irredundant families of maximal subgroups of finite solvable groups

Let m be a family of maximal subgroups of a group g. we say that m is irredundant if its intersection is not equal to the intersection of any proper subfamily of 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 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.