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Let $G$ be a finitely generated solvable-by-finite linear group. We present an algorithm to compute the torsion-free rank of $G$ and a bound on the Pr{u}fer rank of $G$. This yields in turn an algorithm to decide whether a finitely generated subgroup of $G$ has finite index. The algorithms are implemented in MAGMA for groups over algebraic number fields.
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We consider groups of finite Morley rank with solvable local subgroups of even and mixed types. We also consider miscellaneous aspects of small groups of finite Morley rank of odd type.
We prove a general dichotomy theorem for groups of finite Morley rank with solvable local subgroups and of Prufer p-rank at least 2, leading either to some p-strong embedding, or to the Prufer p-rank being exactly 2.
In 1878, Jordan showed that a finite subgroup of GL(n,C) contains an abelian normal subgroup whose index is bounded by a function of n alone. Previously, the author has given precise bounds. Here, we consider analogues for finite linear groups over algebraically closed fields of positive characteristic l. A larger normal subgroup must be taken, to eliminate unipotent subgroups and groups of Lie type and characteristic l, and we show that generically the bound is similar to that in characteristic 0 - being (n+1)!, or (n+2)! when l divides (n+2) - given by the faithful representations of minimal degree of the symmetric groups. A complete answer for the optimal bounds is given for all degrees n and every characteristic l.
We lay down the fundations of the theory of groups of finite Morley rank in which local subgroups are solvable and we proceed to the local analysis of these groups. We prove the main Uniqueness Theorem, analogous to the Bender method in finite group theory, and derive its corollaries. We also consider homogeneous cases as well as torsion.
We classify a large class of small groups of finite Morley rank: $N_circ^circ$-groups which are the infinite analogues of Thompsons $N$-groups. More precisely, we constrain the $2$-structure of groups of finite Morley rank containing a definable, normal, non-soluble, $N_circ^circ$-subgroup.
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