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Register a new userSmall groups of finite Morley rank with involutions
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Authors
Adrien Deloro
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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.
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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 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.
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.
The present survey aims at being a list of Conjectures and Problems in an area of model-theoretic algebra wide open for research, not a list of known results. To keep the text compact, it focuses on structures of finite Morley rank, although the same questions can be asked about other classes of objects, for example, groups definable in $omega$-stable and $o$-minimal theories. In many cases, answers are not known even in the classical category of algebraic groups over algebraically closed fields.
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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