Converting subgroups of generators of a free group into free generators through directed graphs
Year: 2025
Type: article
Abstract: This research paper looks to address the following: Let H be a subgroup of π½π½ 2 .We are given a set of generators of subgroup H {h 1, h 2, β¦, h k } which are not necessarily free.The goal is to find {h' 1, h' 2, β¦, 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, β¦, 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. Binary OperationsGiven a set πΊπΊ, a binary operation on the set πΊπΊ is a function that takes as input pairs of elements of πΊπΊ (the order of the pair matters) and gives as output an element of πΊπΊ.For example, addition is a binary operation on the set of integers.The input is a pair of numbers, and the output is the sum of those numbers.In this case, the order in of the numbers are in the input pair does not matter.For example, 3 + 5 = 5 + 3.In cases like this one, where the order in of the numbers are in the input pair does not matter, we say that the operation is commutative.An alphabet is a set whose elements are referred to as letters.For example, the lower-case letters a to z, form an alphabet.An example of a finite sequence of these letters is abnear.Given two finite sequences of letters, we can create a third finite sequence of letters by placing the second sequence after the second one.This process is called concatenation.For example, the concatenation of the sequences qwe and utf is qweutf.Note that concatenation of finite sequences is a binary operation on the set of sequences.This operation is not commutative; the concatenation of qwe and utf but in reverse order would give utfqwe and not qweuft.In the rest of this article, binary operations will be denoted by *, and the result of applying the binary operation to a pair of elements ππ and ππ will be denoted by ππ * ππ or simply by ππππ.
Source: Journal of Student Research
Institution: Georgia Institute of Technology
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: diamond
OpenAlex ID: W4413564893