Let $K$ and $L$ be groups, let $n$ be a positive integer, let $\rho$$:$ $K$ $\to$$S$$_n$ be a homomorphism and let $H$ be the direct product of $n$ copies of $L$. Construct an injective homomomorphism $\psi$ from $S$$_n$ into $Aut$$($$H$$)$ by letting the elements of $S$$_n$ permute the $n$ factors of $H$. The composition $\psi$ $\circ$ $\rho$ is a homomomorphism from $K$ into $Aut$$($$H$$)$. The $wreath$ $product$ of $L$ by $K$ is the semi direct product $H$$\rtimes$ $K$ with respect to this homomomorphism and is denoted by $L$ $\wr$ $K$.
$(i)$Assume that $K$ and $L$ are finite groups and $\rho$ is the left regular representation of $K$. Find $|$$L$ $\wr$ $K$$|$ in terms of $|$$K$$|$ and $|$$L$$|$.
$(ii)$Let $p$ be prime, let $K$ $=$ $L$ $=$ $Z$$_p$ and let $\rho$ be the left regular representation of $K$.Prove that $Z$$_p$$\wr$$Z$$_p$ is a non abelian group of order $p$$^p$$^-$$^1$ and id isomorphic to Sylow $p$$-$subgroup of $S$$_p$$^2$.