Linear groups

The paper is based on material reviewed in RefZhMat during 1971–1977 and continues a survey of the same name which appeared in this series in 1971. In light of the Tits alternative two old areas have now acquired special importance-free linear groups and solvable linear groups; they are given special attention. Other areas are discussed in somewhat more detail than previously.Translated from Itogi Nauki i Tekhniki, Algebra, Topologiya, Geometriya, Vol. 16, pp. 35–89, 1978.     

Linear groups and computation

We present an exposition of our ongoing project in a new area of applicable mathematics: practical computation with finitely generated linear groups over infinite fields. Methodology and algorithms available for this class of groups are surveyed. We illustrate the solution of hard mathematical problems by computer experimentation. Possible avenues for further progress are discussed.     

Linear groups as right multiplication groups of quasifields

For quasifields, the concept of parastrophy is slightly weaker than isotopy. Parastrophic quasifields yield isomorphic translation planes but not conversely. We investigate the right multiplication groups of finite quasifields. We classify all quasifields having an exceptional finite transitive linear group as right multiplication group. The classification is up to parastrophy, which turns out to be the same as up to the isomorphism of the corresponding translation planes.     

Linear optimization over permutation groups

For a permutation group given by a set of generators, the problem of finding “special” group members is NP-hard in many cases, e.g., this is true for the problem of finding a permutation with a minimum number of fixed points or a permutation with a minimal Hamming distance from a given permutation. Many of these problems can be modeled as linear optimization problems over permutation groups. We develop a polyhedral approach to this general problem and derive an exact and practically fast algorithm based on the branch & cut-technique.     

Linear preserver problems and algebraic groups

Supported by University of Waterloo Grant and the Alexander von Humboldt Foundation     

Linear spaces with flag-transitive automophism groups

Dedicated to Jacques Tits on the occasion of his sixtieth birthday     

Linear core conditions in residually finite groups

LetG be a residually finite or pro-finite group. We say thatG satisfies the linear core condition with constantc if all finite index (open) subgroups ofG contain a subgroup of index at mostc which is normal inG. Answering a question of L. Pyber we give a complete characterisation of finitely generated residually finite and pro-finite groups satisfying a linear core generated residually finite and pro-finite groups satisfying a linear core condition. In the case of infinitely generated groups we prove that such groups are abelian-by-finite. Research supported by the Hungarian National Research Foundation (OTKA), grant no. 16432 and F023436.     

Linear groups defined by decomposable tensor equalities

Let R be the ring of integers of some finite algebraic extension of the rationals Q of degree n. A necessary and sufficient condition for s elements of R to be an R-basis is given, in terms of the Hermite normal form of a certain n ×ns integral matrix depending on the elements, and on the structure constants of R.     

Linear algebraic groups and countable Borel equivalence relations

If is an equivalence relation on a standard Borel space , then we say that is Borel reducible to if there is a Borel function such that . An equivalence relation on a standard Borel space is Borel if its graph is a Borel subset of . It is countable if each of its equivalence classes is countable. We investigate the complexity of Borel reducibility of countable Borel equivalence relations on standard Borel spaces. We show that it is at least as complex as the relation of inclusion on the collection of Borel subsets of the real line. We also show that Borel reducibility is -complete. The proofs make use of the ergodic theory of linear algebraic groups, and more particularly the superrigidity theory of R. Zimmer.