Abstract: A group is a set of elements together with a binary operation and four defined properties: closure, associativity, existence of the unit element and existence of the inverse of each element. At the most basic level, group theory systematizes the notion of symmetry, that can be applicable whether of geometric objects, crystals, roots of equations or a great variety of other examples. A way to study the structure of a group is to look at its subgroups; but sometimes it is not possible to describe ALL the subgroups of a group. So it can be useful to find some suitable family of subgroups whose behaviour respect to some property has a strong influence on the structure of the whole group. In this talk I will show you the family I was studying during my Phd, the family of subgroups of infinite rank.