I'm reading "Presentations of Groups," by D. L. Johnson.
Tl;dr:
How do you find the invariant factors of $G/G'$ from a presentation of a group $G$?
The Details:
In Chapter 3 of the book ibid., on Abelian groups, there are the following theorems.
Theorem 1: The cyclic group of order $n\in\Bbb N$ has a presentation $\langle x\mid x^n\rangle$.
Theorem 2: The Abelian group $$A=\Bbb Z_{d_1}\times\dots\times\Bbb Z_{d_r}\times\underbrace{\Bbb Z\times\dots\times\Bbb Z,}_{n-r\text{ times,}}\tag{1}$$ where $r, n$ and the $d_i$ are integers with $d_i\ge 2$ and $n\ge r$, has a presentation $$\langle X\mid P, C\rangle,$$ where $$\begin{align} X&=\{x_i\mid i\in \overline{1, n}\}, \\ P&=\{x_i^{d_i}\mid i\in\overline{1,r}\}, \\ C&=\{[x_i, x_j]\mid i, j\in \overline{1, n}, i<j\}, \end{align}$$ where $[a,b]=aba^{-1}b^{-1}$ is the commutator of $a, b$.
Theorem 3: Let $F$ be free on $X=\{x_i\mid i\in\overline{1, n}\}$ and let $R=\{[x_i, x_j]\mid 1\le i<j\le n\}$. Then:
(i) $\overline{R}=F'$ (and I'm not sure what $\overline{R}$ is; please tell me in the comments),
(ii) $F/F'=\langle X\mid R\rangle$,
(iii) $F/F'$ is free Abelian of rank $n$, and
(iv) $F/F'$ is isomorphic to the direct product of $n$ infinite cyclic groups.
Theorem 4: Let $X$ be as above. If $\langle X\mid R\rangle$ is a presentation for a group $G$, then $\langle X\mid R, S\rangle$ is a presentation for $G/G'$, where $S=\{[x_i, x_j]\mid 1\le i<j\le n\}$.
Definition 1: Suppose we have a presentation $$\langle x_1,\dots , x_n\mid r_1, \dots, r_m\rangle\tag{2}$$ for a group $G$, and let $a_{ij}\in\Bbb Z$ be the sum of the exponents with which the generator $x_i$ appears in the relator $r_j$. We call $A=(a_{ij})$ the relation matrix of the presentation.
Theorem 5: Let $A$ be the relation matrix of a presentation $(2)$ of a group $G$ and let $h_i$ be the highest common factor of the $i$-rowed minors of $A$ (and $h_0=1$). If $h_{r+1}$ is the first of these to be zero, then $G/G'$ is isomorphic to the direct product of $r$ finite cyclic groups, of orders $\frac{h_i}{h_{i-1}}$ with $i\in\overline{1, r}$, and the free Abelian group of rank $n-r$.
The Question(s):
NB: It may seem like I'm asking multiple questions, but I'm really only asking one, viz., . . .
Question 1: Given the theorems above, how do I find the invariant factors of $G/G'$ from a given presentation of $G$?
To avoid things being too broad and for concrete examples, the book asks us to . . .
Find the invariant factors of $G/G'$ when $G$ is given by the following seven presentations, respectively:
$\langle a, b, c,d,e\mid ab=c, bc=d, cd=e, de=a, ea=b\rangle$.
$\langle a,b,c,d,e,f\mid ab=c, bc=d, cd=e, de=f, ef=a, fa=b\rangle$.
$\langle x, y, z\mid (xyx)^2, x^3=y^3, (zxy)^4\rangle$.
$\langle x, y, z\mid (xy)^2, z^4, [x,z], [y, z]\rangle$.
$\langle x, y, z\mid x^2, y^2, z^2, (xy)^3, (yz)^3, [x, z]\rangle$.
$\langle x, y\mid x^ry=y^rx, x^n=1\rangle, r, n\in \Bbb N$.
$\langle x, y, z\mid [x, y]=x^t, [y,z]=y^t, [z,x]=z^t\rangle, t\in \Bbb Z$.
Thoughts:
I think you just apply Theorem 5, but surely it can't be that simple.
Question 2 (optional): Would someone do at least one of $1$-$7$ for me, please, just for illustration?
Also, it seems to me like AbelianInvariants(G) does the job in GAP since $G/G'\stackrel{?}{\cong}\operatorname{Abelianisation}(G)$, but I would prefer a more theoretical understanding of what's going on than just that.
Please help :)
Please fix Definition 1: "generator $x_i$ appears in the relator $r_i$" one of the subscripts should be $j.$
From the context the author is saying that $\overline R$ is the normal subgroup of $F$ generated by $R$.
The answer to your question is: Yes, just apply Theorem 5. It is that simple.
7) The relation matrix is $$ \begin{matrix} t & 0 & 0 \\ 0 & t & 0 \\ 0 & 0 & t \\ \end{matrix} $$ so the group is $C_{|t|} \times C_{|t|} \times C_{|t|}$ for $|t|>1$. I'll leave the other cases, $|t|<2$, for you.