This is essentially an exercise from Jacobson's Basic Algebra I. (p.79, ex.11)
Exercise 11 says that the canonical action of the wreath product of $G$ and $H$ on the product space can be defined by $(f,h)(t,s)=(f(s)t,hs)$. I tried to check the associativity, which is the only nontrivial part of the assertion, but
$$((f_1,h_1)(f_2,h_2))(t,s)=(f_1(h_1f_2),h_1h_2)(t,s)=((f_1(h_1f_2))(s)t,h_1h_2t)$$
which is $(f_1(s)f_2(h_1^{-1}(s))t,h_1h_2t)$ and
$$(f_1,h_1)((f_2,h_2)(t,s))=(f_1,h_1)(f_2(s)(t),h_2(s))=(f_1(h_2(s))(f_2(s)(t)),h_1h_2t)$$
so the two calculations gives different results!
What mistakes did I make? I'm starting to think that the definition $(f,h)(t,s)=(f(s)t,hs)$ doesn't make sense..
Some help would be nice. Thanks in advance!