I saw two different definitions of a nilpotent group, but I'm not really sure how these definitions are equivalent.
The first one is from Basic Abstract Algebra (Robert Ash):
A central series for $G$ is a normal series $1 = G_0 \trianglelefteq G_1 \trianglelefteq … \trianglelefteq G_r = G$ such that $G_i/G_{i-1} \subseteq Z(G/G_{i-1})$ for every $i = 1, … ,r$. An arbitrary group $G$ is said to be nilpotent if it has a central series.
The second one is from Advanced Modern Algebra (Rotman):
The descending central series of a group $G$ is $$ G = \gamma_1(G) \supseteq \gamma_2(G) \supseteq …,$$
where $\gamma_{i+1}(G) = [\gamma_i(G),G].$ A group $G$ is called nilpotent if the lower central series reaches $\{1\}$; that is, if $\gamma_n(G)=\{1\}$ for some $n$.
I can't really see how these definitions are equivalent, because they are defining it in very different ways… so I would appreciate it if anybody could clarify this for me.
Thanks in advance
Best Answer
Observe that
$$\frac{G_{i+1}}{G_i}\subseteq Z\left(\frac{G}{G_i}\right)\iff \left[\frac{G_{i+1}}{G_i},\frac{G}{G_i}\right]\subseteq\frac{G_i}{G_i}\iff [G_{i+1},G]\subseteq G_i. \tag{$\circ$}$$
Thus by induction
$$[[\cdots[[G_i,\overbrace{G],G],\cdots],G}^{\large i}]\subseteq G_0=1.$$
In particular, if $G_n=G$ then $\gamma_{n+1}(G)=1$. So existence of an ascending series implies the descending series terminates. Conversely, the descending series is also an ascending series if it actually has $1$ at its base, since its terms $G_i:=\gamma_{n+1-i}(G)$ satisfy the right-hand side of $(\circ)$.