Undergrad The quotient group of a group with a presentation

[Mr Davis 97]

Suppose that we know that $G=\langle S \mid R\rangle$, that is, $G$ has a presentation. If $N\trianglelefteq G$, what can be said about $G/N$? I know that for example, if $G=\langle x,y \rangle$, then $G/N = \langle xN, yN \rangle$. But is there anything that can be said about the relations in $G$as they might correspond to relations for $G/N$, or is there no correspondence in the relations?

 

[fresh_42]

Suppose that we know that $G=\langle S \mid R\rangle$, that is, $G$ has a presentation. If $N\trianglelefteq G$, what can be said about $G/N$?

That it is a group.

I know that for example, if $G=\langle x,y \rangle$, then $G/N = \langle xN, yN \rangle$. But is there anything that can be said about the relations in $G$as they might correspond to relations for $G/N$, or is there no correspondence in the relations?

A relation is a word with letters from the set of generators which multiplies to $1$. Now $\pi\, : \,G \twoheadrightarrow G/N$ is a group homomorphism, so $\pi(R)=1_{G/N}$ where the left hand side is a word with letters from the images of the generators. Say $R=a^nb^m$ then $N=(a^nb^m)N=(aN)^n(bN)^m$. You could say that $\bar{R}=\pi(R)$ is a relation in $G/N$.