wreath product
WreathProduct
2002-10-03 09:47:57
2005-12-31 21:06:29
Definition
wreath product
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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
\[
b f(\gamma) := f(b^{-1}\gamma),
\]
for any $f\in A^{\Gamma}$ and $\gamma\in\Gamma$.
The \emph{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
\[(f,b)(f',b') = (fbf', bb').\]
Note that by definition of the action of $B$ on $A^{\Gamma}$,
\[(fbf')(\gamma) = f(\gamma)f'(b^{-1}\gamma).\]
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 $(f,b),~(f',b')\in A\wr_{\Gamma} B$.
Let $b_i$ denote $b^{-1}(i)$. Then
\begin{eqnarray*}
(f,b)(f',b') &=& \bigl((f(1),~f(2),~\ldots,~f(n)),~b\bigr) \bigl((f'(1),~f'(2),~\ldots,~f'(n)),~b'\bigr) \\
&=& \bigl((f(1),~f(2),~\ldots,~f(n)\bigr)\bigl(f'(b_1),~f'(b_2),~\ldots,~f'(b_n)\bigr),~bb')
\text{(*)} \\
&=& \bigl((f(1)f'(b_1),~f(2)f'(b_2),~\ldots,~f(n)f'(b_n)),~bb'\bigr).
\end{eqnarray*}
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.