Lamplighter group ($C_2\wr \mathbb{Z}$)

The lamplighter group is the restricted wreath product $C_2\wr \mathbb{Z}$. An element consists of a finitely supported function $f:\mathbb{Z}\to {\mathbb{F}}_2$, encoding the on/off lamps, and an integer position $n$. Let ${\tau }_{n}g(k)=g(k-n)$. The group operation isOnly finitely many lamps are on, so the configuration group is the restricted direct sum of groups ${⨁}_{k\in \mathbb{Z}}{C}_2$. The position group shifts their indices, giving the displayed semidirect product.

Let $a=({\delta }_0,0)$ switch the lamp at zero and let $t=(0,1)$ move the position by one. Then $t^{k}a{t}^{-k}=({\delta }_k,0)$ switches the lamp at $k$. Products of these switches create any finite configuration, so $a,t$ are a generating set of a group. Although every switch has order two, $t$ has infinite order; the group itself is infinite and nonabelian. Projection $(f,n)\mapsto n$ has the lamp configuration group as its group kernel.