Wreath product

Remarks

The order of a finite wreath product is $\abs{A \wr_\Omega G} = \abs{A}^{\abs{\Omega}} \abs{G}$.

Example: let $A = C_2$ and let $G = C_2$ act on $\Omega = \set{1, 2}$ by swapping. Then $A \wr_\Omega G$ has order $2^2 \cdot 2 = 8$. It is isomorphic to the dihedral group of order $8$, the symmetry group of a square. The base group $C_2 \times C_2$ consists of the identity, the half-turn and the reflections in the two diagonals. The swap is the reflection in an axis through the midpoints of two opposite sides.

Non-example: the direct product $A^\Omega \times G$ has the same underlying set, but there $G$ does not move the functions at all. Its multiplication differs from that of the wreath product when $A \ne 1$ and $G$ moves some point, because then $g \cdot f \ne f$ for some $g$ and $f$.

For $G = S_m$ acting on $\set{1, \dots, m}$, the group $S_k \wr S_m$ is the subgroup of $S_{km}$ that preserves a fixed partition of $km$ points into $m$ parts of size $k$. The group $C_2 \wr S_n$ is the group of signed permutations, of order $2^n n!$. For an infinite set $\Omega$, the group defined above is the unrestricted wreath product, built from all functions $\Omega \to A$. The restricted wreath product uses only the functions with $f(\omega) = 1$ for all but finitely many $\omega$.