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In this paper we bring together results about the density of subsemigroups ofabelian Lie groups, the minimal number of topological generators of abelian Liegroups and a result about actions of algebraic groups. We find the minimalnumber of generators of a finitely generated abelian semigroup or group ofmatrices with a dense or a somewhere dense orbit by computing the minimalnumber of generators of a dense subsemigroup (or subgroup) of the connectedcomponent of the identity of its Zariski closure.

Topological generators of abelian Lie groups and hypercyclic finitely generated abelian semigroups of matrices