The lamplighter group can be defined by the semidirect product: $$ L_N=(\mathbb{Z} _N) \wr \mathbb{Z} \cong \bigoplus_{i=-\infty}^{\infty}\mathbb{Z}_{N} \rtimes_\phi\mathbb{Z},$$
where $\phi(1)$ "shifts" every element in $\bigoplus_{i=-\infty}^{\infty}\mathbb{Z}_{N} $ to the right by $1$. I tried to compute the abelianization of the group.
I worked out that $[L_N,L_N]$ is the set of elements in $\bigoplus_{i=-\infty}^{\infty}\mathbb{Z}_{N}$ whose coordinates sum to $0$.
Then I got $L_N \big/ [L_N,L_N] = \mathbb{Z}_{N} \oplus \mathbb Z$, is this correct?