Transitivity and persistence properties of the subalgebras of a finite-dimensional Lie algebra

A subalgebra H of  L  is called an α -subalgebra, if H  has the property α. Also, we say that a subalgebra H  of a Lie algebra L  is α -transitive, whenever each α -subalgebra of H  is an α -subalgebra of L  and a subalgebra H  of L  is α -sensitive if for every α -subalgebra K  of H, there is an α -subalgebra A of L  such that A⋂H=K . These concepts are analogous to the concepts of α -transitive and α -sensitive subgroups of finite groups. In this paper, the main results are based on the properties cover-avoidance, maximality, ideality, and c-ideality, and in particular, we examine α -transitive and maximal-sensitive subalgebras. Furthermore, we obtain the influence of these notions on the structure of finite-dimensional Lie algebras and we give some results about supersolvable Lie algebras.