Maximal Subgroups in the Classical Groups Normalizing Solvable Subgroups

Let [Formula: see text] be a classical group over an arbitrary field [Formula: see text], acting on an [Formula: see text]-dimensional [Formula: see text]-space [Formula: see text]. All those maximal subgroups of [Formula: see text] are classified each of which normalizes a solvable subgroup [Formula: see text] of [Formula: see text] not lying in [Formula: see text].

Locally solvable subgroups of PLo(I) are countable

We show every locally solvable subgroup of [Formula: see text] is countable. A corollary is that an uncountable wreath product of copies of [Formula: see text] with itself does not embed into [Formula: see text].

Solvable Subgroup Theorem for simplicial nonpositive curvature

Given a group [Formula: see text] with bounded torsion that acts properly on a systolic complex, we show that every solvable subgroup of [Formula: see text] is finitely generated and virtually abelian of rank at most [Formula: see text]. In particular, this gives a new proof of the above theorem for systolic groups. The main tools used in the proof are the Product Decomposition Theorem and the Flat Torus Theorem.

THE BINARY ADDING MACHINE AND SOLVABLE GROUPS

We prove that any solvable subgroup K of automorphisms of the binary tree, which contains the binary adding machine is an extension of a torsion-free metabelian group by a finite 2-group. If the group K is moreover nilpotent then it is torsion-free abelian.

Kernel systems on finite groups

We introduce a notion of kernel systems on finite groups: roughly speaking, a kernel system on the finite group G consists in the data of a pseudo-Frobenius kernel in each maximal solvable subgroup of G, subject to certain natural conditions. In particular, each finite CA-group can be equipped with a canonical kernel system. We succeed in determining all finite groups with kernel system that also possess a Hall p′-subgroup for some prime factor p of their order; this generalizes a previous result of ours (Communications in Algebra 18(3), 1990, pp. 833-838). Remarkable is the fact that we make no a priori abelianness hypothesis on the Sylow subgroups.

Integrability of doubly-periodic Riccati equation

By the structure of solvable subgroup ofSL(2,ℂ)(see [1]), the integrability and properties of solutions of a Riccati equation with an elliptic function coefficient, which is related to a Fuchsian equation on the torusT2,is studied.