Converting subgroups of generators of a free group into free generators through directed graphs

This research paper looks to address the following: Let H be a subgroup of 2 .We are given a set of generators of subgroup H {h 1, h 2, , h k } which are not necessarily free.The goal is to find {h' 1, h' 2, , h' s }, a set of free generators of H.We will first introduce the concept of groups, then we will introduce the concept of colored directed graphs, and we will finally connect the two to achieve this objective.We will address the problem through the use of Geometric Group Theory [1,2,3].This will be done by converting {h1, h2, , hk} into a colored directed graph, doing two different sequences of consecutive mappings known as folding and collapsing respectively, tracing back an edge path through these mappings, and converting these edgepaths back to words that will result in the free generators of H.In this article, we will explain all the relevant concepts, and work out a couple of examples. Free GroupsIn this section, we review the notions of abstract algebra that we will need in the rest of this paper.We explain the meaning: groups, subgroups, morphisms between groups, isomorphisms, generators of groups, free generators, free groups, and the rank of free groups.