Free Group

Let $G$ and $H$ be groups. We define a word in $G$ and $H$ as a product

$$s_1{s}_2\cdots s_n$$

where each $s_i\in G$ or $s_i\in H$. Any word can be reduced by the following operations:

• remove an instance of the identity elements
• replace a pair $g_1{g}_2$ by its product in $G$ or a pair $h_1{h}_2$ by its product in $H$.

Free Group

The free group $G\ast H$ is the group of all reduced words under the operation of concatenation followed by reduction.

For example, let $G=\mathbb{Z}$ and $H={\mathbb{Z}}_2$. Then the free group $G\ast H$ would be words of the form

$$aaaaaabaabaaaa,$$

i.e., a string of $a$‘s with the occasional $b$.

Representations

One typically gives free groups as a presentation. A presentation of a group $G$ is a set of generators $S$ with a set of relations $R$ and is denoted $⟨S\mid R⟩$.

The simplest example is the cyclic group of order $n$:

$$⟨a\mid a^n=1⟩,$$

where 1 represents the group identity. Many sources simplify this $⟨a\mid a^n⟩$.

Given the representations of $G$ and $H$, the representation of the free group is easy to compute. If $G=⟨S_G\mid R_G⟩$ and $H=⟨S_H\mid R_H⟩$, then

$$G\ast H=⟨S_G\cup S_H\mid R_G\cup H_G⟩.$$

For example, for $\mathbb{Z}=⟨a⟩$ and ${\mathbb{Z}}_2=⟨b\mid b^2=1⟩$, we get that

$$\mathbb{Z}\ast {\mathbb{Z}}_2=⟨a,b\mid b^2=1⟩.$$

Below is a list of common representations:

Group	Presentation
$C_n$, cyclic group	$⟨a\mid a^n⟩$
$D_n$, dihedral group	$⟨r,f\mid r^n,f^2,(rf{)}^2⟩$
$\mathbb{Z}\times \mathbb{Z}$	$⟨a,b\mid ab=ba⟩$
${\mathbb{Z}}_m\times {\mathbb{Z}}_n$	$⟨a,b\mid a^m,b^n,ab=ba⟩$