There's a typo on page 42 of Robinson's, "A Course in the Theory of Groups (Second Edition)", ISBN 987-1-4612-6442-9.
Here's a picture:
It reads as follows:
If $H$ and $K$ are permutation groups on finite sets $X$ and $Y$, show that the order of $\color{red}H\quad \color{red}K$ is $\lvert H\rvert^{\lvert Y\rvert}\lvert K\rvert$.
The typo is the gap between the red $H$ and $K$.
Based on the context, I believe it should be $\color{red}{H\wr K}$. In previous instances of the Wreath product in the book, the symbol $\wr$ is slanted backward a bit, so it takes up about the same amount of space as the gap in question; besides, the result seems to match the properties of the Wreath product listed on Wikipedia.
Is this correct?
Please help :)
When $K$ acts on a set $\Omega$, the (unrestricted) wreath product $H\wr_{\Omega}K$ is the semidirect product of $\prod_{\omega\in\Omega}H$ by $K$, letting $K$ act on the product by permuting the coordinates according to its action. When no set is specified, $H\wr K$ is usually considered to be the (unrestricted) regular wreath product with $K$ acting on itself by translation (depending on how you write your wreath products, it can be the left or the right action...). Rotman uses $\wr_r$ to denote the regular wreath product.
(The restricted wreath product uses the restricted direct product instead, i.e. the subgroup of the cartesian product of almost null elements; Hanna Neumann's book uses $H\mathop{\mathrm{Wr}}_{\Omega}K$ for the unrestricted product, and $H\mathop{\mathrm{wr}}_{\Omega}K$ for the restricted one.)
In the case at hand, it is clear that Robinson means $H\wr_Y K$, given the cardinality.