Action of wreath product on Cartesian product
From Groupprops
Definition
Suppose both and
are being viewed along with specific group actions of each, i.e., they are both being viewed as groups of permutations, with
acting on a set
and
acting on a set
. Then,
comes equipped with a natural action on the Cartesian product
as follows.
First, recall that is the external semidirect product
. To specify an action of this, we will specify how
and
act; then, via the equivalence of internal and external semidirect product, we would have defined the action of a generating set and hence of all of
. Here is the action:
-
acts on
as follows: An element of
is a function
. For an element
, the action of
on
is obtained as
where
is the action of the element
on
. In other words, it permutes each of the fibers according to the
-value on that fiber.
-
acts on
as follows: An element
sends
to
where
denotes the image of
under the action of
.
Note that we need to verify a compatibility condition to show that this is well defined.