In Rotman's group theory book, a wreath product is defined as
$$D\wr_\Omega Q := K \rtimes Q$$
where $\rtimes$ is the semidirect product, $K = \prod_{\omega \in \Omega} D_\omega$, and $D_\omega \cong D$ for every $\omega$ in the finite set $\Omega$. $Q$ and $D$ are said to be just 'groups'.
The problem I'm having with this definition is that, according to the definition of semidirect product, a group $G$ is the semidirect product of two subgroups $K$ and $Q$ iff:
- $KQ = G$
- $K \lhd G$
- $K \cap Q = 1$.
I am not at all sure the wreath product fulfills these conditions, and I don't quite know how to check it. They all seem to imply that both $K$ and $Q$ are subgroups of $D\wr_\Omega Q$, but this group hasn't been defined in any other way than the semidirect product itself, so I'm stuck in a circular problem... What am I missing here?