I'm trying to solve this problem from my abstract algebra course:
The lower central series of a group $G$ is defined by means of $L_0=G$, $L_{n+1}=[G,Ln]=\langle[g,h]:g\in G,h\in L_n\rangle$, for any $n\geq 0$.
a) Prove that $L_n\unlhd G$ for any $n\geq 0$.
b) Show that we have indeed a descending series: $L_0\unrhd L_1 \unrhd > L_2 \cdots$.
c) Prove that for any $n$, $L_n/L_{n+1}\leq Z(G/L_{n+1})$.
d) Prove that the group $G$ is nilpotent if and only if $L_n=1$ for some $n$.
The work I've done so far:
a) I first started trying to solve the a) question using induction. The base case is true ($L_0=G\unlhd G$), so now I assume it verifies for some $n$ and try to prove that it also verifies for $n+1$ then. Despite this, I've not been able to see how to prove it, so I'll need help here.
b) This one was easy considering true the a) statement: using induction again, It's clear that $L_0\unrhd L_1$, since $L_1=[G,G]=G'$. So I try to see if for any $n$, $L_n\unrhd L_{n+1}$. Considering $[g,h]=ghg^{-1}h^{-1}\in L_{n+1}$, notice that by definition $h^{-1}\in L_n$, and since it's normal for a), then $ghg^{-1}\in L_n$ too, so we conclude that $L_n\unrhd L_{n+1}$.
For questions c) and d) I still don't know where to start.
Any help will be appreciated, thanks in advance.