= Lamplighter group
{title2=$C_2\wr\mathbb Z$}
{wiki}
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> is
$$
(f,n)(g,m)=(f+\tau_n g,n+m).
$$
Only finitely many lamps are on, so the configuration <group> is the <restricted direct sum of groups> $\bigoplus_{k\in\mathbb Z}C_2$. 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 $t^kat^{-k}=(\delta_k,0)$ 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>.
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