Irredundant families of maximal subgroups of finite solvable groups

Let \mathcal{M} be a family of maximal subgroups of a group G. We say that \mathcal{M} is irredundant if its intersection is not equal to the intersection of any proper subfamily of \mathcal{M}. The maximal dimension of G is the maximal size of an irredundant family of maximal subgroups of G. In this paper we study a class of solvable groups, called \mathcal{M}-groups, in which the maximal dimension has properties analogous to that of the dimension of a vector space such as the span property, the extension property and the basis exchange property.