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[Music] today we are going to talk about free groups now this is not a topic that you usually encounter in first course on group theory for example but it is nevertheless a very important notion and we will try to understand it as intuitively as possible initially and then slowly work our way towards the formal definition ok so here is the main motivation so firstly we know what generators of a group mean okay so what is the set of generators of a group so given a set s so if suppose g is a group so let g be a group and s be any subset of g then we say s generates the whole group g if the following is true if ah so what we will call the smallest subgroup generated by s is all of g ok now what is the subgroup generated by s so lets unravel that little notation there so this is what is called the subgroup generated by the subset s so let us define that so this is just defined to be the smallest subgroup of g which contains the set s so it is defined as follows so here is definition this
