Wreath product

The wreath product is an product that operates on groups.

Definition
Given a group $$(G,\times)$$ and a permutation group $$(H,*)$$ such that $$(H,*) \subseteq S_n$$, then $$G\wr H$$ is a group $$(K,\cdot)$$ where:

$$K = \left\{(g,h)\mid g\in G^n, h\in H\right\}$$

and:

$$((g_1,\dots,g_n),h)\cdot((g_1',\dots,g_n'),h')=((g_{h'(1)}\times g_1',\dots,g_{h'(n)}\times g_n'),h*h')$$

Put simply the operation permutes the tuple $n$ in the first argument by the permutation $S_{0}$ in the second argument, and then combines the two tuples pairwise with the usual operations.

This definition can be generalized to apply to arbitrary groups, by observing that due to Cayley's theorem every group is isomorphic to a permutation group.

Properties

 * For groups $g$ and $$H \subseteq S_n$$: $$|G\wr H|=|G|^n|H|$$.
 * If $$G \subseteq S_m$$ and $$H \subseteq S_n$$ then $$G\wr H$$ is isomorphic to a subgroup of $h'$.