When can I define a map from a group presentation to $\mathbb{Z}$?

Question

I'm trying to wrap my head around group presentations and I came up with the question:

When can I define maps out of a presentation?

So, for example, given a presentation $G=\langle S\mid R\rangle$ for some generators $S$ and relations $R$, when can I define a homomorphism $\phi:G\to\mathbb{Z}$?

I'm trying to map generators to generators but I also think relations need to coincide as well so I'm not sure.

You can always define the homomorphism $G \rightarrow 0$. – CyclotomicField Mar 17 '21 at 01:13 — CyclotomicField, Mar 17 '21 at 01:13
Well, yeah, but I was hoping for a little more than that ;) – Xochitl Garcia Mar 17 '21 at 01:23 — Xochitl Garcia, Mar 17 '21 at 01:23

score 4 · Answer 1 · answered Mar 17 '21 at 01:21

Let $H$ be any group. Suppose $f:S\to H$ is a function which you can extend to $S\cup S^{-1}$ by letting $f(s^{-1})=f(s)^{-1}$. If for every $r=s_1s_2...s_n\in R$ we have $f(s_1)f(s_2)...f(s_n)=e_H$ (intuitively, all elements of $R$ belong to the "kernel" of $f$) then $f$ can be uniquely extended to a homomorphism $\hat{f}:G\to H$. This is the fundamental theorem of group presentations.

When can I define a map from a group presentation to $\mathbb{Z}$?

1 Answers1