Wreath product ($H\wr G$)

The restricted wreath product of groups $H$ and $G$ iswhere $G$ acts on the finitely supported functions $G\to H$ by translation.

The base group acts independently on each coordinate of a Cartesian power, and the top group permutes coordinates. This is distinct from the imprimitive action on a disjoint union of blocks. It is primitive when the base action is primitive and nonregular and the top group is transitive. The full product-action overgroup is $S_{|\Delta |}\wr S_r$. Socles of almost-simple, diagonal or holomorph bases produce respectively product-action, compound-diagonal or holomorph-compound types.

If a group $K$ acts on a finite set $X$, it acts on the direct product of groups $H^X$ by permuting its factors. The resulting semidirect product is the permutation wreath product. For $K=S_m$ acting on $m$ two-point blocks and $H=C_2$, it describes independent within-block swaps combined with a permutation of blocks, and has order $2^{m}m!$. The action on $X$ is part of the definition; it need not be the regular action used in a regular wreath product.

The lamplighter group is the restricted wreath product $C_2\wr \mathbb{Z}$. It is generated by one lamp switch and the translation of the integer line.