Definition: A group $G$ is called nilpotent if there exists a chain of subgroups $N_0, N_1,\ldots, N_k$ such that $$\{e\} = N_0 \le N_1 \le N_2 \le … \le N_k = G$$ and for $0\le i\le k-1$,
- $N_i \vartriangleleft G$, i.e. $N_i$ is normal in $G$,
- $N_{i+1}/N_i \subset Z(G/N_i)$
where $Z(G)$ denotes the center of some group $G$, and $e$ is the identity of $G$.
What motivated this definition? What is so special about $N_{i+1}/N_i \subset Z(G/N_i)$? It feels so random and out of the blue!
Best Answer
I like the nLab description, which phrases nilpotent groups in terms of towers of central extensions. In this guise it's a generalization of being abelian.
An extension of a group $G$ is a group $E$ such that $G$ is a quotient of $E$. It's actually slightly more--it's a short exact sequence $1 \to A \to E \to G \to 1$, effectively a "witness" to the fact that $G$ is a quotient of $E$.
A central extension is a group extension as above where additionally $A$ is central in $E$, i.e. (the image of) every element of $A$ commutes with everything in $E$, including all of $A$ itself, so $A$ in particular is abelian. As an example, the central extensions of $G=1$ are precisely the abelian groups.
What are the central extensions of abelian groups? What about the central extensions of central extensions of abelian groups, etc.? Well, whatever they are, we call them nilpotent groups.
The central series can be thought of as a "witness" to this iterated construction. The successive quotients are the central subgroups we're extending by.
Another reason this is a nice definition is that solvable groups are pretty transparent: they're iterated abelian extensions, rather than central extensions, a somewhat weaker constraint. That tracks back to solvability in Galois theory.