Let $A$ and $B$ be groups, and let $B$ act on the set $\Gamma$. Define the action of $B$ on the direct product $A^{{\Gamma}}$ by

for any $f\in A^{{\Gamma}}$ and $\gamma\in\Gamma$. The wreath product of $A$ and $B$ according to the action of $B$ on $\Gamma$, denoted $A\wr_{{\Gamma}}B$, is the semidirect product of groups $A^{{\Gamma}}\rtimes B$.

Let us pause to unwind this definition. The elements of $A\wr_{{\Gamma}}B$ are ordered pairs $(f,b)$, where $f\in A^{{\Gamma}}$ and $b\in B$. The group operation is given by

Note that by definition of the action of $B$ on $A^{{\Gamma}}$,

The action of $B$ on $\Gamma$ in the semidirect product permutes the elements of a tuple $f\in A^{{\Gamma}}$, and the group operation defined on $A^{{\Gamma}}$ gives pointwise multiplication. To be explicit, suppose $\Gamma$ is an $n$-tuple, and let $(f,b),~{}(f^{{\prime}},b^{{\prime}})\in A\wr_{{\Gamma}}B$. Let $b_{i}$ denote $b^{{-1}}(i)$. Then

Notice the permutation of the indices in (*).

A moment’s thought to understand this slightly messy notation will be illuminating, and might also shed some light on the choice of terminology.