorem. Let S be the branch curve in CP of a generic projection of a Veronese surface. Then π1(CP −S) is an extension of a solvable group by a symmetric group. A group with the property mentioned in the theorem is “almost solvable” in the sense that it contains a solvable normal subgroup of finite index. We pose the following question. Question. For...

A subgroup of the linear translation complement of a translation plane is geometrically irreducible if it has no invariant lines or subplanes. A similar definition can be given for "geometrically primitive". If a group is geometrically primitive and solvable then it is fixed point free or metacyclic or has a normal subgroup 2a+b a of order w where w divides the dimension of the vector space. Si...