I was trying to understand the above Corollary but I have a problem, namely why in the second to last line $A_0 \leq \zeta_p(G)$? Any ideas?
Definitions
By recurrence we define $[x,_0\, y]=x$; $[x, _{i+1}\, y]=[[x, _i\, y],\, y]$
$\zeta_n(G)$ is $n$-th center of $G$.
$[A_0,_p,x] = [[A_0,_{p-1},x],x]=1$, so $x$ centralizes $[[A_0,_{p-1},x]$, and $N$ does also, so $C_G([A_0,_{p-1},x])$ contains $\langle N,x \rangle = G$. i.e. $[[A_0,_{p-1},x] \in \zeta_1(G)$.
Then we get $x$ centralizes $[[A_0,_{p-2},x]$ modulo $\zeta_1(G)$, so $[[A_0,_{p-2},x] \le \zeta_2(G)$, etc.