presentation of a group

Definition

A presentation of a group $G$, written $G=⟨S|R⟩$ where $S\subset G$ is a set of generators and $R$ is a list of relations such that $R$ generates $\ker \left({\prod }^{\ast }\mathbb{Z}\cdot s⟶G\right)$

Every group has a representation

This is a way to understand groups based on the free group of a set of generators $S$.

Examples

• $\mathbb{Z}=⟨a⟩:=⟨a|\varnothing ⟩$
• $\mathbb{Z}\ast \mathbb{Z}=⟨a,b⟩$
• ${\mathbb{Z}}^n=⟨a|a^n⟩$
• $D_{2n}=⟨r,s|r^2,s^2,srsr⟩$ (note $srs=r^{-1}$)