Let say I have an infinite elementary abelian $p$-group $E$ (i.e. with presentation $E= \langle x_1,x_2,x_3,... \mid x_i^p=1, \ x_i x_j = x_j x_i \rangle$).
How do I find the Fitting subgroup of the wreath product $G := \mathbb{Z}_p \wr E$, or more precisely, how could I prove that $\operatorname{Fitt}(G) = G$?
What my thinking is, find a series of normal subgroups with increasing nilpotency class, so something like $\mathbb{Z}_p \times E$, $(\mathbb{Z}_p \oplus \mathbb{Z}_p) \times E, \ldots$ but these don't seem to be normal in $G$ and I can't think of any better ones.
Any ideas?