Consider the subset of $SL_3(\Bbb Z)$ consisting of matrices of the form
$$\begin{pmatrix} 1 & a & c\\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix}$$
for $a,b,c\in\Bbb Z$.
I need to show that it admits the presentation:
$$\langle A, B, C\mid AC=CA, BC=CB, ABA^{-1}B^{-1}=C\rangle.$$
I am not sure what exactly I need to show. I tried to follow the definition given in class, but didn't really understand it.
Definition 1.36: Let $G$ be a group and $S$ a generating set. Let $R\subseteq F(S)$. Denote by $\pi$ the morphism $F(S)\to G$. We say that $G$ admits the presentation $\langle S\mid R\rangle$ if $R$ normally generates $\ker \pi$, that is, if $\ker\pi$ is the smallest normal subgroup containing $R$.
Remark 1.38: We can also build a group with any presentation we choose: given a set $S$ and a set $R$ of words in $S$, the quotient $F(S)/\langle\langle R\rangle\rangle$ obviously admits the presentation $\langle S\mid R\rangle$. Note that this is a way to specify a group algebraically, not geometrically.