F3 The Free Group With 3 Generators

Definition of a Free Group with 3 Generators:

A group, G, is called Free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses.

We call each written element that is a product of the group elements and their inverses "words".

More specifically, let a, b, c be elements in G. The subgroup generated by a,b,c, <a,b,c>, is the smallest subgroup containing a, b, and c. This group is the Free Group with 3 generators; a, b, and c. (G = <a, b, c>) This group can also be called F3.

Example - <a, b, c> = < e, a, b, c, a^-1, b^-1, c^-1, a^n*b^n*c^n,
b^n*c^n*a^n, c^n*b^n*a^n, aabc, abbc, abcc, … >

Claim: That F3 is a group.

Verification: According to the group axioms, we need to be able to show that F3 is closed under its respective binary operation, it contains an identity element, and there exists an inverse element.

Well, by definition, F3 is a group since it is a way to describe an already defined group. In this case, a group G generated by 3 elements, a, b, and c. Since words themselves are written in terms of the group elements and their inverses or the identity, this implies that the group already has fulfilled the group axioms.

Additionally, the cosets of a free group with 3 generators can be described based on the subgroup that is isomorphic to it. In other words, the cosets based on F3 look something like this:

Let F3 be a free group generated by three generators, and let H be a subgroup of F3. The Left Cosets of F3 then have the form of:

0 + H
1 + H
2 + H
3 + H

and so on.