Composition series and chief series of $p$-group composition series and chief series of $p$-group. How to solve the following Problem? Thanks. Let $G$ be a group of order $p^n$, $p$ prime. Prove every chief factor and every composition factor is of order $p$. Is every composition series a chief series of $G$?

Hints:

Let $\,G\;,\;\;|G|=p^n\;,\; p\;$ a prime, be a group, then use that $\,|Z(G)|>1\;$ and a little induction to show that for any $\,0\le k\le n\;,\;\;G\;$ has a _normal_ subgroup of order $\;p^k\;$ . This already answers (almost...) questions (1)-(2) .

As for the last question: take the dihedral group $\;G:=\\{s,t\;;\;s^2=t^4=1\;,\;sts=t^3\\}\;$ of order 8, and check the series

$$1\le\langle t^2\rangle\le\langle t\rangle\le G\;\ldots$$