Radjavi, Heydar; matematični kolokvij junij 2002

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Transitivity Questions

Heydar Radjavi

Dalhousie University, Halifax, Canada

20. junij 2002

Let F be a family of functions from a set X into itself. F is said to be transitive if for every x and y in X there is a member f in F such that f(x) = y. Transitive groups of permutations on a set are perhaps first examples of this frequently used concept, which everybody encounters in a first course in university algebra.

We shall restrict attention to the case where F is a family of linear operators with some structure, for example, an algebra, a linear space, or a semigroup. An obvious and trivial modification in the above definition is necessary: x cannot be zero. In some cases, for example, the case of groups of invertible operators, it is also necessary to have y nonzero.

A celebrated theorem of Burnside gives the clean and final answer for algebras of operators on a finite-dimensional vector space V over an albegraically closed field: the only transitive algebra is the set of all operators on V. When the structure of F is less rich, the situation is not so simple. There are interesting questions, some answered, some not, some of which will be discussed.

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