# How do I find the abelianization of a group determined by generators and relations?

For example let $G=\langle a,b,c|abcbc=a^2cb^2=a^2cba^{-1}ca=1\rangle$

I know that $G^{ab} \cong \mathbb{Z}^n / \langle 𝜙(𝑤_1) = · · · = 𝜙(𝑤_𝑘) = 0\rangle.$ Where $𝜙$ maps $w$ to the column \begin{pmatrix} y_1\\\ y_2\\\ \vdots \\\ y_i \\\ \vdots \\\ y_k \end{pmatrix}with $y_i$ being the sum of the powers for $x_i$ in $w$. So I get this matrix as an $Im$ of $L:ℤ^k→ ℤ^n$ mapping $e_i→𝜙(w_i)$ \begin{pmatrix} 1 & 2 & 2 \\\ 2 & 2 & 1 \\\ 2 & 1 & 2 \end{pmatrix} I know that there are bases $𝑓_1, . . . , 𝑓_𝑘 ∈ ℤ^k$ and $𝑒_1, . . . , 𝑒_𝑛 ∈ ℤ^k$ so that $𝐿𝑓_𝑖 = \lambda_𝑖𝑒_i$ if $i\leq n$ and $0$ otherwise

and i know that $\mathbb{Z}^n / Im 𝐿 = \prod^n_{i=1} \mathbb{Z}/\lambda 𝑖\mathbb{Z}.$ But i'm not quite sure how to use it.

• I don't understand your question. It would help if the math was a little bit cleaned up, see here for some help. (Also : I think the (or at least, a) correct spelling is "abelianization".) – Arnaud D. Apr 10 '18 at 13:52
• This similar question could also be of interest. – Arnaud D. Apr 10 '18 at 13:55
• Thanks, i didn't know the correct spelling. I have some problems writing negative powers with MathJax so i dont know how to fix some things – Alexander Kraynov Apr 10 '18 at 14:02
• I cleaned it up as much as i can. I hope it's better now. – Alexander Kraynov Apr 10 '18 at 14:16
• But that group is already abelian. Actually, it is the trivial group. – Tobias Kildetoft Apr 10 '18 at 14:26