TITLE: Definition of a nilpotent group.
QUESTION [12 upvotes]: 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

REPLY [9 votes]: Allow me first to change the notation in your first definition, just so as not to give us both a headache:

A central series for G is a normal series $G = G_1 \unrhd G_2 \unrhd \dots \unrhd G_n = \{1\} $ such that $G_i/G_{i+1}\subseteq Z(G/G_{i+1})$ for every $i=1,...,n-1$. An arbitrary group G is said to be nilpotent if it has a central series.

Check I haven't changed anything (modulo potential typos).
For one direction: show that, if the descending central series of $G$ terminates, then it is a central series (set $G_i = \gamma_i(G)$ and show that $G_i/G_{i+1}\subseteq Z(G/G_{i+1})$ using the definition of $\gamma$). (In spite of the name, the descending central series is only a central series when it terminates.)
For the other direction: suppose you have any central series $G_i$ (as above) for $G$. Show that the descending central series descends at least as fast as your central series, i.e. that $\gamma_i(G) \subseteq G_i$ (you may want to start with $i = 2$), and hence that the descending central series terminates. (We sometimes say that the descending central series is the fastest descending central series.)