Wreath product

Summary

The wreath product $A \wr_\Omega G$ builds a new group from a group $A$ and a group $G$ that acts on a set $\Omega$. It is the semidirect product of the group of all functions $\Omega \to A$ with $G$, where $G$ moves the functions by moving the points of $\Omega$. Wreath products describe the symmetries of objects made of several copies of a smaller object. Each group extension $E$ of a group $B$ by a group $A$ embeds in the unrestricted regular wreath product $A \wr B$.