If a central series is considered as $G = G_0 \supset G_1 \supset \cdots \supset G_m = \{1\}$ such that $G_{i+1} \triangleleft G_i$ and $G_i/G_{i+1} \subset Z(G/G_{i+1})$ then,
Show that finite p-groups admit central series.
What I have so far is this.
If the order of group G is 1 - nothing to prove
If the order of group G is p, then certainly $\{1\} \triangleleft G$ and $G/\{1\} \subset Z(G/\{1\})$
If the order of group G is p^n, then it has a nontrivial center, $Z(G_n) \neq \{1\}$. Choose an element of the center, $a \in Z(G_n)$ s.t. $ord(a) = p$, then let $G^{(n)} := $ (because $p \mid p^n$ and using Cauchy's thm)
Then $G^{(2)} \triangleleft G_2$ and $ord(G_2/G^{(2)})=p$ (using the class equation)
This is where I seem to run out of steam. Does anyone have any suggestions?
Thanks a lot!