Free Groups

Subjects: Group Theory
Links: Free Product of Groups, Cyclic Groups

Let $G$ be a group. If $S$ is a subset o f$G$ such that the subgroup $⟨S⟩$ generated by $G$ is all of $G$, then $S$ is said to generate $G$, and the elements of $S$ are called generators for $G$.

Given any object $\sigma$, we can form an infinite cyclic group generated by $\sigma$, called the free group generated by $\sigma$ and denoted by $F(\sigma )$, as follows: $F(\sigma )$ is the set $\{\sigma \}\times \mathbb{Z}$ with the operation being $(\sigma ,m)(\sigma ,n):=(\sigma ,k+m)$. We identify $\sigma$ with the element $(\sigma ,1)$, and we use the notation ${\sigma }^n$ for elements of $F(\sigma )$.

We degine the free group on $S$ denoted by $F(S)$, to be the free product of all infinite cyclic groups generated by elements of $S$: $$F(S) := \coprod_{\sigma \in S} F(\sigma).$$There is a natural injection $\iota :S\hookrightarrow F(S)$, defined by sending each $\sigma \in S$ to the word $\sigma \in F(S)$. In the case $S=\{{\sigma }_1,\dots ,{\sigma }_n\}$ is a finte set, we often denote $F(S)$ as $F({\sigma }_1,\dots ,{\sigma }_n)$.

Characteristic Property of the Free Group: Let $S$ be a set. For any group $\phi :S\to H$, there exists a unique homomorphism $\mathrm{\Phi }:F(S)\to H$ extending $\phi$

\usepackage{tikz-cd}
\usepackage{amsfonts, amsmath, amssymb}

\begin{document}
\begin{tikzcd}[row sep=2cm, column sep=2cm]
F(S) \arrow[dr, dashed, "\Phi"] & \\
S \arrow[u, hook,"\iota_\alpha"] \arrow[r, "\varphi"'] & H
\end{tikzcd}
\end{document}

Cor: the Free group on $S$ is the unique group (up to isomorphism) satisfying the characteristic property

Any group $G$ is said to be a free group if there is some subset $S\subseteq G$ such that the homomorphism $F(S)\to G$ induced by the inclusion $S\hookrightarrow G$ is an isomorphism.

Prop: A group $G$ is free iff it has a generating set $S\subseteq G$ such that every element $g\in G$ other than the identity has a unique expression asa product of the form $$g = \sigma_1^{n_1}\dots, \sigma_k^{n_k},$$ where ${\sigma }_i\in S$, $n_i\in \mathbb{N}$, and ${\sigma }_i\ne {\sigma }_{i+1}$ for each $i=1,\dots ,k-1$.