# Wreath product

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The wreath product of two groups $A$ and $B$ is constructed in the following way. Let $A^B$ be the set of all functions defined on $B$ with values in $A$. With respect to pointwise multiplication, this set is a group which is the complete direct product of $|B|$ copies of $A$ ($|B|$ denotes the cardinality of $B$); $B$ acts on $A^B$ as a group of automorphisms in the following way: if $b \in B$, $\phi \in A^B$, then $\phi^b(x) = \phi(xb^{-1})$ for $x \in B$. With respect to this operation, one can form the semi-direct product $W$ of $B$ and $A^B$, that is, the set of all pairs $(b,\phi)$, where $b \in B$, $\phi \in A^B$, with multiplication operation given by $$(b,\phi) (c,\psi) = (bc, \phi^c \psi) \ .$$
The resulting group $W$ is called the Cartesian (or complete) wreath product of $A$ and $B$, and is denoted by $A \mathop{Wr} B$ (or $A \bar{\wr} B$, a denotation of Ph. Hall). If instead of $A^B$ one takes the smaller group $A^{(B)}$ consisting of all functions with finite support, that is, functions taking only non-identity values on a finite set of points, then one obtains a subgroup of $W$ called the wreath product (direct wreath product, discrete wreath product) of $A$ and $B$; it is denoted by $A \mathop{wr} B$ (or $A \wr B$). Both wreath products are widely used for constructing various examples of groups.